# Global viscosity solutions to Lorentzian eikonal equation on globally
hyperbolic space-times

Siyao Zhu<EMAIL_ADDRESS>Hongguang Wu<EMAIL_ADDRESS>Xiaojun Cui
<EMAIL_ADDRESS>Department of Mathematics, Nanjing University, Nanjing 210003,
P.R. China  Department of Mathematics, Changzhou University, Changzhou 213164,
P.R. China  Department of Mathematics, Nanjing University, Nanjing 210003,
P.R. China

###### Abstract

In this paper, we show that any globally hyperbolic space-time admits at least
one globally defined distance-like function, which is a viscosity solution to
the Lorentzian eikonal equation. According to whether the time orientation is
changed, we divide the set of viscosity solutions into some subclasses. We
show if the time orientation is consistent, then a viscosity solution has a
variational representation locally. As a result, such a viscosity solution is
locally semiconcave, as the one in the Riemannian case. Also, if the time
orientation of a viscosity solution is non-consistent, we analyse its peculiar
properties which make this kind of viscosity solutions are totally different
from the ones where the Hamiltonians are convex.

###### keywords:

Global viscosity solution , Lorentzian eikonal equation , weak KAM theory

##  1 Introduction

Since the landmark work of Penrose [15][16], the study of Lorentzian causality
theory has attracted many researchers in recent decades. In this theory,
Lorentzian geodesic flow is a central research object. Eikonal equation, which
can be regarded as a special Hamilton-Jacobi equation, plays an important role
in the study of Lorentzian geodesic flow. Actually, if the eikonal equation
admits a smooth solution, the integral curves to the gradient field are
geodesics, e.g. [2, Lemma B.1]. Unfortunately, smooth solutions for such
partial differential equations are usually absent. At this moment, viscosity
solutions introduced by Crandall and Lions [5] show powerful advantages than
other weak solutions (e.g., Viscosity solutions usually have better well-
posedness in Dirichlet problem of partial differential equations).

On the other hand, a field closely related to viscosity solutions to the
Hamilton-Jacobi equation induced by a Tonelli Lagrangian is the so-called weak
KAM theory. Weak KAM theory, introduced by Fathi [10], establishes a
connection between Aubry-Mather theory and viscosity solutions to the
Hamilton-Jacobi equation. Infromly, backward action-minimizing curves, the
main object in Aubry-Mather theory, are exactly calibrated curves of some
viscosity solutions. In other words, viscosity solutions are nothing but the
weak KAM solutions of backward type. This can give an explicit variational
representation for global viscosity solutions to the Hamilton-Jacobi equation.
For example, in [4], we know the viscosity solutions of eikonal equation are
the so-called dl(distance-like) functions on Riemannian manifold. Such a
variational representation can provide us a geometrical intuition for the
gradient dynamics of the viscosity solutions. For some recent works on weak
KAM theory, Lorentzian geodesic flows and Lorentzian Aubry-Mather theory, we
refer interested readers to [4, 2, 7, 14, 9, 11, 18, 19] and the references
therein. Inspired by these works, it is interesting to construct a space-time
counterpart of weak KAM theory for viscosity solutions to the eikonal
equation. However, classical weak KAM theory depends heavily on the positive-
definiteness of the Lagrangian. On space-time, non-positive-definiteness of
Lorentzian metric makes problem complicated and challenging. Furthermore, non-
completeness is a common phenomenon on a space-time, it makes things worse.

As far as we know, there are only a few works on this topic. Cui and Jin [7]
study space-time with the regular cosmologcial time function $\tau$ and show
that $-\tau$ is a viscosity solution to Lorenzian eikonal equation. Jin and
Cui [14] establish the existence of global viscosity solution to Lorentzian
eikonal equation on the Abelian cover of a class A Lorentzian 2-torus. In [3],
Bernard and Suhr propose a cone field theory for generalized space-time and
establish the equivalence between global hyperbolicity and the existence of a
steep temporal function, where the steep temproal function is a smooth (of
course, viscosity) subsolution to Lorentzian eikonal equation in our setting.
It is not surprising that viscosity solutions in Riemannian case and space-
time case may have different properties. For example, in Riemannian case,
viscosity solutions are always locally semiconcave, but even in 2-dimensional
Minkowski space-time, a viscosity solution may not be locally semiconcave.
Actually, it is easy to check that $f(x,y)=|x|$ is a viscosity solution to
eikonal equation in 2-dimensional Minkowski space-time. Obviously, $f$ is not
locally semiconcave in any neighbourhood of $x=0$. In this paper, we will give
a much more complete theory on viscosity solutions of eikonal equation on a
globally hyperbolic space-time. More precisely, we classify the set of
viscosity solutions to the Lorentzian eikonal equation according to whether
the time orientation of the solution is consistent. We show that a viscosity
solution has a (local) variational representation when the time orientation is
consistent. Such a variational representation ensures that the viscosity
solution has the similar properties as in the Riemannian case, for instance,
local semiconcavity and some properties of weak KAM type. We also study the
properties of viscosity solution when the time orientation is non-consistent.
In this case, viscosity solutions exhibit many peculiar properties, which are
totally different from the ones in the Riemannian case.

##  2 Preliminaries and main results

We will use classical terminology as in [2, 11, 4, 7, 14]. Let $(M,g)$ be a
space-time (i.e., a connected, time-oriented, smooth Lorentzian manifold with
the Lorentzian metric $g$), where the signature for the Lorentzian metric is
$(-,+,\cdots,+)$. A point $p\in M$ is usually called an event from the
viewpoint of general relativity. For each $p\in M$, let $T_{p}M$ be the
tangent space at $p$ and $TM$ be the tangent bundle of $M$. A tangent vector
$V\in T_{p}M$ is called timelike, spacelike or lightlike if $g(V,V)<0$,
$g(V,V)>0$ or $g(V,V)=0$ ($V\neq 0$), respectively. Also, a tangent vector
$V\in T_{p}M$ is called causal if $V$ is either timelike or lightlike and is
called non-spacelike if $g(V,V)\leq 0$. The set of null vectors consists of
zero vector and lightlike vectors. The time-orientation of $(M,g)$ ensures
that $M$ admits a continuous, nowhere vanishing, timelike vector field $X$.
$X$ is used to separate the causal vectors at each base point into two classes
called past-directed and future-directed respectively. A causal tangent vector
$V\in T_{p}M$ is said to be past-directed (respectively, future-directed) if
$g(X(p),V)>0$ (respectively, $g(X(p),V)<0$). Then a piecewise $C^{1}$ curve
$\gamma:I\rightarrow M$ ($I\subset R$ is an interval of the real line) is
called causal, timelike, lightlike and past-directed or future-directed if the
tangent vector $\dot{\gamma}(s)$ is causal, timelike, lightlike and past-
directed or future-directed at every differentiable point $\gamma(s)$.

For two events $p,q\in M$, if there exists a future-directed timelike
(respectively, non-spacelike) curve from $p$ to $q$, we say $p,q$ are
chronologically related (respectively, causally related) and denote by $p\ll
q$ (respectively, $p\leq q$). For a given $p\in M$, define chronological past
$I^{-}(p)$, chronological future $I^{+}(p)$, causal past $J^{-}(p)$, causal
future $J^{+}(p)$ as follows:

$\displaystyle I^{-}(p)=\\{q\in M:q\ll p\\},$ $\displaystyle I^{+}(p)=\\{q\in
M:p\ll q\\},$ $\displaystyle J^{-}(p)=\\{q\in M:q\leq p\\},$ $\displaystyle
J^{+}(p)=\\{q\in M:p\leq q\\}.$

Analogously, for subsets $A\subseteq M$, we can define $I^{-}[A]$, $I^{+}[A]$,
$J^{-}[A]$ and $J^{+}[A]$. For example, $I^{-}[A]:=\\{q\in M|q\ll
s~{}~{}\text{for some}~{}~{}s\in A\\}$. We call a subset $A$ of $M$ is an
achronal (respectively, acausal) set if no two points of $A$ are
chronologically related (respectively, causally related ).

###### Definition 2.1

A space-time $(M,g)$ is said to be causal if there are no causal loops. A
space-time $(M,g)$ is globally hyperbolic if it is causal and the sets
$J^{+}(p)\cap J^{-}(q)$ are compact for all $p,q\in M$.

Throughout this paper, we always assume that $(M,g)$ is a globally hyperbolic
space-time. We are going to use two metrics on $M$, one is the Lorentzian
metric $g$, the other is an auxiliary complete Riemannian metric $h$. The
length functionals associated to $g$ and $h$ are denoted by $L(\cdot)$ and
$L_{h}(\cdot)$ respectively. Let $\Omega_{x,y}$ denote the path space of all
future-directed causal curves $\gamma:[0,1]\rightarrow M$ with $\gamma(0)=x$
and $\gamma(1)=y$, for a piecewise $C^{1}$ causal curve
$\gamma\in\Omega_{x,y}$, $L(\gamma)$ is given by

$\displaystyle
L(\gamma):=\int_{0}^{1}\sqrt{-g(\dot{\gamma}(s),\dot{\gamma}(s))}ds.$

Correspondingly the distance functions associated to $g$ and $h$ are denoted
by $d(\cdot,\cdot)$ and $d_{h}(\cdot,\cdot)$ respectively. More precisely,
given any $x,y\in M$ with $x\leq y$, then $d(x,y)$ is given by

$\displaystyle
d(x,y)=\sup\limits_{\gamma}\\{L(\gamma):\gamma\in\Omega_{x,y}\\}.$

The norms associated to $g$ and $h$ are denoted by $|\cdot|$ and $|\cdot|_{h}$
respectively. Recall that for a causal vector $V\in TM$, $|V|=\sqrt{-g(V,V)}$.
Let $\nabla$ (respectively, $\nabla_{h}$) be the gradient induced by the
Lorentzian metric $g$ (respectively, Riemannian metric $h$). There is a
timelike Lorentzian eikonal equation associated to the Lorentzian metric $g$:

$g(\nabla u,\nabla u)=-1.$ (2.1)

In this paper, we devote to viscosity solutions of equation (2.1). Before
stating our results, we need to introduce some more definitions for
convenience. The next two definitions are about to generalized gradients and
viscosity solutions to eikonal equation (2.1).

###### Definition 2.2

[6, Definition 3.1.6] Let $f:M\rightarrow R$ be a continuous function on
$(M,g)$. A vector $V\in T_{p}M$ is called to be a subgradient (resp.,
supergradient) of $f$ at $p\in M$, if there exist a neighborhood $O$ of $p$
and a $C^{1}$ function $\phi:O\rightarrow R$, differentiable at $p$, with
$f(p)=\phi(p)$, $\phi(p)\leq f(p)$ (resp., $\phi(p)\geq f(p)$) for every $x\in
O$ and $\nabla\phi(p)=V$.

We denote by $\nabla^{-}f(p)$ the set of subgradients of $f$ at $p$,
$\nabla^{+}f(p)$ the set of supergradients of $f$ at $p$. With the help of
$\nabla^{-}f(p)$ and $\nabla^{+}f(p)$, we can state that the definition of
viscosity solution as follows:

###### Definition 2.3

A continuous function $f$ is called a viscosity subsolution of equation (2.1)
if for any $p\in M$,

$g(V,V)\leq-1~{}~{}\mbox{for every}~{}~{}V\in\nabla^{+}f(p).$

Similarly, $f$ is called a viscosity supersolution of equation (2.1) if for
any $p\in M$,

$g(V,V)\geq-1~{}~{}\mbox{for every}~{}~{}V\in\nabla^{-}f(p).$

A continuous function $f$ is a viscosity solution of equation (2.1) if it is a
viscosity subsolution and a viscosity supersolution simultaneously.

Let $Lip_{loc}(M)$ be the set of locally Lipschitz (with respect to the
Riemannian metric $h$) functions on $M$ and

$\mathcal{S}(M):=\\{f\in Lip_{loc}(M)|f\mbox{ is global viscosity solution to
equation (\ref{E1})}\\}.$

In this paper, $\mathcal{S}(M)$ is the main object that we want to study
systematically. Concretely, we expect to get a classification of
$\mathcal{S}(M)$ and give a variational representation of elements in
$\mathcal{S}(M)$. For this purpose, we need the following definitions.

###### Definition 2.4

[7, Definition 2.8] Let $\phi:M\rightarrow R$ be a locally Lipschitz function
defined on the space-time $(M,g)$, then a vector $V\in T_{p}M$ is called a
limiting gradient if there exists a sequence $p_{k}\subset M\setminus\\{p\\}$
with $\lim\limits_{k\rightarrow\infty}p_{k}=p$ such that $\phi$ is
differentiable at $p_{k}$ for each $k\in\mathbb{N}$, and
$\lim\limits_{k\rightarrow\infty}\nabla\phi(p_{k})=V$. Here, the first limit
is taken in the sense of manifold topology on $M$; the second limit is taken
in the sense of any fixed chart that contains $p$. Since the first limit is
taken, we know that when $k$ is sufficiently large, $p_{k}$ goes into that
chart. The second limit does not depend on the choice of chart. We denote
$\nabla^{*}\phi(p)$ to be the set of all limiting gradients of $\phi$ at $p$.

It should be mentioned that for any $f\in\mathcal{S}(M)$, if there exists a
sequence $\\{x_{i}\\}_{i}\subseteq M\backslash\\{x\\}$ such that
$x_{i}\rightarrow x$ as $i\rightarrow\infty$ and $f$ is differentiable at
$x_{i}$ for each $i$, then the set $\\{\nabla_{h}f(x_{i})\\}_{i}$ has a bound
depending only on $x$. Furthermore, let $\phi$ be the map such that
$\phi(\nabla_{h}f(\cdot))=\nabla f(\cdot)$ for any smooth function $f$ on $M$.
Obviously, $\phi$ is a linear transformation on each fiber of $TM$ and the
norm of $\phi$ is locally bounded. Thus, the set $\\{\nabla f(x_{i})\\}_{i}$
is contained in a compact subset of $TM$. More precisely,
$\lim\limits_{i\rightarrow\infty}\nabla f(x_{i})\in\nabla^{*}f(x)$.

###### Definition 2.5

For a Lipschitz continuous function $f$ on $(M,g)$ where $\nabla f$ is
timelike as long as it exists,

  * •

We say the time orientation of $f$ is future-directed at $x$ if any limiting
gradient $V\in\nabla^{*}f(x)$ is future-directed timelike, the time
orientation of $f$ is past-directed at $x$ if any limiting gradient
$V\in\nabla^{*}f(x)$ is past-directed timelike.

  * •

We say the time orientation of $f$ changes at $x$ if there exist two timelike
vectors $V,W$ in $\nabla^{*}f(x)$ such that $V$ is past-directed and $W$ is
future-directed.

  * •

We say the time orientation of $f$ is consistent if for any two points $x,y\in
M$, the vectors in both $\nabla^{*}f(x)$ and $\nabla^{*}f(y)$ have the same
time orientation. Otherwise, we say the time orientation of $f$ is non-
consistent.

######  Definition 2.6 (Forward calibrated curve)

For any $u\in\mathcal{S}(M)$, a future-directed timelike curve
$\alpha:[0,t]\rightarrow M$ is said to be a forward calibrated curve of $u$ if
$u(\alpha(0))=u(\alpha(s))-s$ for any $s\in[0,t]$.

######  Definition 2.7 (Partial Cauchy Surface)

Let $A\subseteq M$ be an achronal set, the $\text{edge}(A)$ consists of all
points $p\in\bar{A}$, such that every neighborhood $U$ of $p$ contains a
timelike curve form $I^{-}(p,U)$ to $I^{+}(p,U)$ which does not meet $A$.
Where $I^{-}(p,U)$ denotes the chronological past of $p$ with respect to the
space-time $(U,g|_{U})$. Similarly, $I^{+}(p,U)$ denotes the chronological
future of $p$ with respect to $(U,g|_{U})$. The set A is said to be edgeless
if $\text{edge}(A)=\emptyset$. A subset $S$ of $(M,g)$ is said to be a partial
Cauchy surface if $S$ is acausal and edgeless in $M$.

It is well known that the level sets of Busemann function are partial Cauchy
surfaces on a globally hyperbolic space-time and any Busemann function is a
viscosity solution of equation (2.1) on its domain. This motivates us to
consider whether $u_{s}$ is a partial Cauchy surface for any
$s\in\text{Image}(u)$. Here and in the following $\text{Image}(u)$ denotes the
image set of $u$. In what follows, for any $u\in\mathcal{S}(M)$ and $s\in R$,
we denote the level set of $u$ by $u_{s}$, i.e., $u_{s}=\\{x\in M:u(x)=s\\}$.
In general, $u_{s}$ is not a Cauchy surface even if $M$ is globally
hyperbolic. We left such an example in Section 4. Our main results of this
paper are stated as follows.

###### Theorem 1

If $(M,g)$ is a globally hyperbolic space-time, then
$\mathcal{S}(M)\neq\emptyset$. In other words, the eikonal equation admits at
least one global locally Lipschitz viscosity solution.

In the following we will give the consequents of $u\in\mathcal{S}(M)$ when the
time orientation of $u$ is consistent. Without loss of generality, we only
consider the case where time orientation of $u$ is always past-directed. More
precisely:

###### Theorem 2

For any $u\in\mathcal{S}(M)$ for which the time orientation of $u$ is always
past-directed, there exists a $t_{0}>0$, which is dependent on $x$, we have

  * •

For each $s\in\text{Image}(u)$, $u_{s}$ is a partial Cauchy surface.

  * •

For any $0<t<t_{0}$, we have

$u(x)=\inf\limits_{x\leq y,d(x,y)=t}u(y)-t.$ (2.2)

  * •

The function $u$ is locally semiconcave on $(M,g)$.

Actually, under the same conditions of Theorem 2, viscosity solutions possess
some properties of weak KAM type like the ones in the Riemannian case.

###### Theorem 3

For any $u\in\mathcal{S}(M)$ for which the time orientation is always past-
directed, then

  * •

$u(x)$ is differentiable at $x\in M$ if and only if there exists a unique
future-directed timelike ray $\gamma_{x}:[0,T)\rightarrow M$ that satisfies

$\displaystyle\gamma_{x}(0)=x,~{}~{}u(\gamma_{x}(t))=u(x)-t~{}~{}\mbox{for
every }t\in[0,T)$ (2.3)

and

$\displaystyle\dot{\gamma_{x}}(0)=-\nabla u(x)$ (2.4)

where $T$ is the maximal existence time of $\gamma_{x}$.

  * •

For any $x\in M$, $u$ is differentiable at any $\gamma_{x}(t)$ for $t\in(0,T)$
and $\dot{\gamma_{x}}(t)=-\nabla u(\gamma_{x}(t))$.

The results of the above theorems are based on the fact that the time
orientation of $u$ is consistent. One may wonder what happens when the time
orientation of $u$ is non-consistent. Because of distinctiveness of causal
structure and non-positive-definiteness of Lorentzian metric, the properties
of viscosity solutions to the equation (2.1) are greatly different from the
ones in the Riemannian case. Although the situation in this case is much more
complicated, as the following results will show, we can still get some results
under some assumptions. In order to classify the elements concretely in
$\mathcal{S}(M)$, for any $u\in\mathcal{S}(M)$, we introduce the peculiar set
$P_{u}$ by

$P_{u}:=\\{x\in
M|\nabla^{+}u(x)=\emptyset~{}\text{and}~{}\nabla^{-}u(x)=\emptyset\\}.$

###### Theorem 4

For any $u\in\mathcal{S}(M)$, when $P_{u}=\emptyset$, we have:

  * •

If the time orientation of viscosity solution $u$ changes at point $\bar{x}$,
then $\nabla^{-}u(\bar{x})\neq\emptyset$.

  * •

The time orientation of $u$ changes at most once along any inextendilbe causal
curve.

By Theorem 4, when the time orientation of $u$ changed along some future-
directed inextendible causal curve, let $C_{u}$ be the time orientation
changing set of $u$,

$C_{u}:=\\{x\in M|~{}\text{there are}~{}V,W\in\nabla^{*}u(x)~{}\text{such
that}~{}V~{}\text{is past-directed causal vector},~{}W~{}\text{is future-
directed causal vector}\\}.$

The following theorem is the characterization of the set $C_{u}$.

###### Theorem 5

For any $u\in\mathcal{S}(M)$, if $P_{u}=\emptyset$ and the time orientation of
$u$ is non-consistent, then $C_{u}$ is a partial Cauchy surface.

The remainder of this paper is organized as follows. In Section 3, we
construct a function $u^{+}$ and prove $u^{+}\in\mathcal{S}(M)$, which
completes the proof of theorem 1. In Section 4, we study the properties of
level sets of $u$ when the time orientation of $u$ is consistent. In Section
5, we prove $u$ has a variational representation when the time orientation of
$u$ is consistent and as a consequence, $u$ is locally semiconcave. Based on
the results in these two sections, Theorem 2 is proved. In Section 6, we
establish the weak KAM properties of $u$ when the time orientation of $u$ is
consistent. This completes the proof of Theorem 3. In Section 7, we study the
function $u\in\mathcal{S}(M)$ when the time orientation of $u$ is non-
consistent and thus prove Theorem 4. We prove the set $C_{u}$ is a partial
Cauchy surface in Section 8.

##  3 Construction of global viscosity solutions to equation (2.1)

In this section, we firstly construct a globally defined function $u^{+}$ by a
steep time function $\tau$ introduced in [3]. Secondly, we will prove the
well-posedness and the semiconcavity of $u^{+}$ by using the method of support
function. Finally, we will show that $u^{+}$ is a viscosity solution to
equation (2.1), which completes the proof of Theorem 1.

###  3.1 The construction of $u^{\pm}$

Recall that $(M,g)$ is globally hyperbolic, $d(\cdot,\cdot)$ is continuous
with respect to both two variables and satisfies the reverse triangle
inequality:

$d(p,q)+d(q,r)\leq d(p,r)~{}~{}~{}\text{for}~{}~{}p,q,r\in
M~{}~{}\text{and}~{}~{}p\leq q\leq r.$

Let $\gamma:[a,b)\rightarrow M$ be a past-directed (respectively, future-
directed) causal curve. The point $p\in M$ is said to be a past (respectively,
future) endpoint of $\gamma$ corresponding to $t=b$ if

$\lim\limits_{t\rightarrow b^{-}}\gamma(t)=p.$

A causal curve is said to be past (respectively, future) inextendible if it
has no past (respectively, future) endpoint. A causal curve
$\gamma:(a,b)\rightarrow M$ is said to be inextendible if it is both past and
future inextendible.

######  Definition 3.1 (Cauchy Surface)

A Cauchy surface $S$ is a subset of $M$ for which every inextendible causal
curve intersects exactly once.

By [3, Theorem 3], there exists a steep temporal function $\tau:M\rightarrow
R$ such that $\tau$ is strictly increasing along any future-directed causal
curve and

$|d\tau(V)|\geq\max\\{\sqrt{-g(V,V)},|V|_{h}\\}$ (3.1)

for all timelike vector $V\in TM$. [3, Corollary 1.8] implies that there
exists a diffeomorphism $M\cong R\times N$ such that

$\tau:M\cong R\times N\rightarrow R,~{}~{}p\cong(s,x)\rightarrow s.$

For $s\in R$, let $\tau_{s}$ be the level set of $\tau$, i.e.,
$\tau_{s}:=\\{p\in M|\tau(p)=s\\}$.

###### Remark 3.1

For an inextendible causal curve $\gamma:(a,b)\rightarrow M$, by the
completeness of $h$, we have $\lim\limits_{t\rightarrow
a^{+}}\tau(\gamma(t))=-\infty$ and $\lim\limits_{t\rightarrow
b^{-}}\tau(\gamma(t))=\infty$.

For $s\in R$, each $\tau_{s}$ is a Cauchy surface. For any $p\in
I^{-}[\tau_{s}]$, let $d(p,\tau_{s}):=\sup\limits_{z\in\tau_{s}}d(p,z)$.
Similarly, let $d(\tau_{s},p):=\sup\limits_{z\in\tau_{s}}d(z,p)~{}\text{for
any}~{}p\in I^{+}[\tau_{s}]$.

###### Remark 3.2

Fixed $s\in R$, then for any $p\in I^{-}[\tau_{s}]$, $d(p,\tau_{s})$ is
finite. Indeed, by (3.1), for any future-directed causal curve $\gamma$
connected $p$ to $\tau_{s}$, $\sqrt{-g(\dot{\gamma},\dot{\gamma})}\leq
d\tau(\dot{\gamma})$. Integrating on both sides, we have
$L(\gamma)\leq\tau(s)-\tau(p)<\infty$.

###### Definition 3.2

For fixed $s>0$, $x_{0}\in\tau_{0}$ and for any $p\in I^{-}[\tau_{s}]$,
define:

  * •

$u^{+}_{\tau_{s}}(p):=d(x_{0},\tau_{s})-d(p,\tau_{s})=\sup\limits_{z\in\tau_{s}}d(x_{0},z)-\sup\limits_{z\in\tau_{s}}d(p,z)$,

  * •

$u^{+}(p):=\lim\limits_{s\rightarrow+\infty}u^{+}_{\tau_{s}}(p)$.

Similarly, for fixed $s<0$, $x_{1}\in\tau_{0}$ and for any $p\in
I^{+}[\tau_{s}]$, define:

  * •

$u^{-}_{\tau_{s}}(p):=d(\tau_{s},x_{1})-d(\tau_{s},p)=\sup\limits_{z\in\tau_{s}}d(z,x_{1})-\sup\limits_{z\in\tau_{s}}d(z,p)$,

  * •

$u^{-}(p):=\lim\limits_{s\rightarrow-\infty}u^{-}_{\tau_{s}}(p)$.

By the definition of $u^{+}_{\tau_{s}}(p)$, the continuity of
$u^{+}_{\tau_{s}}(p)$ can be derived from the continuity of $d(p,\tau_{s})$.
Obviously, $d(p,\tau_{s})$ is a lower semicontinuous function on
$I^{-}[\tau_{s}]$ since it is the supremum of a family of continuous
functions, so to get the continuity of $d(p,\tau_{s})$, we only need to prove
the upper semicontinuity.

Let $U$ be a causal convex neighborhood of $p$, we need the following lemma:

###### Lemma 3.1

[8, Lemma 3.2] If $U$ is small enough, there is a constant $C>0$, $T>0$ with
the following property: If $p\in U$ and $s>2T$ then every maximal unit speed
geodesic segment $\alpha$ from $p$ to $\tau_{s}$ satisfies
$\|\alpha^{\prime}(0)\|_{h}\leq C$.

Lemma 3.1 shows that the set of initial tangent vectors of these maximal
geodesic segments stays in a compact subset of tangent bundle $TU$.

Now suppose $d(p,\tau_{s})$ is not upper semicontinuous at $p$ in $U$. Then
there exists a sequence $p_{n}\in U$ with $p_{n}\rightarrow p$ but
$\lim\limits_{n\rightarrow\infty}d(p_{n},\tau_{s})>d(p,\tau_{s})$. For each
$n$, let $\alpha_{n}$ be a maximal timelike geodesic segment from $p_{n}$ to
$\tau_{s}$ parametrized by $h$-arc length. By Limit Curve Lemma [2, Lemma14.2]
and Lemma 3.1, there exists a subsequence ${\alpha_{n_{k}}}$ which converges
to a timelike curve $\alpha$ from $p$ to $\tau_{s}$, so we have

$d(p_{n},\tau_{s})=L(\alpha_{n_{k}})\rightarrow L(\alpha)\leq d(p,\tau_{S}).$

This contradiction implies the continuity of $u^{+}_{\tau_{s}}(p)$ on $U$.

###  3.2 A priori estimate of $u^{+}$

In this subsection we will show that both $u^{+}$ and $u^{-}$ are well-
defined. Actually, we only need to prove that $u^{+}(p)$ is well-defined and
$u^{-}(p)$ can be treated analogously. By the Arzela-Ascoli theorem, it is
sufficient to show that $u^{+}_{\tau_{s}}$ are locally equi-Lipschitz.

###### Lemma 3.2

For $p,q\in I^{-}[\tau_{s}]$ with $p\leq q$, the function $u^{+}_{\tau_{s}}$
satisfies

$u^{+}_{\tau_{s}}(q)-u^{+}_{\tau_{s}}(p)\geq d(p,q).$

###### Proof 1

By the reverse triangle inequality of Lorentzian distance function $d$, for
any $q\leq z\in\tau_{s}$, we have

$d(p,q)+d(q,z)\leq d(p,z)$

Then by the definition of $u^{+}_{\tau_{s}}$, it is easy to see that

$\displaystyle u^{+}_{\tau_{s}}(q)-d(p,q)$ $\displaystyle=$ $\displaystyle
d(x_{0},\tau_{s})-d(q,\tau_{s})-d(p,q)$ $\displaystyle=$ $\displaystyle
d(x_{0},\tau_{s})-\sup\limits_{z\in\tau_{s}}d(q,z)-d(p,q)$ $\displaystyle\geq$
$\displaystyle d(x_{0},\tau_{s})-\sup\limits_{z\in\tau_{s}}d(p,z)$
$\displaystyle=$ $\displaystyle u^{+}_{\tau_{s}}(p).$

###### Proposition 3.1

$u^{+}_{\tau_{s}}(x)$ are locally equi-Lipschitz functions, and there exist a
subsequence $s_{n}$ and a Lipschitz function $u^{+}(x)$ such that

$\lim\limits_{s_{n}\rightarrow+\infty}u^{+}_{\tau_{s_{n}}}(x)=u^{+}(x)~{}~{}\text{for
any}~{}~{}x\in M.$

Moreover, $u^{+}(x)$ is a locally semi-concave function on $M$. In particular,
when $s_{n}\rightarrow\infty$, $I^{-}[\tau_{s_{n}}]\rightarrow M$.

In order to prove Proposition 3.1, we need to introduce the concept of support
function.

###### Definition 3.3

For each $s>0$, $p\in I^{-}[\tau_{s}]$, there exist $q\in\tau_{s}$ and
$\gamma\in\Omega_{p,q}$ such that $d(p,q)=d(p,\tau_{s})=L(\gamma)$. Choosing
$p_{1}\in\gamma$ and a neighborhood $O$ of $p$ satisfying $O\subseteq
I^{-}(p_{1})$. Define

$\displaystyle u^{+}_{p,s}:O\rightarrow[-\infty,+\infty]$

with

$\displaystyle
u^{+}_{p,s}(x)=d(x_{0},\tau_{s})-d(x,p_{1})-d(p_{1},q)~{}~{}\text{for
any}~{}~{}x\in O.$

Due to the reverse triangle inequality, it is easy to see that for any $s>0$,
$u^{+}_{p,s}$ given by Definition 3.3 is a continuous upper support function
for $u^{+}_{\tau_{s}}(x)$ at $p$, i.e., $u^{+}_{p,s}(x)\geq
u^{+}_{\tau_{s}}(x)$ for all $x$ near $p$ with equality when $x=p$. With the
help of $u^{+}_{p,s}(x)$, we can prove that $u^{+}_{\tau_{s}}(x)$ are locally
equi-Lipschitz.

Put $d_{s}(x)=d(x,\tau_{s})$, it is sufficient to show that the functions
$u^{+}_{\tau_{s}}$ are Lipschitz continuous with a uniform Lipschitz constant
for all $s$ big enough. Since

$u^{+}_{\tau_{s}}(x)-u^{+}_{\tau_{s}}(y)=d_{s}(y)-d_{s}(x),$

we only need to show that $d_{s}$ are equi-Lipschitz coninuous functions. To
prove this result, the following lemma is useful.

###### Lemma 3.3

[2, Lemma 14.20] Let U be an open convex domain in $R^{n}$ and $f:U\rightarrow
R$ a continuous function. Suppose that for each $p$ in $U$ there is a smooth
lower support function $f_{p}$ defined in a neighborhood of $p$ such that
$\|d(f_{p})p\|_{h}<L$. Then $f$ is Lipschitz continuous with Lipschitz
constant $L$, i.e., for all $x,y\in U$ we have

$|f(x)-f(y)|\leq L|x-y|_{h}.$ (3.2)

Note $-u^{+}_{p,s}(x)$ defined in Definition 3.3 is a lower support function
to $-u^{+}_{\tau_{s}}$. To apply Lemma 3.3, we need to establish an estimate
for the lower support functions $-u^{+}_{p,s}(x)$. Thanks to lemma 3.3 in [8],
there exists a constant $L$ independent of $s$ such that $u^{+}_{p,s}(x)$ is
locally Lipschitz for each $s$. What is more, the constant $L$ can be written
as $L=GC$, where $G=\sup\\{g_{i,j}(x):x\in U,1\leq i,j\leq n\\}$ and $C$ is as
in Lemma 3.1. Then by the Arzela-Ascoli theorem, there exist a subsequence
$s_{n}$ and a locally Lipschitz function $u^{+}(x)$ such that

$\lim\limits_{s_{n}\rightarrow+\infty}u^{+}_{\tau_{s_{n}}}(x)=u^{+}(x)~{}~{}\text{for
any}~{}~{}x\in M.$

So far, we have obtained the local Lipschitzness of $u^{+}$. In the following
we will show that $u^{+}$ is a locally semiconcave function. Local
semiconvexity (semiconcavity) is defined as follows.

###### Definition 3.4

[7, Definition 2.6] Let $O$ be an open subset of $M$. A function
$\psi:O\rightarrow R$ is said to be semiconvex if there exists a $c>0$ such
that for any constant-speed geodesic path $\gamma(t)$, $t\in[0,1]$,
$\gamma(t)\in O$,

$\psi(\gamma(t))\leq(1-t)\psi(\gamma(0))+t\psi(\gamma(1))+c\frac{t(1-t)}{2}d^{2}_{h}(\psi(\gamma(0)),\psi(\gamma(1))).$
(3.3)

A function $\psi:M\rightarrow R$ is said to be locally semiconvex if for each
$p\in M$ there is a neighborhood $O$ of $p$ in $M$ such that (3.3) holds true
as soon as $\gamma(t)\in O$ ($t\in[0,1]$); or equivalently if (3.3) holds for
some fixed positive number $c$ as long as $\gamma$ stays in a compact subset
$K$ of $O$.

Similar definitions for semiconcavity and local semiconcavity are obtained in
an obvious way by reversing the sign of the inequality in (3.3).

To show $u^{+}(x)$ is locally semiconcave, we need the following lemma.

###### Lemma 3.4

[1, Lemma 3.2] Let $U$ be an open convex domain in $R^{n}$ and $f:U\rightarrow
R$ a continuous function. Assume for some constant $c$ and for all $p\in U$
that $f$ has a smooth upper support function $f_{p}$ at $p$, i.e.
$f_{p}(x)\geq f(x)$ for all $x$ near $p$ with equality holding when $x=p$,
such that $D^{2}f_{p}(x)\leq cI$ near $p$. Then $f-\frac{c}{2}\|x\|^{2}_{E}$
is concave in $U$, thus $f$ is semiconcave and twice differentiable almost
everywhere in $U$. In this lemma, $\|\cdot\|_{E}$ denotes the Euclidean norm
on $R^{n}$.

Proof of Proposition 3.1 Without loss of generality, we can assume that
$\bar{O}$ in Definition 3.3 is a compact subset of $M$. For any $x\in O$, let
$\gamma_{x}$ be the segment between $x$ to $p_{1}$. Then by Lemma 3.4 and the
definition of $u^{+}_{p,s}(x)$, we only need to use comparison theory to give
an estimation of the Hessian of the distance function $d(x,p_{1})$$(x\in O)$
in terms of upper and lower bounds of the timelike sectional curvatures of
2-planes containing $\dot{\gamma}_{x}(0)$ and the length of $\gamma_{x}$,
where the Hessian of $d(x,p_{1})$ is defined in terms of the Levi-Civita
connection with respect to $g$. Since $\gamma$ (see Definition 3.3) is maximal
segment between $x$ to $q$, the internal of $\gamma$ is free of cut points. By
[9, Proposition 3.2], we can choose $O$ such that the distance function
$d(x,p_{1})$ is a smooth function with respect to $x\in O$. When $p_{1}$ is
close to $x$, there is a compact subset $K_{1}\subseteq TM$ that will contain
all the tangent vectors $\dot{\gamma}_{x}|_{x\in O}$. Thus, both the timelike
sectional curvatures of two planes containing $\dot{\gamma}_{x}(0)$ and the
length of $\gamma_{x}$ are bounded from above. By the method in [14, Theorem
5.10], there exists a $c(O)>0$ depending only on $O$ such that
$D^{2}d(x,p_{1})\geq-c(O)I$ for $x\in O$. Then we can finally conclude that
$u^{+}(x)$ is locally semiconcave by Arzela-Ascoli theorem.

###  3.3  $u^{+}$ is global viscosity solution to equation (2.1)

In this subsection, we will prove the function $u^{+}$ defined in Definition
3.2 is a viscosity solution of Lorentzian eikonal equation (2.1). The proof is
slightly modified from the one in [7, Proposition 2.1].

###### Lemma 3.5

If $u^{+}$ is differentiable at $p$, then $g(\nabla u^{+}(p),\nabla
u^{+}(p))=-1$ and $\nabla u^{+}(p)$ is a past-directed vector.

###### Proof 2

Assume that $u^{+}$ is differentiable at $p$. Choosing any smooth future-
directed causal curve $\gamma:[0,T)\rightarrow M(T>0)$ with $\gamma(0)=p$, we
have

$\displaystyle u^{+}(\gamma(t))-u^{+}(p)$
$\displaystyle=u^{+}(\gamma(t))-u^{+}(\gamma(0))$
$\displaystyle=\lim\limits_{s\rightarrow\infty}u^{+}_{\tau_{s}}(\gamma(t))-\lim\limits_{s\rightarrow\infty}u^{+}_{\tau_{s}}(\gamma(0))$
$\displaystyle=\lim\limits_{s\rightarrow\infty}(d(\gamma(0),\tau_{s})-d(\gamma(t),\tau_{s}))$
$\displaystyle\geq d(\gamma(0),\gamma(t))$
$\displaystyle\geq\int_{0}^{t}\sqrt{-g(\dot{\gamma}(l),\dot{\gamma}(l))}dl$

for every $t\in[0,T)$. Dividing by $t$ on both sides, we get

$\frac{u^{+}(\gamma(t))-u^{+}(p)}{t}\geq\frac{1}{t}\int_{0}^{t}\sqrt{-g(\dot{\gamma}(l),\dot{\gamma}(l))}dl.$
(3.4)

Letting $t\rightarrow 0_{+}$, the differentiability of $u^{+}$ at $p$,
together with inequality (3.4), we have

$du^{+}(p)(\dot{\gamma}(0))\geq\sqrt{-g(\dot{\gamma}(0),\dot{\gamma}(0))}.$

Note that $du^{+}(p)(\dot{\gamma}(0))=g(\nabla u^{+}(p),\dot{\gamma}(0))$,
this means that

$g(\nabla
u^{+}(p),\dot{\gamma}(0))\geq\sqrt{-g(\dot{\gamma}(0),\dot{\gamma}(0))}$ (3.5)

for any future-directed causal vector $\dot{\gamma}(0)$. To continue the proof
of Lemma 3.5, we need the following two claims.

Claim 1. $\nabla u^{+}(p)$ is a past-directed timelike vector.

First of all, by (3.5) we have $\nabla u^{+}(p)\neq 0$. Suppose $\nabla
u^{+}(p)$ is spacelike, then there exists a smooth future-directed timelike
curve $\bar{\gamma}:[0,\epsilon)\rightarrow M$ such that $\bar{\gamma}(0)=p$
and $g(\dot{\bar{\gamma}}(0),\nabla u^{+}(p))=0$, this means that $\nabla
u^{+}(p)$ and $\dot{\bar{\gamma}}(0)$ are orthogonal with respect to the
Lorentzian metric $g$. On the other hand, form (3.5), we get $g(\nabla
u^{+}(p),\dot{\bar{\gamma}}(0))>0$, this contradicts the orthogonal
hypothesis. Thus, $\nabla u^{+}(p)$ is a causal vector. Furthermore, $g(\nabla
u^{+}(p),V)\geq 0$ for any future-directed causal vector $V\in T_{p}M$ implies
$\nabla u^{+}(p)$ is past-directed. To show $\nabla u^{+}(p)$ is indeed a
timelike vector, we need the following lemma

###### Lemma 3.6

Suppose $V\in T_{p}M$ is a nonzero past-directed non-spacelike vector and
$g(V,W)\geq\sqrt{-g(W,W)}$ for any future-directed causal vector $W$, then $V$
is not lightlike.

The proof of Lemma 3.6 is omitted and we refer to [7, Lemma 2.5].

Claim 2. $|\nabla u^{+}(p)|=1$.

First we choose a future-directed smooth causal curve
$\delta:[0,\epsilon)\rightarrow M$ with $\delta(0)=p$,
$\dot{\delta}(0)=-\nabla u^{+}(p)$. Then by inequality (3.5),

$\displaystyle|\nabla u^{+}(p)|^{2}=-g(\nabla u^{+}(p),\nabla
u^{+}(p))=g(\nabla
u^{+}(p),\dot{\delta}(0))\geq\sqrt{-g(\dot{\delta}(0),\dot{\delta}(0))}=|\nabla
u^{+}(p)|.$

Hence, either $|\nabla u^{+}(p)|\geq 1$ or $|\nabla u^{+}(p)|=0$. By Claim 1,
we obtain

$|\nabla u^{+}(p)|\geq 1.$ (3.6)

Notice that there exists a maximal geodesic segment between $p$ and $\tau_{s}$
denoted by $\gamma_{p,s}$, i.e., $\gamma_{p,s}$ is a future-directed timelike
(unit-speed) ray $\gamma_{p,s}:[0,d(p,\tau_{S})]\rightarrow M$ with
$\gamma_{p,s}(0)=p$, such that $d(p,\tau_{s})=L(\gamma_{p,s})$ and for
$0<t_{1}<t_{2}<s$,

$\displaystyle d(\gamma_{p,s}(t_{1}),\gamma_{p,s}(t_{2}))=t_{2}-t_{1}.$

Then we get

$\displaystyle\frac{u^{+}(\gamma_{p,s}(t))-u^{+}(\gamma_{p,s}(0))}{t-0}=\frac{\lim\limits_{s\rightarrow\infty}d(\gamma_{p,s}(0),\tau_{s})-d(\gamma_{p,s}(t),\tau_{s})}{t-0}=1$

for every $t\in(0,s)$. Letting $t\rightarrow 0^{+}$, by the the reverse
Cauchy-Schwarz inequality for causal vectors ([17, 2.2.1]) and $\gamma_{p,s}$
is a timelike (unit-speed) ray, we have

$1=du^{+}(p)(\dot{\gamma_{p,s}}(0))=g(\nabla
u^{+}(p),\dot{\gamma_{p,s}}(0))\geq|\nabla
u^{+}(p)||\dot{\gamma_{p,s}}(0)|=|\nabla u^{+}(p)|.$ (3.7)

Combining inequalities (3.6) and (3.7), Claim 2 follows.

So far, the proof of Lemma 3.5 is finished.

For a set $B$ in a vector space, the convex hull of $B$, $coB$, is the
smallest convex set containing $B$. From convex analysis, we know the
following lemma:

###### Lemma 3.7

If $\psi$ is a locally semiconvex (resp., semiconcavity) function on manifold
$M$, then it is locally Lipschitz (under any reasonable metric), and
$\nabla^{-}\psi(p)$ (resp., $\nabla^{+}\psi(p)$) is nonempty for any $p\in M$.
In this case,
$\nabla^{-}\psi(p)(resp.,\nabla^{+}\psi(p))=co\nabla^{*}\psi(p)\subset
T_{p}M$.

For a proof of Lemma 3.7, we refer to [6, Theorem 3.3.6], where the limiting
gradient is called reachable gradient.

###### Lemma 3.8

Let $(M,g)$ be a globally hyperbolic space-time, then the function $u^{+}$
defined in definition 3.2 is a viscosity solution to Lorentzian eikonal
equation (2.1) .

###### Proof 3

By the definition of viscosity solution, we need to show $u^{+}$ is
subsolution and supersolution of equation (2.1). Firstly, for any
$V\in\nabla^{+}u^{+}(p)$, by Lemma 3.7,

$\displaystyle V\in\nabla^{+}u^{+}(p)=co\nabla^{*}u^{+}(p).$

Together with Lemma 3.5, we can conclude that

$g(V,V)\leq-1.$

It means that $u^{+}$ is a subsolution of the equation (2.1).

On the other hand, since $u^{+}$ is locally semiconcave, $u^{+}$ is
differentiable at $p\in M$ whenever $\nabla^{-}u^{+}(p)\neq\emptyset$. Thus,
Lemma 3.5 implies that $u^{+}$ is a supersolution.

Therefore $u^{+}$ is indeed a viscosity solution of the equation (2.1) and
this completes the proof of Theorem 1.

###### Remark 3.3

Using the same method, we can finally conclude that $u^{-}$ defined in
Definition 3.2 is also a global viscosity solution to Lorentzian eikonal
equation (2.1).

##  4 The level sets of viscosity solution of Lorentzian eikonal equation

In this section, for any $u\in\mathcal{S}(M)$ we shall study the level sets of
$u$ when the time orientation of $u$ is consistent. Firstly, we give an
equivalent characterization of time orientation of $u$. Secondly, we show that
$u_{s}$ is a partial Cauchy surface for each $s\in\text{Image}(u)$. Finally,
we give an example, which shows that in general the level set $u_{s}$ is not a
Cauchy surface. Recall that the time orientation of $u$ is consistent means
that for any $x,y\in M$, $V\in\nabla^{*}u(x)$ and $W\in\nabla^{*}u(y)$, $V$
and $W$ have the same time orientation. Without loss of generality, we always
assume that the time orientation of $u$ is past-directed and
$s\in\text{Image}(u)$ in this section.

###  4.1 Equivalent characterization of time orientation of viscosity
solutions

###### Lemma 4.1

Let $u\in\mathcal{S}(M)$, then the time orientation of $u$ is always past-
directed if and only if $u$ is increasing monotonically along any inextendible
future-directed causal curve.

###### Proof 4

Assume that the time orientation of $u$ is always past-directed and let
$\zeta$ be an inextendible future-directed causal curve. We firstly consider
the case that $u$ is differentiable almost everywhere on $\zeta$. For any
$a<b\in\text{dom}(\zeta)$, we have

$\displaystyle u(\zeta(b))-u(\zeta(a))$
$\displaystyle=\int_{a}^{b}d_{\zeta(s)}u(\dot{\zeta}(s))ds$
$\displaystyle=\int_{a}^{b}g(\nabla u,\dot{\zeta})|_{\zeta(s)}ds$
$\displaystyle>0.$

If $u$ is not differentiable almost everywhere on $\zeta$, taken an open
subset $U\subseteq M$ such that $\zeta|_{[a,b]}\subseteq U$. Since the time
orientation of $u$ is past-directed, we can find a sequence of future-directed
causal curves $\zeta_{i}$ satisfying

  * •

$\zeta_{i}\subseteq U$ for each i.

  * •

$\zeta_{i}\rightarrow\zeta$ in the $C^{0}$-topology.

  * •

$u$ is differentiable almost everywhere on $\zeta_{i}$.

Furthermore,

$\displaystyle u(\zeta(b))-u(\zeta(a))$
$\displaystyle=\lim\limits_{i\rightarrow\infty}\int_{a}^{b}d_{\zeta_{i}(s)}u(\dot{\zeta}_{i}(s))ds$
$\displaystyle=\lim\limits_{i\rightarrow\infty}\int_{a}^{b}g(\nabla
u,\dot{\zeta}_{i})|_{\zeta_{i}(s)}ds$ $\displaystyle>0.$

By the arbitrariness of $a$, $b$ and $\zeta$, $u$ is increasing monotonically
along any inextendible future-directed causal curve. For the rest of the
proof, we propose the claim.

Claim: Let $x$ be a differentiable point of $u$, then there exists a
neighborhood $U$ of $x$ such that for any differentiable point $y\in U$, the
time orientation of $u$ at $y$ is consistent with the time orientation of $u$
at $x$.

Proof of Claim. Actually, if the claim is not true, we can get a sequence
$\\{x_{i}\\}$ with $x_{i}\rightarrow x$ and the time orientation of $u$ at
$x_{i}$ is different from the one at $x$. Then there exists a timelike vector
in $\nabla^{*}u(x)$, which is opposite to the time orientation of $u$ at $x$.
This contradicts the hypothesis that $x$ is a differentiable point of $u$.
Furthermore, by the analysis above, $u$ increases locally monotonically along
any future-directed causal curve passing through $x$ if $u$ is differentiable
at $x$ and the time orientation of $u$ is past-directed.

If there exists a future-directed timelike vector $V\in\nabla^{*}u(x)$, by the
definition of $\nabla^{*}$, we can get a sequence of differentiable points
$x_{i}$ such that $\nabla u(x_{i})$ is future-directed timelike for each
$x_{i}$. By the Claim, we can easily find a future-directed causal curve
$\gamma$ in a neighborhood of $x_{i}$ such that $u$ is decreasing
monotonically on $\gamma$. One can extend $\gamma$ to an inextendible causal
curve. It is a contradiction to the fact that $u$ is increasing monotonically
along any inextendible future-directed causal curve. Up to now, the proof of
Lemma 4.1 is complete.

###  4.2 The level sets of $u$ are partial Cauchy surfaces

###### Lemma 4.2

Fix $s\in\text{Image}(u)$, $u_{s}$ is a closed acausal set.

###### Proof 5

Indeed, $u_{s}$ is a closed subset by the continuity of $u$. We only need to
show that $u_{s}$ is a acausal set. For $x_{1},x_{2}\in u_{s}$, if there
exists a future-directed causal curve $\theta$ connecting $x_{1}$ and $x_{2}$,
without loss of generality, we can assume that $\theta(0)=x_{1}$,
$\theta(1)=x_{2}$ and $u$ is differentiable almost everywhere on $\theta$. By
Lemma 4.1, we have

$\displaystyle u(x_{2})>u(x_{1})$

This contradicts the fact $x_{1},x_{2}\in u_{s}$.

###### Lemma 4.3

For each $s\in\text{Image}(u)$, $\text{edge}(u_{s})=\emptyset$ .

###### Proof 6

On the contrary, for any arbitrarily fixed $s\in\text{Image}(u)$ and any
$p\in\text{edge}(u_{s})$, by the definition of $\text{edge}(u_{s})$, there
exists a future-directed timelike curve $\beta:[a,b]\rightarrow M$ with
$\beta(a)\ll p\ll\beta(b)$ and $\beta(t)\cap u_{s}=\emptyset$ for any
$t\in[a,b]$. By Lemma 4.1, $u$ is strictly monotonically increasing along any
future-directed causal curve. Thus we must have
$u(\beta(0))<u(p)<u(\beta(t))$, this inequality together with the continuity
of $u$ implies that there exists a $t_{0}\in[a,b]$ such that
$u(\beta(t_{0}))=u(p)$. This is impossible, since $\beta$ is chosen such that
$\beta(t)\cap u_{s}=\emptyset$ for any $t\in[a,b]$.

Up to now, by Definition 2.7, $u_{s}$ is a partial Cauchy surface for each
$s\in\text{Image}(u)$. To conclude this section we provide an example, which
shows that in general the level sets $u_{s}$ are not Cauchy surfaces.

Example. In the 2-dimensional Minkowski space-time $(R^{2},g)$, let
$M=I^{+}((-1,0))\cap I^{-}((1,0))$ be the induced space-time. Clearly, $M$ is
globally hyperbolic. It is easy to see that $u(x,y)=x$ is a globally defined
viscosity solution of equation (2.1) on $M$, but level sets $u_{t}$ are only
partial Cauchy surfaces except $t=0$. See Figure 1.

$x$$y$$O$$\scriptstyle(-1,0)$$\scriptstyle(1,0)$$\scriptstyle(0,-1)$$\scriptstyle(0,1)$$\scriptstyle
x=a$$\scriptstyle(a,0)$ Figure 1: A space-time admits a viscosity solution
$u(x,y)$, where the level sets of $u$ are not Cauchy surfaces in general.

##  5 A variational representation of $u$

In this section, we will give a variational representation of $u$ when the
time orientation of $u$ is always past-directed. First of all, we propose a
variational problem and prove the existence of minimizers. In the second
place, we prove equality (2.2) and the existence of forward calibrated curve
of $u$. In the end, we show that $u$ is a locally semiconcave function. These
results in company complete the proof of the Theorem 2. Our proof is motivated
by the weak KAM theory for positively definite Lagrangian systems [10]. To go
into the details, we first fix a notation. In this section, $\Omega_{x}^{t}$
denotes the set of all the past-directed piecewise smooth timelike curve
$\gamma:[0,t]\rightarrow M$ with $\gamma(t)=x$.

###  5.1 The existence of minimizer to the variational problem

Define

$\tilde{u}(t,x):=\inf\limits_{\gamma\in\Omega_{x}^{t}}\left[u(\gamma(0))-\int_{0}^{t}\sqrt{-g(\dot{\gamma}(s),\dot{\gamma}(s))}ds\right].$
(5.1)

Firstly, we shall prove the existence of minimizers in this variational
problem.

###### Lemma 5.1

For any $x\in M$, there exist a $t_{0}>0$ depending on $x$ and a past-directed
timelike curve $\gamma_{x}\in\Omega_{x}^{t}$ such that

$\tilde{u}(t,\gamma_{x}(t))=u(\gamma_{x}(0))-\int_{0}^{t}\sqrt{-g(\dot{\gamma}_{x}(s),\dot{\gamma}_{x}(s))}ds$
(5.2)

for any $t\leq t_{0}$.

###### Proof 7

Before proceeding, we need to introduce the concept of domain of dependence,
or Cauchy development.

######  Definition 5.1 (Cauchy development)

Given a closed subset $S$ of $(M,g)$, the past Cauchy development or domain of
dependence of $S$, $D^{-}(S)$, is defined as the set of all points $q$ such
that every future inextendible causal curve from $q$ intersects $S$. The
future domain of dependence of $S$, $D^{+}(S)$, is defined in a similar way by
exchanging the roles of the future and the past. The domain of dependence is
the set $D(S)=D^{-}(S)\cup D^{+}(S)$.

Since $u_{s}$ is a partial Cauchy surfaces for each $s\in\text{Image}(u)$, it
is easy to conclude that $D^{-}(u_{s})\setminus u_{s}\neq\emptyset$ and
$D^{+}(u_{s})\setminus u_{s}\neq\emptyset$. In the following, we will prove a
useful claim.

Claim: Let $\Sigma$ be a partial Cauchy surface. For $x\in D^{-}[\Sigma]$, set
$I^{+}_{\Sigma}(x)=\Sigma\cap I^{+}(x)$, $J^{+}_{\Sigma}(x)=\Sigma\cap
J^{+}(x)$. Then $J^{+}_{\Sigma}(x)=\overline{I^{+}_{\Sigma}(x)}$ is compact.

Proof of Claim: Recall that for $x\in D^{-}[\Sigma]$, any future-directed
causal curve $\gamma$ with $\gamma(0)=x$ must strike $\Sigma$. Reparametrize
all inextendible future-directed causal geodesics by $h$-arclength, then these
geodesics satisfy a system of second order ordinary differential equations.
Since $\Sigma$ is a partial Cauchy surface, we can define a map $T_{\Sigma}$
sending any $\dot{\gamma}(0)\in U_{h}T_{x}M\cap\mathcal{C}_{x}$ to the unique
intersection point of $\gamma$ with $\Sigma$. Where $U_{h}T_{x}M$ is the unit
tangent sphere at $x$ with respect to Riemannian metric $h$, $\mathcal{C}_{x}$
is the future-pointing light cone at $x$. Applying the theorem of Picard-
Lindelöf [13, p.8], it is easy to see that $T_{\Sigma}$ is continuous. Since
the $U_{h}T_{x}M\cap\text{int}\mathcal{C}_{x}$ is diffeomorphic to an open
ball in Euclidean space,

$J^{+}_{\Sigma}(x)=T_{\Sigma}(U_{h}T_{x}M\cap\mathcal{C}_{x})=\overline{T_{\Sigma}(U_{h}T_{x}M\cap
int\mathcal{C}_{x})}=\overline{I^{+}_{\Sigma}(x)}$

is compact. Hence, the claim is proved.

Now we continue the proof of Lemma 5.1. For any $x\in M$ and
$\gamma\in\Omega_{x}^{t}$, $L(\gamma)\leq u(\gamma(0))-u(x)$, and consequently
$\tilde{u}(t,x)\geq u(x)$. Meanwhile, since $u$ is locally Lipschitz with
respect to $h$, there exists a compact neighborhood $O$ of $x$ such that the
set of all limiting gradient $V\in\nabla^{*}u(x)(x\in O)$ lies in a compact
subset of $TM$. Thus $\text{dist}_{h}(V,\partial\mathcal{C}_{x})>\delta>0$,
where $\delta>0$ is a positive constant depending only on $x$ and
$\partial\mathcal{C}_{x}$ is the boundary of future-pointing light cone
$\mathcal{C}_{x}$. By Lemma 4.1, for any future-directed causal curve
$\gamma:[0,t]\rightarrow M$ with $\gamma(0)=x$,
$u(\gamma(t))-u(\gamma(0))>K(t,\delta)>0$, where $K(t,\delta)$ is a positive
constant depends only on $t,\delta$. Therefore, there exists an
$s\in\text{Image}(u)$ such that $x\in D^{-}(u_{s})\setminus u_{s}$. For any
$z\in J^{+}_{u_{s}}(x)$, we set $t_{x,z}=\frac{1}{2}d(x,z)$ and
$t_{0}=\max\limits_{z\in J^{+}_{u_{s}}(x)}t_{x,z}$. For the set $\\{p\in
I^{+}(x)|d(x,p)=t_{0}\\}$, there exists a point $y\in u_{s}$ such that
$d(x,y)=t_{0}$. Recall that $J^{+}_{u_{s}}(x)$ is a compact set by the above
claim, then for any $V_{i}\in U_{h}T_{x}M\cap\mathcal{C}_{x}$, there exists a
$T_{V_{i}}>0$, such that $\gamma(T_{V_{i}})\in J^{+}_{u_{s}}(x)$ and

$J^{+}(x)\cap J^{-}[u_{s}]=\bigcup\limits_{i}\gamma_{V_{i}}[0,T_{V_{i}}].$

Furthermore, There exists a $T>0$ independent of $i$, such that
$\bigcup\limits_{i}\gamma_{i}[0,T_{V_{i}}]\subseteq\bigcup\limits_{i}\gamma_{i}[0,T]$.
Since the $U_{h}T_{x}M\cap\text{int}\mathcal{C}_{x}$ is diffeomorphic to an
open ball in Euclidean space, we have

$J^{+}(x)\cap J^{-}[u_{s}]\subseteq\bigcup\limits_{i}\gamma_{V_{i}}[0,T]$

is a compact subset of $M$. Since the Lorentzian length functional $L(\gamma)$
does not rely on parameterization of $\gamma$, for the minimizer $\gamma$ of
(5.1), we can always assume that $\gamma$ is a unit speed maximal geodesic,
then it is easy to check that

$\displaystyle\tilde{u}(t_{0},x)=\inf\limits_{x\leq
y,d(x,y)=t_{0}}u(y)-t_{0}.$ (5.3)

Then for any $t\leq t_{0}$, the minimizer of (5.3) must be contained in
$\\{y|d(x,y)=t_{0}\\}\cap J^{-}[u_{s}]$, which is a compact subset of $M$.
Because of the upper semi-contimuity of $L(\cdot)$ and the existence of
maximal geodesic between any two points in $M$, we can finally find
$\gamma_{x}$ satisfies equation (5.2).

In the rest of this section, for any $x\in M$, we still use $\gamma_{x}^{t}$
to denote one of the minimizers in equality (5.1).

###  5.2  Variational representation and forward calibrated curve

In this subsection, we shall show that the function $u$ has a local
representation and a forward calibrated curve. Based on Lemma 5.1, we have the
following lemma.

###### Lemma 5.2

For any $x\in M$ and $0<t\leq t_{0}$, we have

$\displaystyle u(x)=\tilde{u}(t,x)=\inf\limits_{x\leq y,d(x,y)=t_{0}}u(y)-t.$
(5.4)

###### Proof 8

Indeed, the second equality has been proved in Lemma 5.1. We only need to show
that $u(x)=\tilde{u}(t,x)$ for any $t\in[0,t_{0}]$. For $0<t_{1}<t_{2}<t_{0}$,
by Lemma 5.1, we have two past-directed timelike curves $\gamma_{x}^{t_{1}}$
$\gamma_{x}^{t_{2}}$. We define a past-directed timelike curve
$\gamma_{x}^{t_{1},t_{2}}:[0,t_{2}]\rightarrow M$ with
$\gamma_{x}^{t_{1},t_{2}}(s)=\gamma_{x}^{t_{1}}(\frac{t_{1}}{t_{2}}s)$ for any
$s\in[0,t_{2}]$. Since the Lorentzian length functional $L(\gamma)$ does not
rely on parameterization of $\gamma$, it can be concluded that

$\displaystyle\tilde{u}(t_{2},x)-\tilde{u}(t_{1},x)\leq
u(\gamma_{x}^{t_{1},t_{2}}(0))-L(\gamma_{x}^{t_{1},t_{2}})-u(\gamma_{x}^{t_{1}}(0))-L(\gamma_{x}^{t_{1}})=0.$

Similarly, it can be inferred that

$\displaystyle\tilde{u}(t_{2},x)-\tilde{u}(t_{1},x)\geq 0.$

Thus, $\tilde{u}(t,x)$ is a constant with respect to $t$. Letting
$t\rightarrow 0$, we can finally conclude that

$\tilde{u}(t,x)=u(x)$

Up to now, we have shown that $u(x)=\inf\limits_{x\leq y,d(x,y)=t}u(y)-t$ for
any $t\in[0,t_{0}]$. Without loss of generality, in the rest of this paper ,
we always assume that $\gamma_{x}^{t_{0}}$ is a unit speed maximal geodesic
segment. For any $s\in[0,t_{0}]$, we define

$\breve{\gamma}^{t_{0}}_{x}(s)=\gamma^{t_{0}}_{x}(t_{0}-s).$

Then we have a future-directed unit speed timelike curve
$\breve{\gamma}^{t_{0}}_{x}:[0,t_{0}]\rightarrow M$ satisfying

$\displaystyle
u(\breve{\gamma}_{x}^{t}(s_{2}))-u(\breve{\gamma}_{x}^{t}(s_{1}))=s_{2}-s_{1},$
(5.5)

for any $0\leq s_{1}\leq s_{2}\leq t_{0}$. For $\breve{\gamma}_{x}(t_{0})$,
using Lemma 5.2 again, we can extend
$\breve{\gamma}_{x}^{t_{0}}:[0,t_{0}]\rightarrow M$ to a future-directed
timelike curve (denoted by $\breve{\gamma}_{x}^{t_{0}+t_{1}}$)
$\breve{\gamma}_{x}^{t}:[0,t_{0}+t_{1}]\rightarrow M$. Obviously,
$\breve{\gamma}_{x}^{t_{0}+t_{1}}$ satisfies equality (5.5) for any
$s\in[0,t_{0}+t_{1}]$. Do above steps inductively, one can easily get a
future-directed inextendible timelike curve, denoted by $\gamma_{x}$.
Actually, since $u$ is a local Lipschitz function, for any $x\in M$, $u$ will
never blow up at $x$, then the future endpoint of $\gamma_{x}$ must not belong
to $M$.

In the rest of this paper, we always use $\gamma_{x}:[0,T)$ denote a forward
calibrated curve of $u$ at $x$, where $T$ is the maximal existence time of
$\gamma_{x}$. By definition 2.6, $\gamma_{x}$ is a forward calibrated curve of
$u$ at $x$. In the following, we shall show that $\gamma_{x}$ is a ray.
Without loss of generality, we always assume that $u$ is differentiable almost
everywhere on any timelike curve. Fix arbitrarily $t_{1},t_{2}\in[0,T)$, for
any timelike curve $\eta$ connected $\gamma_{x}(t_{1})$ to
$\gamma_{x}(t_{2})$, by the reverse Cauchy-Schwarz inequality, we have

$L(\gamma_{x})|_{(t_{1},t_{2})}=u(\gamma_{x}(t_{2}))-u(\gamma_{x}(t_{1}))\geq\int
g(\nabla u(\dot{\eta}),\eta)=L(\eta)$

This means that the inextendible timelike curve $\gamma_{x}$ is maximal on
each segment. In a word, $\gamma_{x}$ is an inextendible forward calibrated
ray.

###  5.3 Local semiconcavity of $u$

In this subsection, we will show that $u$ is locally semiconcave when the time
orientation of $u$ is consistent. Actually, with the help of variational
representation of $u$, one can directly calculate the semiconcavity of $u$ in
a local chart. Here we use the method in Proposition 3.1.

###### Proposition 5.1

Let $u\in\mathcal{S}(M)$, suppose that the time orientation of $u$ is always
past-directed, then for any $x\in M$, there exists an $s\in\text{Image}(u)$
such that up to a constant,

$u(x)=-d(x,u_{s}).$

Proposition 5.1 can be seen as a space-time counterpart of the results
obtained in [4], where the author showed that any viscosity solution is
somehow a kind of distance like function in the Riemannian case. Distance like
functions, introduced by Gromov, play an important role in studying the
geometry and topology of some kind of non-compact manifold. For more details,
readers can refer [4, 12]. The proof of Proposition 5.1 is based on the
following lemmas.

###### Lemma 5.3

For any unit vector $V\in\nabla^{*}u(x)$, there exist a constant $c>0$ and a
unique future-directed timelike curve $\gamma_{x,V}:[0,c]\rightarrow M$,
satisfies

$\gamma_{x,V}(0)=x,~{}~{}~{}u(\gamma_{x,V}(t))=u(\gamma_{x,V}(0))+t~{}~{}\mbox{for
every}~{}~{}t\in[0,c]$

and

$\dot{\gamma}_{x,V}(0)=-V.$

###### Proof 9

For any $V\in\nabla^{*}u(x)$, by the definition of $\nabla^{*}u(x)$, there
exists a sequence of events $x_{i}\in M$ with $x_{i}\rightarrow x$, such that
$u$ is differentiable at $x_{i}$ and $\nabla u(x_{i})\rightarrow V$. For each
$x_{i}\in M$, by equality (5.5), there exists a future-directed timelike curve
$\gamma_{i}:[0,t_{i}]\rightarrow M$, such that

$u(\gamma_{i}(t))-u(\gamma_{i}(0))=t$

for every $t\in[0,t_{i}]$. Diving by $t$ and letting $t\rightarrow 0$, we have

$\displaystyle g(\nabla u(x_{i}),\dot{\gamma}_{x_{i}}(0))=1.$

On the other hand, by the reverse Cauchy-Schwarz inequality,

$g(\nabla u(x_{i}),\dot{\gamma}_{x_{i}}(0))\geq|\nabla
u(x_{i})||\dot{\gamma}_{x_{i}}(0)|=1$

Thus,

$\dot{\gamma}_{i}(0)=-\nabla u(x_{i}).$

For such a sequence of timelike curves $\gamma_{i}$, without loss of
generality, we can always assume that $\gamma_{i}(0)$ are contained in a
compact neighborhood of $x$. Then, we have:

###### Lemma 5.4

There exists a constant $c>0$ such that
$\bigcup_{i=1}^{\infty}(\gamma_{i}([0,c]),\dot{\gamma}_{i}([0,c]))$ is
contained in a compact subset $K_{c}$ of $TM$.

###### Proof 10

The proof is based on the following two claims:

Claim 1: There exists a $c>0$ such that
$\bigcup_{i=1}^{\infty}(\gamma_{i}([0,c]))$ is contained in a compact subset
$K^{c}$ of $M$.

Proof of Claim 1. Suppose Claim 1 is false, then for any $c>0$, there exists a
subsequence $\\{\gamma_{i_{k}}\\}$ of $\\{\gamma_{i}\\}$, labeled by
$\gamma_{k}:=\gamma_{i_{k}}$, such that for each $k$, one can find a point
$y_{k}\in\gamma_{k}([0,c])$ with

$u(y_{k})\leq u(x_{k})+c$

and the set $\\{y_{k}\\}$ is not contained in any compact subset of $M$.

Reparameterize all $\gamma_{k}$ with respect to arc length induced by the
metric $h$ and denoted by $\tilde{\gamma_{k}}$ with
$\tilde{\gamma_{k}}(0)=\gamma_{k}(0)=x_{k}$. Then it is easy to see, for each
$k$, there exists a unique $a_{k}\in R_{+}$ such that
$y_{k}=\tilde{\gamma_{k}}(a_{k})$. By assumption on $\\{y_{k}\\}$,
$a_{k}\rightarrow\infty$ as $k\rightarrow\infty$. By the limit curve lemma [2,
Lemma 14.2] $\\{\tilde{\gamma_{k}}|_{[0,a_{k}]}\\}$ converges uniformly (with
respect to $h$), passing to a subsequence if necessary, on any compact
interval of $[0,\infty)$ to a future-directed inextendible causal curve
$\tilde{\gamma}:[0,\infty)\rightarrow M$. Thanks to the continuity of $u$, for
each $t\in[0,\infty)$ we have

$u(\tilde{\gamma}(t))=\lim\limits_{k\rightarrow\infty}u(\tilde{\gamma}_{k}(t))\leq\limsup\limits_{k\rightarrow\infty}u(y_{k})\leq\limsup\limits_{k\rightarrow\infty}u(x_{k})+c=u(\tilde{\gamma}(0))+c.$

Since the time orientation of $u$ is past-directed, $u$ is strictly increasing
along any future-directed causal curve, i.e.,
$u(\tilde{\gamma}(t))>u(\tilde{\gamma}(0))$. This contradicts the
arbitrariness of $c$. On the other hand, by the classical ODE theory, the
maximal existence time of ODE is lower semi-continuous with respect to initial
value. Then we can find a uniform $c>0$ such that
$\bigcup_{i=1}^{\infty}(\gamma_{i}([0,c]))$ is contained in a compact subset.

Claim 2: For any fixed compact set $K\subset M$ and any $x\in K$, let
$\gamma_{x}:[0,c]\rightarrow M$ be a future-directed timelike curve with
$\gamma_{x}(0)=x$ and $u(\gamma_{x}(t))=u(\gamma_{x}(0))+t$. Then
$\\{\dot{\gamma}_{x}(0):x\in K\\}\subset TM$ is bounded. That is to say,
$\\{\dot{\gamma}_{x}(0):x\in K\\}$ has compact closure in $TM$.

Proof of Claim 2. The proof is similar to claim 1 and again we parameterize
$\gamma_{k}$ with respect to arc length induced by $h$. If
$\\{\dot{\gamma}_{x}(0):x\in K\\}$ is not bounded, there exists a family of
inextendible future-directed timelike curves
$\gamma_{\ell}:[0,\infty)\rightarrow M$, parameterized with respect to
$h$-arclength, such that

$d(\gamma_{\ell}(0),\gamma_{\ell}(\bar{c}))=u(\gamma_{\ell}(\bar{c}))-u(\gamma_{\ell}(0))$
(5.6)

for some $\bar{c}\in[0,\infty)$, $\gamma_{\ell}(0)\rightarrow x\in K$ and the
sequence of $h$-unit vectors $\dot{\gamma_{\ell}}(0)$ converges to an $h$-unit
null vector $V$. Let $\bar{\gamma}:[0,\infty)\rightarrow M$ be the limit curve
of $\gamma_{\ell}$. Obviously, $\bar{\gamma}$ satisfies $\bar{\gamma}(0)=x$
and $\dot{\bar{\gamma}}(0)=V$. Then $\bar{\gamma}$ is necessarily a null ray.
By (5.6), we have

$\displaystyle d(\bar{\gamma}(0),\bar{\gamma}(\bar{c}))$
$\displaystyle=\lim\limits_{\ell\rightarrow\infty}d(\bar{\gamma}_{\ell}(0),\bar{\gamma}_{\ell}(\bar{c}))$
$\displaystyle=\lim\limits_{\ell\rightarrow\infty}u(\bar{\gamma}_{\ell}(\bar{c}))-u(\bar{\gamma}_{\ell}(0))$
$\displaystyle=u(\bar{\gamma}(\bar{c}))-u(\bar{\gamma}(0)).$

The curve $\bar{\gamma}$ is null means that $\bar{\gamma}(\bar{c})\in
u_{\bar{\gamma}(0)}$, which is obviously impossible since the level sets of
$u$ are acausal sets.

By Claim 1 and Claim 2, Lemma 5.4 is proved.

Now we continue the proof of Lemma 5.3. By Lemma 5.4,
$\bigcup_{i=i_{c}}^{\infty}(\gamma_{i}([0,c]),\dot{\gamma}_{i}([0,c]))$ is
contained in a compact subset $K_{c}$ of $TM$. It is worth noting that
$\\{\gamma_{i}\\}$ satisfy the geodesic equation, this together with the
compactness of $K_{c}$ and the smoothness of the Lorentzian metric, the family
of maps $(\gamma_{i},\dot{\gamma}_{i}):[0,c]\rightarrow TM$ is uniformly
Lipschitz. Then by Arzela-Ascoli theorem, the family of maps
$(\gamma_{i},\dot{\gamma}_{i})$ must converge uniformly on $[0,c]$ to a
continuous map $(\gamma,\dot{\gamma}):[0,c]\rightarrow TM$. Clearly $\gamma$
is a future-directed causal curve with $\gamma(0)=x$. By the continuity of the
Lorentzian distance function $d$, the maximality of $\\{\gamma_{i}\\}$ and the
upper semicontinuity of the length functional with respect to the
$C^{0}$-topology, we have, for each $t\in[0,c]$,

$\displaystyle d(\gamma(0),\gamma(t))$
$\displaystyle=\lim\limits_{i\rightarrow\infty}d(\gamma_{i}(0),\gamma_{i}(t))$
$\displaystyle=\lim\limits_{i\rightarrow\infty}L(\gamma_{i}|_{[0,t]})$
$\displaystyle\leq L(\gamma)|_{[0,t]}$ $\displaystyle\leq
d(\gamma(0),\gamma(t))=d(x,\gamma(t)).$

Thus $\gamma$ is maximal on each segment. By the continuity of $u$ and the
$C^{0}$ convergence of $\gamma_{i}$,

$u(\gamma(t))=\lim\limits_{i\rightarrow\infty}u(\gamma_{i}(t))=\lim\limits_{i\rightarrow\infty}u(\gamma_{i}(0))+t=u(x)+t.$

This is to say $\gamma$ is the unique timelike (unit-speed) curve with
$\gamma(0)=x$, $\dot{\gamma}(0)=-V$.

Finally, define $\gamma_{x,V}:=\gamma$, then it is easily seen that
$\gamma_{x,V}$ satisfied all the properties we need. So the proof of Lemma 5.3
is complete.

###### Lemma 5.5

Let $u\in\mathcal{S}(M)$, suppose that the time orientation of $u$ is always
past-directed, then for any $x\in M$, there exists an $s\in R$, which is
sufficiently close to $u(x)$, such that $d(x,u_{s})=s-u(x)$.

###### Proof 11

Choosing a reachable unit vector $V\in\nabla^{*}u(x)$. By Lemma 5.3, there
exists a unique future-directed maximal geodesic segment
$\beta:[0,t_{0}]\rightarrow M$ with $\beta(0)=x$ and $\dot{\beta}(0)=V$ and
$u(\beta(t))-u(x)=t$. Since $s$ is sufficiently close to $u(x)$, there exists
a unique $t_{1}$ such that $u(\beta(t_{1}))=s$, $t_{1}=s-u(x)$ and

$d(x,\beta(t_{1}))=\mbox{length}(\beta|[0,t_{1}])=t_{1}=s-u(x).$

Since $\beta(t_{1})\in u_{s}$, we have $d(x,u_{s})\geq s-u(x)$. If
$s_{1}:=d(x,u_{s})>s-u(x)$, applying Fubuni’s theorem, we can find an
absolutely continuous curve (parameterized by arc-length)
$\gamma:[0,s_{2}]\rightarrow M$ with $\gamma(0)=x\in u_{u(x)}$,
$\gamma(s_{2})\in u_{s}$ such that $0\leq
s_{1}-s_{2}\leq\frac{1}{2}(s_{1}-s+u(x))$ and $u$ is differentiable almost
everywhere along $\gamma$ (with respect to the 1-dimensional Lebesgue
measure). Then $u\circ\gamma:[0,s_{2}]\rightarrow R$ is still a Lipschitz
function, thus $u\circ\gamma$ is differentiable almost everywhere (with
respect to the 1-dimensional Lebesgue measure on $M$). By the reverse Cauchy-
Schwarz inequality for causal vectors, we have

$\displaystyle s-u(x)$ $\displaystyle=u(\gamma(s_{2}))-u(\gamma(0))$
$\displaystyle=\int_{0}^{s_{2}}g(\nabla u,\dot{\gamma})|_{\gamma(t)}dt$
$\displaystyle\geq\mbox{length}(\gamma|_{[0,s_{2}]})$ $\displaystyle=s_{2}$
$\displaystyle\geq s_{1}-\frac{1}{2}(s_{1}-s+u(x))$ $\displaystyle>s-u(x).$

This contradiction proves $d(x,u_{s})=s-u(x)$.

Proof of Proposition 5.1. For any $x\in M$, by Lemma 5.1, there exists
$s\in(a,b)$ such that $x\in D^{-}(u_{s})$ and $u(x)$ has a variational
representation (2.2). Then by lemma 5.5, for another $x_{0}\in D^{-}(u_{s})$,
we have

$d(x,u_{s})=s-u(x),$ (5.7)

and

$d(x_{0},u_{s})=s-u(x_{0}).$ (5.8)

Let equality (5.8) minus equality (5.7), we obtain,

$d(x_{0},u_{s})-d(x,u_{s})=u(x)-u(x_{0}).$

This means that

$u(x)=-d(x,u_{s})+d(x_{0},u_{s})+u(x_{0}).$

Proof of Theorem 2. The first two conclusions of Theorem 2 can be obtained
from Lemma 4.2, Lemma 4.3 and Lemma 5.2. To complete the proof of Theorem 2,
we only need to show that $u(x)$ is locally semiconcave on $(M,g)$. Actually,
thanks to Proposition 5.1 and Proposition 3.1, we can construct support
function for $u(x)$ and apply Lemma 3.4 to obtain this result.

###### Remark 5.1

Our results show that when the time orientation of viscosity solution is
consistent, viscosity solution is locally semiconcave. Actually, it is not
difficult to show that when the time orientation of a function $u\in
Lip_{loc}(M)$ is consistent, $u\in\mathcal{S}(M)$ if and only if $u$ is a
locally semiconcave function satisfying (2.1) almost everywhere.

##  6 Weak KAM properties of $u$

In this section, we will show that $u$ satisfies some particular properties,
so-called weak KAM properties [10] when the time orientation of $u$ is
consistent. Similar properties for Busemann functions and regular cosmological
time functions are obtained on space-time [7]. In this section, we always
assume that the time orientation of $u$ is past-directed.

Recall that if $u^{+}$ is differentiable at $x\in M$, then by Lemma 5.3 and
subsection 5.2, there exists a forward calibrated ray
$\gamma_{x}:[0,T)\rightarrow M$ satisfies

$\gamma_{x}(0)=x,u(\gamma_{x}(t))=u(\gamma_{x}(0))+t~{}~{}~{}\mbox{for
every}~{}~{}~{}t\in[0,T)$

and

$\dot{\gamma_{x}}(0)=-\nabla u(x).$

###### Proposition 6.1

If there is a unique future-directed timelike curve
$\gamma_{x}:[0,T)\rightarrow M$, such that

$\gamma_{x}(0)=x,~{}~{}u(\gamma_{x}(t))=u(\gamma_{x}(0))+t~{}~{}~{}\mbox{for
every}~{}~{}~{}t\in[0,T).$ (6.1)

Then $u$ is differentiable at $x$.

###### Proof 12

By Proposition 5.1 and Theorem 2, for any $x\in M$ there exists some $s\in R$
such that $u(x)=-d(x,u_{s})$ is a locally semiconcave function. Furthermore,
$\nabla^{+}u(x)\neq\emptyset$ for every $x\in M$. Suppose $u$ is non-
differentiable at $x$, then $\nabla^{+}u(x)$ is not a singleton. By Lemma 3.7,
$\nabla^{*}u(x)$ is not a singleton as well. Finally, using Lemma 5.3 the
curves satisfying equation (6.1) must not be unique. This completes the proof.

###### Proposition 6.2

For any $x\in M$, let $\hat{\gamma}_{x}(t):[0,T)\rightarrow M$ be a future-
directed timelike curve with $\hat{\gamma}_{x}(0)=x$, and

$u(\hat{\gamma}(t))=u(\hat{\gamma}(0))+t~{}~{}\mbox{for
every}~{}~{}t\in[0,T).$ (6.2)

Then, $u^{+}$ is differentiable at $\hat{\gamma}_{x}(t)$ and
$\dot{\hat{\gamma}}_{x}(t)=-\nabla u(\hat{\gamma}_{x}(t))$ for any
$t\in(0,T)$.

###### Proof 13

If there is a $T>s_{0}>0$ such that $u^{+}$ is non-differentiable at
$\hat{\gamma}_{x}(s_{0})$, by Lemma 5.3, there exists a future-directed
timelike ray $\check{\gamma}_{\hat{\gamma}_{x}(s_{0})}:[s_{0},T)\rightarrow M$
with
$\check{\gamma}_{\hat{\gamma}_{x}(s_{0})}(s_{0})=\hat{\gamma}_{x}(s_{0})$, and

$u(\check{\gamma}_{\hat{\gamma}_{x}(s_{0})}(s))=u(\hat{\gamma}_{x}(s_{0}))+(s-s_{0})~{}~{}\mbox{for
every}~{}~{}s\in[s_{0},T).$ (6.3)

For the sake of convenience, let $\hat{\gamma}(t)=\hat{\gamma}_{x}(t)$, for
any $t\geq 0$, $\check{\gamma}(s)=\check{\gamma}_{\hat{\gamma}_{x}}(s)$ for
any $s\geq s_{0}$. By the above assumption and
$\dot{\hat{\gamma}}(s_{0})\neq\dot{\check{\gamma}}(s_{0})$, then the curve
$\check{\gamma}\ast\hat{\gamma}|_{[0,s_{0})}:[0,T)\rightarrow M$ has a corner
at $\hat{\gamma}(s_{0})$. Here and in the following, $\ast$ denotes the
conjunction of curves. By the transitivity of chronological relation $\ll$ [2,
p.55], for $0<s_{0}<s_{1}$, we have
$x=\hat{\gamma}(0)\ll\hat{\gamma}(s_{0})\ll\check{\gamma}(s_{1})$. Since $M$
is globally hyperbolic, there is a maximal future-directed timelike geodesic
$\sigma:[0,s_{1}]\rightarrow M$ such that $\sigma(0)=x$,
$\sigma(s_{1})=\check{\gamma}(s_{1})$. By the smoothness of geodesic and
reverse triangle inequality,

$d(\sigma(0),\sigma(s_{1}))=d(\hat{\gamma}(0),\check{\gamma}(s_{1}))>d(\check{\gamma}(s_{0}),\check{\gamma}(s_{1}))+d(\hat{\gamma}(0),\hat{\gamma}(s_{0}))=s_{1}.$
(6.4)

By Lemma 3.2 and inequality (6.4), we obtain

$\displaystyle u(\check{\gamma}(s_{1}))$ $\displaystyle\geq
u(x)+d(x,\check{\gamma}(s_{1}))$ $\displaystyle>u(x)+s_{1}.$ (6.5)

On the other hand, by (6.2) and (6.3), we have

$\displaystyle u(\check{\gamma}(s_{1}))$
$\displaystyle=u(\check{\gamma}(s_{0}))+(s_{1}-s_{0})$
$\displaystyle=u(x)+s_{0}-(s_{1}-s_{0})$ $\displaystyle=u(x)+s_{1}.$ (6.6)

This contradiction proves Proposition 6.2.

Proof of Theorem 3. The last conclusion of Theorem 3 follows from Proposition
6.2. If $u$ is differentiable at $x$, by Proposition 6.2, $u$ is
differentiable everywhere on $\gamma_{x}(t)$ for any $t\in[0,T)$. By
subsection 5.2, we have

$\nabla u(\gamma_{x}(t))=-\dot{\gamma}_{x}(t)$

for any $t\in[0,T)$. This is to say, if $u$ is differentiable at $x$, there is
only one future-directed timelike ray $\gamma_{x}:[0,T)\rightarrow M$ with
$\gamma_{x}(0)=x$, $\dot{\gamma}(0)=-\nabla u(x)$ and

$u(\gamma(0))=u(\gamma(s))-s~{}~{}\mbox{for any}~{}~{}s\in[0,T).$

This together with Proposition 6.1 establish the first conclusion of Theorem
3.

##  7 Classification of globally viscosity solutions of equation (2.1)

Up to now, we discussed the case when the time orientation of the viscosity
solution is consistent. In this section, we want to study the situation when
the time orientation of viscosity solution is non-consistent.

###### Lemma 7.1

Assume that $u\in\mathcal{S}(M)$ and $P_{u}=\emptyset$, then for any
inextendible future-directed causal curve $\gamma$ satiefying $u$ is
differentiable almost everywhere on $\gamma$, $u$ can not achieve its maximum
at $\gamma(t)$ for any $t\in\text{dom}(\gamma)$.

###### Proof 14

Assume that there exists a future-directed causal curve
$\gamma:(a,b)\rightarrow M$ and a $t_{0}\in(a,b)$ such that $u$ have attained
a local maximizer at $\gamma(t_{0})$ along $\gamma(t)$. For convenience, set
$\bar{x}=\gamma(t_{0})$. Since $u$ is viscosity solution to equation (2.1),
$\bar{x}$ must not be a differentiable point of $u$. Recall that
$P_{u}=\emptyset$, we only need to consider the following two cases: $u$ is
locally semiconvex at $\bar{x}$ or $u$ is locally semiconcave at $\bar{x}$.

If $u$ is locally semiconvex at $\bar{x}$, then for any
$V\in\nabla^{-}u(\bar{x})$, we can find a neighborhood $O$ of $\bar{x}$ and a
function $\Phi\in C^{1}(O)$ with $u(x)\geq\Phi(x)$ in $O$,
$u(\bar{x})=\Phi(\bar{x})$ and $\nabla\Phi(\bar{x})=V$. Consequently, for
$x\in O\cap\gamma(t)$,

$\Phi(x)\leq u(x)<u(\bar{x})=\Phi(\bar{x}).$

then $\Phi$ have attained a local maximizer at $\bar{x}$ along $\gamma(t)$.
Since $\Phi$ is smooth function on $M$, then
$g(\nabla\Phi(\bar{x}),\dot{\gamma}(t_{0}))=0$. This means that $V$ is
orthogonal to $\dot{\gamma}(t_{0})$. On the other hand, $\nabla^{-}u(\bar{x})$
must contain timelike vectors, This contradiction show that $u$ is not locally
semiconvex at $\bar{x}$.

If $u$ is local semiconcave at a neighborhood $O$ of $\bar{x}$, by Lemma 3.7,

$\nabla^{+}u(\bar{x})=co\nabla^{*}u(\bar{x})$

Since $\nabla^{*}u(\bar{x})$ contains both future-directed vectors and the
past-directed vectors, $co\nabla^{*}u(\bar{x})$ must contain either space-like
vectors or $0$ vector. Let $\xi$ be a space-like or $0$ vector in
$co\nabla^{*}u(\bar{x})$, then

$g(\xi,\xi)\geq 0.$ (7.1)

On the other hand, by the definition of viscosity subsolution, for any
$V\in\nabla^{+}u$,

$g(V,V)\leq-1.$ (7.2)

This contradicts equation (7.1) and thus Lemma 7.1 is proved.

Proof of Theorem 4. Recall $u\in\mathcal{S}(M)$ and $P_{u}=\emptyset$. Let
$\gamma$ be an inextendible future-directed causal curve. By Lemma 4.1 and
Lemma 7.1, we get the following three conclusions:

  * •

$u$ is monotonically increasing along $\gamma(t)$ when the time orientation of
$u$ is always past-directed.

  * •

$u$ is monotonically decreasing along $\gamma(t)$ when the time orientation of
$u$ is always future-directed.

  * •

When the time orientation of $u$ has changed at $\gamma(t_{0})$, $u$ decreases
monotonically along $\gamma(t)$ for $t<t_{0}$ and increases monotonically
along $\gamma(t)$ for $t>t_{0}$.

In the last conclusion, if $u$ is not differentiable almost everywhere on
$\gamma$, by the method in Lemma 4.1, we can find a sequence of
$\\{\gamma_{i}\\}$ such that $\gamma_{i}\rightarrow\gamma$ in $C^{0}$ topology
and $u$ is differentiable on $\gamma_{i}$ almost everywhere. Then by taking
$i\rightarrow\infty$, one can obtain the expected conclusion. Furthermore,
$\gamma(t_{0})\in C_{u}$. Conversely, If the time orientation of $u$ has
changed at a point $\bar{x}$, by the analysis above and Lemma 7.1, we know
that $u$ is locally semiconvex at a neighborhood of $\bar{x}$. Meanwhile, once
the time orientation of $u$ is past-directed, it will be always past-directed.
Thus, the time orientation of $u$ is changed at most once along any
inextendible future-directed causal curve.

Up to now, our results show that when the time orientation of viscosity
solution is consistent, the properties of the viscosity solution are
essentially not much different from the case in Riemannian manifold. But when
the time orientation of viscosity solution is non-consistent, viscosity
solutions have many peculiar properties.

According to Theorem 4, we can draw the following conclusions

###### Proposition 7.1

Let $\phi$, $\varphi\in\mathcal{S}(M)$ and the time orientation of each of
them is consistent, suppose the time orientation of $\phi$ and $\varphi$ are
identical, then $\min\\{\phi,\varphi\\}\in\mathcal{S}(M)$.

###### Proof 15

Proposition 7.1 is a straightly conclusion from Theorem 4. Firstly, by Theorem
2, both $\phi$ and $\varphi$ are locally semiconcave. By [6, Proposition
1.1.3], we know $\min\\{\phi,\varphi\\}$ is locally semiconcave on $M$. Thus
by Remark 5.1, we only need to show that $\min\\{\phi,\varphi\\}$ satisfies
the equation (2.1) at its differentiable points. Let
$M_{1}:=\\{x:\phi(x)\neq\varphi(x)\\}$ and
$M_{2}:=\\{x:\phi(x)=\varphi(x)\\}$. Denote the interior of $M_{2}$ by
$\text{int}(M_{2})$. Let $U:=M_{1}\cup\text{int}(M_{2})$, then
$\min\\{\phi,\varphi\\}$ satisfies the equation (2.1) at its differentiable
points in $U$. Obviously, $U$ is an open and dense subset, by [6, Proposition
3.3.4], $\min\\{\phi,\varphi\\}$ satisfies the equation (2.1) at any
differentiable point. So far, we know that $\min\\{\phi,\varphi\\}$ is really
a viscosity solution to equation (2.1).

At the end of this section, we also provide an example, which can illustrate
the differences of viscosity solutions in the case of Riemannian manifold and
space-time.

Example. Let $(M,g)$ be the 2-dimensional Minkowski space-time, and
$f(x,y)=\sqrt{2}|x|-|y|$. Clearly, $f$ is Lipschitz on $M$. By a simple
calculation,

$\displaystyle\nabla
f=\left\\{\begin{array}[]{ll}\displaystyle(\sqrt{2},1){}{}{}{}{}{}{}&x>0,y>0,\\\
\displaystyle(\sqrt{2},-1){}{}{}{}{}&x>0,y<0,\\\
\displaystyle(-\sqrt{2},1){}{}{}{}{}&x<0,y>0,\\\
\displaystyle(-\sqrt{2},-1){}{}{}{}{}&x<0,y<0.\end{array}\right.$

It is easy to check that $f$ is indeed a globally defined viscosity solution
to equation (2.1). Note that $f$ is neither semi-concave nor semi-convex on
any neighborhood of $(0,0)$, i.e., $(0,0)\in P_{f}$. See the Figure 2.

Figure 2: A globally defined viscosity solution to equation (2.1) on the
2-dimensional Minkowski space-time.

##  8 Characterization of $C_{u}$

In this section we will study more precisely the mathematical structure of
$C_{u}$. Let $\Gamma(M)$ be the collection of all inextendible future-directed
causal curves on $M$. In the following, for any $\gamma\in\Gamma(M)$, by the
proof of Theorem 4, we always assume that $u$ is differentiable on $\gamma$
almost everywhere. By the definition of $C_{u}$ and Theorem 4, for any $x\in
C_{u}$, one can easily get an inextendible future-directed causal curve
$\gamma(t)$ satisfying $\gamma(0)=x$ and $u$ is differentiable on $\gamma$
almost everywhere. What is more, $u$ is decreasing on $\gamma(t)$ for any
$t<0,u$ is increasing on $\gamma(t)$ for any $t>0$. The time orientation of
$u$ is future-directed at $\gamma(t)$ for any $t<0$, the time orientation of
$u$ is past-directed at $\gamma(t)$ for any $t>0$. Fix $t>0$, define

$J_{t,x,\gamma}:=J^{+}(\gamma(-t))\cap J^{-}(\gamma(t)).$
$C_{t,x,\gamma}:=J^{+}(\gamma(-t))\cap J^{-}(\gamma(t))\cap C_{u}$

and

$C_{\eta,t,x,\gamma}:=\\{\eta\in\Gamma(M)|\eta(\cdot)\subset
J_{t,x,\gamma}\cup J^{+}(x)\cup J^{-}(x)\\},$

then we have the following conclusion.

###### Lemma 8.1

For $u\in\mathcal{S}(M)$ with $P_{u}=\emptyset$, and for any $x\in C_{u}$,
$\eta\in C_{\eta,t,x,\gamma}$, there exists a $t_{0}\in R$, such that either
$\eta(t_{0})=x$ or $\eta(t_{0})\in J_{t,x,\gamma}\setminus J^{\pm}(x)$.
Furthermore, $u$ is decreasing on $\eta(t)$ for any $t<t_{0},u$ is increasing
on $\eta(t)$ for any $t>t_{0}$.

###### Proof 16

The proof of lemma is divided into the following two claims.

Claim 1: For any $\zeta\in\Gamma(M)$ with $\zeta(0)=x$, the time orientation
of $u$ is future-directed at $\zeta(-s)$, the time orientation of $u$ is past-
directed at $\zeta(s)$ for any $s>0$.

Proof of Claim 1. We choose a sequence $t_{i}\rightarrow 0^{-}$ such that $u$
is differentiable at $\gamma(t_{i})$ for each $t_{i}$. By the claim in Lemma
4.1, for any future-directed causal curve $\eta:[a,b]\rightarrow M$ with
$\eta(b)=\gamma(t_{i})$, $u$ is decreasing on $\eta(s)$ when $s$ is
sufficiently close to $b$. Note that $I^{-}(\gamma(t_{i}))\rightarrow
I^{-}(x)$ as $t_{i}\rightarrow 0^{-}$, for any future-directed causal curve
$\eta:[a,b]\rightarrow M$ with $\eta(b)=x$, $u$ is decreasing on $\eta(s)$
when $s$ is sufficiently close to $b$. Using the same analysis for $I^{+}(x)$,
it is easy to see that for any inextendible future-directed causal curve
$\zeta$ with $\zeta(0)=x$, the time orientation of $u$ is future-directed at
$\zeta(-s)$, the time orientation of $u$ is past-directed at $\zeta(s)$ for
any $s>0$. Furthermore, the time direction of $u$ only changed at $x$ along
$\zeta$.

Claim 2: For any $\eta\in C_{\eta,t,x,\gamma}$ with $\eta(s_{1})=\gamma(-t)$,
$\eta(s_{2})=\gamma(t)$ for some $s_{1}<s_{2}$, then there exists a unique
$s_{1}<s_{0}<s_{2}$ such that $u$ change its time orientation at
$\eta(s_{0})$.

Proof of Claim 2. If $\eta(s_{0})=x$, one can use Claim 1. Otherwise, assume
that there exists an inextendible future-directed causal curve $\bar{\eta}$
with $\bar{\eta}(s_{1})=\gamma(-t)$, $\bar{\eta}(s_{2})=\gamma(t)$ for some
$s_{1}<s_{2}$ and the time orientation of $u$ is consistent along
$\bar{\eta}$, by the transitivity of chronological relation $\ll$, it is easy
to see that the time orientation of $u$ is future-directed at
$\bar{\eta}(s_{2})$ and the time orientation of $u$ is past-directed at
$\bar{\eta}(s_{1})$. This is a contradiction. It is not difficult to see there
exists a $s_{0}\in R$ with $s_{1}<s_{0}<s_{2}$ such that $u$ changes its time
orientation at $\eta(s_{0})$ and $\eta(s_{0})\in J_{t,x}\setminus J^{\pm}(x)$.

For any $\eta\in C_{\eta,t,x,\gamma}$, again by transitivity of chronological
relation $\ll$, there exists a $t_{0}\in R$, such that either $\eta(t_{0})=x$
or $\eta(t_{0})\in J_{t,x}\setminus J^{\pm}(x)$. Moreover, $u$ is decreasing
on $\eta(t)$ for any $t<t_{0},u$ is increasing on $\eta(t)$ for any $t>t_{0}$.

Proof of Theorem 5. Note that $C_{u}$ consists of all $\eta_{i}(t_{i})$. By
Theorem 4, it is easy to see that $C_{u}$ is an acausal set. Indeed, if
$x,y\in C_{u}$ with $x\leq y$, we can find a future-directed causal curve
connects $x$ and $y$, by the definition of $C_{u}$, $u$ changes its time
orientation at $x$ and $y$, This contradicts Theorem 4.

Let $\\{x_{i}\\}_{i}\subset C_{u}$ and $d_{h}(x_{i},x)\rightarrow 0$, by the
definition of $C_{u}$, there exists $\\{\gamma_{i}\\}_{i}\subset\Gamma(M)$
with $\gamma_{i}(0)=x_{i}$ and $u$ is decreasing on $\gamma_{i}(t)$ for any
$t<0$, $u$ is increasing on $\gamma_{i}(t)$ for any $t>0$. Parameterized
$\\{\gamma_{i}\\}_{i}$ with respect to the $h$-arc length, then by limit curve
lemma [2, Lemma 14.2], $\\{\gamma_{i}\\}_{i}\rightarrow\gamma$ uniformly with
respect to $h$ on compact subsets of $R$, and $\gamma(0)=x$ and $u$ is
decreasing on $\gamma(t)$ for any $t<0$, $u$ is increasing on $\gamma_{i}(t)$
for any $t>0$. This means that $x\in C_{u}$, thus $C_{u}$ is a closed set.

Suppose that $\text{edge}(C_{u})\neq\emptyset$. Then for each
$x\in\text{edge}(C_{u})$, we can construct a closed subset
$C_{\eta,t,x,\gamma}$. By Lemma 8.1, any $\eta\in C_{\eta,t,x,\gamma}$ must
intersect $C_{u}$ exactly once. This contradicts the fact
$x\in\text{edge}(C_{u})$. Then $C_{u}$ is a closed acausal set and
$\text{edge}(C_{u})=\emptyset$, i.e., $C_{u}$ is a partial Cauchy surface. In
general, $C_{u}$ is not a Cauchy surface, the following proposition shows that
$C_{u}$ is homeomorphic to Cauchy surface.

###### Proposition 8.1

$C_{u}$ is homeomorphic to a Cauchy surface.

###### Proof 17

Indeed, by [3, Corollary 1.8], the time function $\tau$ is induced by a
complete flow $\Phi^{t}$ on $M$ and for any $x\in\tau_{0}$, the map
$\phi:(t,x)\rightarrow\Phi^{t}(x)$ is a diffeomorphism form $R\times\tau_{0}$
into $M$. For each event $p\in C_{u}$, we have
$\Phi^{-\tau(p)}(p)\in\tau_{0}$. Then we can define the map
$T:C_{u}\rightarrow\tau_{0}$ with

$T(\cdot):=\Phi^{-\tau(\cdot)}(\cdot)$

Clearly, $T$ is a bijection. Let $p_{n},p\in C_{u}$ with $d_{h}(p_{n},p)=0$ as
$n\rightarrow\infty$, we have

$d_{h}(T(p_{n}),T(p))=d_{h}(\Phi^{-\tau(p_{n})}(p_{n}),\Phi^{-\tau(p)}(p))\leq
d_{h}(\Phi^{-\tau(p_{n})}(p_{n}),\Phi^{-\tau(p_{n})}(p))+d_{h}(\Phi^{-\tau(p_{n})}(p).\Phi^{-\tau(p)}(p))$

This means that $C_{u}$ is homeomorphic to $T(C_{u})$. Since $C_{u}$ is a
closed set, then $T(C_{u})$ is also a closed subset in $\tau_{0}$.

For any $x\in T(C_{u})$, there exists a unique $p\in C_{u}$ such that
$T(p)=x$. Consider the following set

$\tilde{d}_{h}(p):=\\{q\in
C_{u}|d_{h}(p,q)<\frac{1}{4}\text{diam}(C_{t,p,\gamma}),~{}~{}q\notin
J^{\pm}(p)\\},$

where $\text{diam}(C_{t,p,\gamma})$ denotes the diameter of $C_{t,p,\gamma}$
with respect to $h$. $T(\tilde{d}_{h}(p))$ is an open subset in $\tau_{0}$.
Then $T(C_{u})$ is an open and closed subset in $\tau_{0}$. Since $\tau_{0}$
is connected and $T(C_{u})\neq\emptyset$, then we can finally get
$T(C_{u})=\tau_{0}$ and $T$ is a homeomorphism.

## Acknowledgments

Xiaojun Cui and Siyao Zhu are supported by the National Natural Science
Foundation of China (Grant No. 12171234). Hongguang Wu is supported by the
National Natural Science Foundation of China (Grant no. 12201073).

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