Present address: Department of Chemistry, University of California, Berkeley,
California 94720, USA

# Vibrational relaxation of hot ground state cations of naphthalene

Geert Reitsma  Serguei Patchkovskii  Judith Dura  Max-Born-Institut für
Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, D-12489
Berlin, Germany  Lorenz Drescher  Max-Born-Institut für Nichtlineare Optik und
Kurzzeitspektroskopie, Max-Born-Straße 2A, D-12489 Berlin, Germany  Jochen
Mikosch  Marc J. J. Vrakking  Oleg Kornilov  Max-Born-Institut für
Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, D-12489
Berlin, Germany<EMAIL_ADDRESS>
###### Abstract

Time-resolved XUV-IR photoion mass spectroscopy of naphthalene conducted with
broadband, as well as with wavelength-selected narrowband XUV pulses reveals a
rising probability of fragmentation characterized by a lifetime of $92\pm 4$
fs. This lifetime is independent of the XUV excitation wavelength and is the
same for all low appearance energy fragments recorded in the experiment.
Analysis of the experimental data in conjunction with a statistical multi-
state vibronic model suggests that the experimental signals track vibrational
energy redistribution on the potential energy surface of the ground state
cation. In particular, populations of the out-of-plane ring twist and the out-
of-plane wave bending modes could be responsible for opening new IR absorption
channels leading to enhanced fragmentation.

##  1 Introduction

Non-adiabatic relaxation of aromatic molecular cations has been investigated
in a number of recent XUV-pump-IR-probe experiments. Time-dependent dication
yields in naphthalene and several other polyaromatic hydrocarbon (PAH)
molecules were measured allowing to capture the relaxation of highly excited
states just below the double ionization threshold 1, 2, 3. These studies
indicate that relaxation timescales increase with the size of the molecule and
with the excitation energy, which was also supported by a time-resolved
photoelectron study of naphthalene 4. The XUV-IR spectroscopy was also applied
to benzene with improved temporal resolution by Galbraith et al 5. The
dynamics of the first six cationic states in benzene were investigated in
detail by utilizing differences in the appearance energies of specific
photoion fragmentation channels which could be activated by resonant IR
photoabsorption in cations produced by the XUV 6.

Common to all these studies is the focus on relaxation of electronically
excited states. In this work we show that dynamics on the cationic ground
state potential energy surface can be captured by XUV-IR photoion studies. We
extend our recent investigation of naphthalene cations 2, using both broadband
as well as wavelength-selected XUV pulses and monitor the production of
fragment ions as a function of XUV-IR delay. The ionization potential (IP) of
naphthalene is 8.12 eV 7. The appearance energy (AE) of the first
fragmentation channel
$\rm{C_{10}H_{8}^{+}}\longrightarrow\rm{C_{10}H_{7}^{+}}+\rm{H}$ is 15.35 eV,
while the $\rm{C_{2}H_{2}}$, $\rm{2H/H_{2}}$, $\rm{C_{4}H_{2}}$, and
$\rm{C_{3}H_{3}}$-loss channels have similar AEs of 15.36, 15.60, 15.65, and
16.09 eV, respectively 7. Multi-step fragmentation processes have
significantly higher AEs, starting at 18.5 eV 8, 7. Therefore channels with
low AEs dominate the XUV-only spectra, in accordance with previous
photodissociation studies 9. Our results demonstrate, that upon absorption of
an extra IR photon hot ground state cations also fragment mostly into these
channels. Therefore, the $\rm{H}$-, $\rm{C_{2}H_{2}}$-, $\rm{2H/H_{2}}$-,
$\rm{C_{2}H_{4}}$-, and $\rm{C_{3}H_{3}}$-loss channels can serve as the
observable for intra-molecular vibrational energy redistribution (IVR)
processes in the electronic ground state.

##  2 Experiment

The experiments reported here were carried out using two experimental
arrangements with broadband and wavelength-selected XUV pulses, respectively,
which were described in Refs. 5, 6 and Ref. 2 In the case of experiments with
broadband XUV pulses, the output of a Ti:Sa laser system (central wavelength
795 nm, pulse duration 25 fs, pulse energy 3 mJ per pulse, repetition rate 1
kHz) is spectrally broadened and temporally compressed by sending it through a
gas-filled hollow core optical fiber. The fiber is 1 m long, has a diameter of
300 $\mu$m, and is statically filled with 1.5 bar of Ne gas. The pulses are
compressed by a chirped mirror compressor and characterized using the SEA-F-
SPIDER technique 10. The beam is then split into two parts, one to generate
high harmonics and one serving as a probe IR beam. Both arms are equipped with
movable wedges, in order to optimize the compression of both pulses. XUV
pulses were generated in Xe gas and the spectrum was filtered using a 300 nm
Al filter, which blocks IR light and XUV light with photon energies below 15
eV and above 72 eV. In the second experimental arrangement the output of the
same Ti:Sa laser is used without further compression. One part is used to
generate high harmonics in Kr gas and the second part is used as IR probe
beam. A Time Delay Compensating Monochromator (TDCM) is used to select a
harmonic with a particular XUV wavelength 11, 2.

In both arrangements the beams are recombined and focused into the interaction
region of a velocity-map-imaging (VMI) spectrometer. The VMI spectrometer is
equipped with a repeller-integrated oven ensuring sufficient target density
and a well-defined interaction region. The VMI is mainly operated in a time-
of-flight (TOF) mode. However, the VMI mode is used for reference experiments,
such as independent measurements of the XUV-IR cross-correlation using Ar
sidebands, which yielded an XUV-IR cross-correlation of 8.6 fs in case of the
broadband source and 35-40 fs for the TDCM setup. The compression of the IR
probe pulse was optimized by enhancing above threshold ionization (ATI) in Ar.
The XUV pulse was optimized by enhancing the intensity of the XUV beam. The
spectral resolution of the TDCM setup was 300 meV. The IR intensity was
measured using the ponderomotive shift of photoionization lines in Ar when the
XUV and the IR pulse were overlapped ($t=0$).

(a) $\rm{C_{10}H_{8}^{2+}}$ (b) $\rm{C_{6}H_{6}^{+}}$ ($\rm{C_{4}H_{2}}$-loss)

(c) $\rm{C_{8}H_{6}^{+}}$ ($\rm{C_{2}H_{2}}$-loss) (d) $\rm{C_{10}H_{7}^{+}}$
($\rm{H}$-loss)

Figure 1: (a) Time-resolved yield of the dication $\rm{C_{10}H_{8}^{2+}}$ in
two-color XUV-IR dissociative photoionization of naphthalene
$\rm{C_{10}H_{8}}$, obtained using the broadband XUV source. The black dashed
line represents the independently determined XUV-IR cross correlation of 8.6
fs. The blue dots are the experimental data. The red line is a fit to the
dynamics of the dication, while the light blue line represents a long-lived
component due to contamination from other channels (see text). Time-resolved
yields of (b) $\rm{C_{6}H_{6}^{+}}$ ($\rm{C_{4}H_{2}}$-loss), (c)
$\rm{C_{8}H_{6}^{+}}$ ($\rm{C_{2}H_{2}}$-loss) and (d) $\rm{C_{10}H_{7}^{+}}$
($\rm{H}$-loss) fragments obtained in the same experiment. The red dots are
the experimental data with the XUV-only sinal subtracted. The blue dots are
the scaled dication signal from part (a). The green dots are the sum of these
two components and the green line is the exponential rise fit to the green
dots. See text for details of the data analysis and fitting.

##  3 Results and Discussion

Figure 1 shows the time-dependent yield of four photoions as a function of the
XUV-IR delay obtained using the broadband source. A positive delay time means
that the IR pulse arrives after the XUV pulse. Part (a) of the figure shows
the transient for the naphthalene dication ($\rm{C_{10}H_{8}^{2+}}$). As
previously reported 1, 2, 4, around $t=0$ the yield of the dication increases
and then decays exponentially. The small long-lived component after $t=0$ is
due to a slight contamination of the measurement by cation fragmentation
channels due to insufficient mass resolution. To extract the relaxation
timescale, the transient was fitted using the following model:

$F_{1}(t)=e^{-}{{}^{\frac{(t-t_{0})^{2}}{2\sigma^{2}}}}\ast\left(H(t-t_{0})\left(A(e^{\frac{-(t-t_{0})}{\tau_{1}}}-e^{\frac{-(t-t_{0})}{\tau_{2}}})+B_{1}\right)\right)+B_{2},$
(1)

where $2\sqrt{2\ln{2}}\sigma$ is the Full Width at Half Maximum (FWHM) of the
XUV-IR cross correlation and $t_{0}$ is the time at which both pulses are
temporally overlapped. The time constants $\tau_{1}$ and $\tau_{2}$ represent
a rise time and a decay time of the transient. $H(t-t_{0})$ is the Heaviside
function and $*$ designates the convolution of the molecular response with a
Gaussian profile representing the XUV-IR cross-correlation of the experiment.

The fast dynamics observed in the dication track the electron-correlation
driven population of the highly excited states of the naphthalene cation 1, 2.
These states are located just below the double ionization threshold and only
one or two IR photons are sufficient to further ionize the molecule. The decay
time of $\tau_{1}=30\pm 1$ fs extracted from the fit is in the good agreement
with the timescale that was reported previously 1. The rise time
$\tau_{2}=7\pm 1$ fs has not been reported previously. We note that this
timescale represents delayed rise of the dication signal. A similar delay has
previously been observed in benzene 5 and phenylalanine 12. Galbraith and co-
authors elaborated that this timescale may correspond either to ultrafast
electronic relaxation of the initially excited state or to dynamics in the
Frank-Condon region.

Figure 1(b-d) (red dots) displays the time-dependent yields of the three most
abundant fragments with low appearance energies: $\rm{C_{6}H_{6}^{+}}$,
$\rm{C_{8}H_{6}^{+}}$, and $\rm{C_{10}H_{7}^{+}}$, corresponding to the
emission of neutral $\rm{C_{4}H_{2}}$, $\rm{C_{2}H_{2}}$, and H, respectively.
The three transients (b-d) show qualitatively similar behavior, which is very
different from that of the dication (Figure 1(a)). Near zero delay,
corresponding to overlap of the XUV and IR pulses, the signal is depleted, but
it then quickly recovers and continues to grow. This indicates that production
of these low AE fragments is enhanced by the IR pulse, when it follows the XUV
excitation.

To analyze the observed time-dependent dynamics we first note, that the
depletion at small delays is expected to reflect the enhancement of the
dication signal. When the XUV photon populates high-lying states, the molecule
rapidly relaxes to the ground state and/or lower excited states with the
electronic excitation energy transferred to vibrational degrees of freedom 2.
This stored energy eventually leads to molecular fragmentation, mainly to the
channels with low AE. However, when the IR pulse further ionizes the cation
before the non-adiabatic relaxation occurs, a part of the population is
transferred to the dication, which has a different fragmentation pattern. This
leads to a depletion of the $\rm{C_{6}H_{6}^{+}}$, $\rm{C_{8}H_{6}^{+}}$, and
$\rm{C_{10}H_{7}^{+}}$ signals. Taking this into account we correct the
experimental data in Figure 1(b-d) (red dots) by adding the dication yield
from the part (a) scaled such as to compensate for the observed depletion. We
fit the resulting sum (green dots) to a rising component described by an
exponential function (green line):

$F_{2}(t)=e^{-}{{}^{\frac{(t-t_{0})^{2}}{2\sigma^{2}}}}\ast\left(H(t-t_{0})\left(C(1-e^{\frac{-(t-t_{0})}{\tau_{3}}})\right)\right),$
(2)

where $C$ is the asymptotic yield of the fragment and $\tau_{3}$ is the
timescale of the exponential rise. To improve the error of the fit we
constructed a multi-dimensional fit model in which we fixed $\sigma$ and
$t_{0}$ to the values obtained from the dication signal and defined the time-
constant $\tau_{3}$ as a global parameter, i.e. fitting the rise time for all
three fragments. This fitting procedure results in a time constant of
$\tau_{3}=92\pm 4$ fs.

We proceed to discuss the origin of the slow rise component corresponding to
enhanced fragmentation induced by the IR pulse. Since the transient dynamics
appears to be the same for all low AE fragments, we propose that also after
the interaction with the IR pulse the fragmentation is of statistical nature
and proceeds on nanosecond timescales from the hot ground state of the
naphthalene cations prepared by the combined XUV+IR ionization. The action of
the IR pulse may thus either be i) to increase the number of hot cations by
ionizing highly excited neutral molecules or ii) to increase the temperature
of the cations prepared by the XUV, leading to an enhancement of the
fragmentation. To distinguish between these two scenarios we employed the
second experimental arrangement (see the Experimental section) using the
wavelength-selected XUV pulses. Controlled tuning of the XUV wavelength over
the range covered by the broadband source helps to distinguish between the
involvement of neutral and cationic species, because neutral excited molecules
can only be prepared when the XUV wavelength is resonant with an existing
transition. In the wavelength-dependent experiment dynamics due to neutral
molecules would only be expected for certain choices of the XUV wavelength,
while the cationic dynamics would be expected, independent of the choice of
the photon energy.

Figure 2: Time-resolved yields of the (a,c,e,g) $\rm{C_{6}H_{6}^{+}}$
($\rm{C_{4}H_{2}}$-loss) and (b,d,f,h) $\rm{C_{8}H_{6}^{+}}$
($\rm{C_{2}H_{2}}$-loss) fragments recorded with the wavelength-selected XUV
pulses using odd harmonics 13 to 19 (20.3, 23.4, 26.5 and 29.6 eV). Red dots
correspond to the measured yields, blue dots correspond to the scaled yield of
the $\rm{C_{10}H_{8}^{2+}}$ dication measured in the same experimental run.
Green dots are the sum of the two yields with appropriate scaling coefficients
and green lines are exponential rise fits. See text for details.

Figure 2 displays time-dependent yields of the $\rm{C_{6}H_{6}^{+}}$
($\rm{C_{4}H_{2}}$-loss) and $\rm{C_{8}H_{6}^{+}}$ ($\rm{C_{2}H_{2}}$-loss)
fragments obtained using odd XUV harmonics 13-19, which correspond to central
photon energies of 20.3, 23.4, 26.5 and 29.6 eV, respectively. Note that the
H-loss channel was insufficiently mass-resolved and therefore excluded from
the analysis. Similar to the Figure 1(b-d), the red dots represent the yields
of (a) $\rm{C_{6}H_{6}^{+}}$ and (b) $\rm{C_{8}H_{6}^{+}}$ fragments, the blue
dots represent the scaled time-dependent dication yield for each harmonic, and
the green dots represnt the sum of these two components. Once more, the
exponential rise model (green line) with the fixed global rise time of
$\tau_{3}=92$ fs (taken from the first experiment) very well describes the
transients for both fragments and for all harmonics. We thus conclude that a
significant contribution from neutral species can be excluded. The $\tau_{3}$
timescale should thus be assigned to relaxation of naphthalene cations towards
molecular states with an increased IR absorption cross section. The energy
deposited by the IR pulse increases the number of cations with temperature
above the dissociation threshold and thus increases the fragmentation yields.
As possible states responsible for the absorption one can consider i) low-
lying electronically excited states which open resonant IR absorption channels
or ii) the vibrationally excited ground state.

The theoretical potential energy surfaces of the lowest six cationic states in
naphthalene and the dynamics on these surfaces have been theoretically studied
by Ghanta et al 13, 14. Within the MCTDH approach the time-dependent
populations of the $\rm{\widetilde{X}}$, $\rm{\widetilde{A}}$,
$\rm{\widetilde{B}}$, $\rm{\widetilde{C}}$, $\rm{\widetilde{D}}$, and
$\rm{\widetilde{E}}$ were calculated upon initial population of the
$\rm{\widetilde{A}}$, $\rm{\widetilde{B}}$, $\rm{\widetilde{C}}$,
$\rm{\widetilde{D}}$, or $\rm{\widetilde{E}}$ states. In most cases the
calculated decay timescales were much shorter than the timescale of $\approx
100$ fs observed in our experiments (see Figure 3 of Ref. 14). The only
exception is transfer of the population between the $\rm{\widetilde{B}}$ and
$\rm{\widetilde{A}}$ states, which proceeds on timescale of about 200 fs, i.e.
slower than observed here. However, we note that the $\rm{\widetilde{B}}$ band
is located at 10.3 eV, which is 5 eV below the lowest fragment appearance
energy. The absorption of three and more IR photons (1.56 eV) would be then
required to enhance fragmentation. Therefore we consider the assignment of the
observed timescale $\tau_{3}$ to relaxation processes in the manifold of
electronically excited states as improbable.

Consequently, we now turn to the analysis of the second proposition, namely
that the observed transients reflect dynamics on the potential energy surface
of the ground electronic state after the electronic relaxation has largerly
been completed. The photoabsorption spectrum for $\rm{C_{10}H_{8}^{+}}$
cations has been studied both experimentally 15 and theoretically 16. The most
prominent feature is the 0-0 transition from the electronic ground state
$\rm{\widetilde{X}^{2}A_{u}(D_{0})}$ to the second excited state
$\rm{{}^{2}\widetilde{B}_{3g}(D_{2})}$ at 675 nm 15. No features were
identified in cold cations at or close to 795 nm - the probe wavelength in our
experiments. The absorption spectrum may change however, when the cations are
highly vibrationally excited, as expected when the electronic energy of the
excited cations is converted to the vibrational energy of the ground state
cations during relaxation via a series of conical intersections.

Figure 3: Cut through the potential energy surfaces of the naphthalene cation
along the $\nu_{10}$ normal mode13. The adiabatic electronic states are
denoted by the corresponding diabatic state label at the neutral equilibrium
geometry ($Q=0$). Note, that adiabatic states do not cross, but can touch each
other due to high degree of degeneracy. The shaded areas indicate ranges of
$Q$ where dipole transitions are accessible for the IR photon energy used in
the present experiment (1.56 eV). The right panel indicates the Boltzmann
factors calculated for temperatures between $1000$ and $2500$ K. The vertical
energy axis is aligned with the axis of the main plot and energies are taken
relative to the minimum of the $\tilde{X}$ cationic ground state ($E_{0}=7.79$
eV). The insert in the top right corner illustrates the structural distortion
along the $\nu_{10}$ ($a_{2u}$) normal mode.  Figure 4: Cut through the
potential energy surfaces of the naphthalene cation along the $\nu_{14}$
($b_{1g}$) normal mode13. See caption of Figure 3.

To examine this assumption we constructed a model based on the vibronic
Hamiltonian of Ghanta et al 13. The Hamiltonian contains all inter-mode and
inter-state couplings for the six lowest diabatic states of the naphthalene
cation. We approximate the transition dipoles between the diabatic states by
the dipole matrix elements between Hartree-Fock molecular orbitals of neutral
naphthalene (cc-pVDZ basis set) at the optimized MP2/cc-pVDZ geometry (the
$Q=0$ point of Ghanta et al 13). In this approximation, the
$\tilde{X}^{2}A_{u}$ diabatic state is strongly coupled to the
$\tilde{B}^{2}B_{2g}$ ($d=6.6$ Debye) and $\tilde{C}^{2}B_{1g}$ ($d=4.8$
Debye) electronic states. The $\tilde{C}$ state is in turn dipole-coupled to
the $\tilde{A}^{2}B_{3u}$ state ($d=7.6$ Debye), while the $\tilde{B}$ state
has no further strong dipole transitions.

We explored the temperature dependence of the IR absorption separately for
each normal vibrational mode of the vibronic Hamiltonian, without considering
combination modes. It is assumed that for each normal mode, the relative
populations of the six adiabatic electronic states follow classical Boltzmann
statistics. At the high temperatures considered here (see below), quantum
vibrational effects are not expected to be important, and are neglected. The
probability of an IR-induced transition between all pairs of adiabatic
electronic states is determined from Fermi’s golden rule, for a random
molecular orientation.

The effective temperature range for our calculation is rationalized as
follows. The XUV photon energies used in our experiment are sufficiently high
to produce highly excited naphthalene cations both below and above the
appearance energy of the $\rm{C_{6}H_{6}^{+}}$ ($\rm{C_{4}H_{2}}$-loss) and
$\rm{C_{8}H_{6}^{+}}$ ($\rm{C_{2}H_{2}}$-loss) fragments. Given the
naphthalene ionization potential of 8.12 eV 17, population of states just
below the cation dissociation threshold of 15.35 eV leads to the internal
energy of 7.23 eV. If we assume an equipartitioning of this energy over all
vibrational degrees of freedom of the molecule, we arrive at an energy of
$7.23/(3*18-6)=0.15$ eV per mode, which corresponds to a cation temperature of
1700 K. Again assuming energy equipartitioning, the absorption of each IR
photon further increases this temperature by about 370 K.

A detailed analysis of the results shows that only excitations of 8 out of the
48 vibrational modes of naphthalene influence the IR-induced transition rate
at temperatures below $2500$ K. The dominant contribution is given by
$\nu_{10}$ (the out-of-plane ring twist) and $\nu_{14}$ (the out-of-plane wave
bend).

A cut through the adiabatic potential energy surfaces along the $\nu_{10}$
normal mode is shown in Figure 3. The two low-lying adiabatic electronic
states (correlating to $\tilde{X}^{2}A_{u}$ and $\tilde{A}^{2}B_{3u}$ at
$Q=0$) are significantly populated for temperatures between $500$ and $2500$ K
as indicated in the inset showing Boltzmann factors for the corresponding
energies. Both states are dipole-coupled to the $\tilde{B}^{2}B_{2g}$ (red),
in the region of $Q$ parameters where the diabatic $\tilde{X}^{2}A_{u}$ state
has a dominant contribution. Both states are also dipole-coupled to the
$\tilde{C}^{2}B_{1g}$ (blue) electronic state; however, only the $\tilde{B}$
state can be accessed with a single IR photon. The calculated rate of the IR
absorption due to the $\nu_{10}$ motion follows the Arrhenius’ law
[$k\propto\exp\left(-T_{a}/T\right)$]. Linear fit to the calculated rates for
$T=250$–$2250$ K in the $T^{-1}$–$\log{k}$ coordinates yield activation
temperature $T_{a}\approx 7.2\times 10^{3}$ K. The harmonic vibrational
frequency for the $\nu_{10}$ mode is $182$($399$) cm-1 [calculated
(experimental)]13. The corresponding vibrational period is $183$($84$) fs.
Both values are compatible with the experimental time constant we determine
($\tau_{3}=92\pm 4$ fs).

A similar analysis for the $\nu_{14}$ normal mode is presented in Figure 4.
Again, the thermally accessible $\tilde{X}$ and $\tilde{A}$ states can be
brought to resonance with the dipole-accessible $\tilde{B}$ state. The second
dipole-allowed transition, to the $\tilde{C}$ state, may also become
accessible at higher temperatures. The Arrhenius activation temperature is now
$T_{a}\approx 6.5\times 10^{3}$ K. The harmonic vibrational frequency for the
$\nu_{14}$ mode is $377$($191$) cm-113, giving a vibrational period of
$88$($175$) fs. Again, these values are compatible with the experimentally
determined timescale.

In the absence of a detailed quantum-dynamics study, it does not appear to be
possible to differentiate between the $\nu_{10}$ and $\nu_{14}$ out-of-plane
vibrational modes as the primary vehicle for the IR-assisted dissociation. In
reality, both are likely active.

The discussion above allows us to interpret the observed slow rise in the two-
color yield of low AE fragments as intra-molecular vibrational energy
redistribution (IVR). Upon ionization the manifold of highly-excited
electronic states of the cation is populated. These states are strongly
coupled to the ground state of the cation via a series of conical
intersections. During and after electronic relaxation the excitation energy
flows into various vibrational modes. The fastest rate at which the energy can
flow into a mode can be estimated from its eigenfrequency. As the populations
of the two relevant modes, the $\nu_{10}$ and $\nu_{14}$ modes, increases the
possibility of IR absorption opens up (see the shaded areas in Figures 3 and
4), leading to further energy deposition into the molecule and to the
enhancement of fragmentation.

##  4 Conclusions

In this work we investigated relaxation dynamics of naphthalene cations by
means of time-resolved XUV-IR photofragment ion mass spectroscopy. We employed
two complementary HHG-based XUV sources: a broadband source with laser pulses
compressed using the hollow core fiber technique, which delivered a superior
experimental time resolution of 8.6 fs; and a wavelength-selected source
employing a time-delay-compensating monochromator, which delivers inferior
time resolution of 35-40 fs, but provides a narrow and tunable XUV spectrum.
The results of measurements in both arrangements broadly agree with each other
and reveal a new exponential rise component in naphthalene cation
fragmentation channels with low appearance energies, namely, the
$\rm{C_{6}H_{6}^{+}}$ ($\rm{C_{4}H_{2}}$-loss), $\rm{C_{8}H_{6}^{+}}$
($\rm{C_{2}H_{2}}$-loss) and $\rm{C_{10}H_{7}^{+}}$ (H-loss) channels, in
addition to the previously observed dynamics in the dication
$\rm{C_{10}H_{8}^{2+}}$. In all experiments the same time constant of $92\pm
4$ fs can consistently describe the exponential rise.

We analyzed possible origins of these dynamics. The fact that the same time
scale is observed for many XUV excitation wavelengths and for all low AE
fragments suggests that highly excited neutral and cationic states can be
excluded from consideration and that the observed experimental signals are
caused by dynamics on the hot ground electronic state of naphthalene cations.
To gain further insight we analyzed the results based on the comprehensive
vibronic Hamiltonian of Ghanta et al 13, 14. We constructed a statistical
model of vibrational excitations, which revealed that activity in the out-of-
plane ring twisting mode ($\nu_{10}$) and the out-of-plane wave bending mode
($\nu_{14}$) may facilitate energy deposition in the molecule by the probe IR
pulse. The vibrational periods of these modes are in good agreement with the
observed timescale.

## Acknowledgements

We thank Franck Lépine for fruitful discussions and Ahmet Ünal for
experimental support. G.R. thanks the Netherlands Organization for Scientific
Research (NWO) for financial support (Rubicon 68-50-1410). O. K. acknowledges
support of the Deutsche Forschungsgeminschaft (KO 4920/1-1).

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# Some inequalities on Finsler manifolds with weighted Ricci curvature bounded
below 111The first author is supported by the National Natural Science
Foundation of China (11871126) and the Science Foundation of Chongqing Normal
University (17XLB022) and the second author is supported by the National
Natural Science Foundation of China (12071423)

Xinyue Cheng & Zhongmin Shen

###### Abstract

We establish some important inequalities under a lower weighted Ricci
curvature bound on Finsler manifolds. Firstly, we establish a relative volume
comparison of Bishop-Gromov type. As one of the applications, we obtain an
upper bound for volumes of the Finsler manifolds. Further, when the
S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-
Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré-
Lichnerowicz inequality by using integrated Bochner inequality, from which we
obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
Keywords: Finsler metric; Ricci curvature; weighted Ricci curvature; geodesic
ball; volume comparison; Poincaré-Lichnerowicz inequality
MR(2010) Subject Classification: 53B40, 53C60, 58C35

##  1 Introduction

Similar to Riemannian case, Finsler manifolds with Ricci curvature bounded
below are always of some amazing properties. For example, the second author
proved the Gromov-Bishop volume comparison theorem in Finsler geometry under
the condition that ${\rm Ric}\geq(n-1)\lambda$. As an application, he obtained
some precompactness and finiteness theorems for Finsler manifolds ([10]).
Later, Ohta proved a new version of the Gromov-Bishop volume comparison
theorem on Finsler manifolds with weighted Ricci curvature bounded below,
${\rm Ric}^{N}\geq K$ for some $K\in R$ and $N\in[n,\infty)$ ([5]). On the
other hand, by using Bochner-Weitzenböck type formula, Wang-Xia obtained a
sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-
Laplacian in Finsler manifolds with weighted Ricci curvature ${\rm Ric}^{N}$
bounded below ([13]). Q. Xia gave an (integrated) $p(>1)$-Bochner-Weitzenböck
formula and the $p$-Reilly type formula on Finsler manifolds. As applications,
she obtained the $p$-Poincaré inequality on an $n$-dimensional compact Finsler
manifold without boundary or with convex boundary under the assumption that
${\rm Ric}^{N}\geq K$ for $N\in[n,\infty]$ and $K\in{R}$ (see [16]).

In this paper, our main aim is to educe some important inequalities on Finsler
manifolds with weighted Ricci curvature ${\rm Ric}^{\infty}$ bounded below.
The paper actually consists of two parts: the first part is dedicated to the
geometry of Finsler manifolds and the second part is dedicated to the analysis
on Finsler manifolds.

Let $(M,F,m)$ be an $n$-dimensional Finsler manifold with a smooth measure $m$
and $dm_{BH}$ denotes the Busemann-Hausdorff volume form of $F$. Then there is
a positive smooth function $\phi$ on $M$ such that

$dm=\phi(x)dm_{BH}.$

Now, let $0\leq\rho_{o}<t_{o}\leq+\infty$ and
$\chi:(\rho_{o},t_{o})\to(0,+\infty)$ be a $C^{\infty}$ function such that
$\lim_{t\to 0^{+}}\chi(t)=0$ if $\rho_{o}=0$ and $\lim_{t\to
t_{o}^{-}}\chi(t)=0$ if $t_{o}<+\infty$. Typical examples are
$\chi(t)=\sin^{n-1}(t)$ or $\chi(t)=t^{n-1}$. Fix $p\in M$. Let
$\rho(x):=d_{F}(p,x)$ be the distance function and $d_{p}:=\sup_{x\in
M}d_{F}(p,x)$. From now on, we always assume that $d_{p}\leq t_{o}$ in our
discussions. Then, under an upper bound on $\Delta\rho$ controlled by a
function $\chi(\rho)$ of $\rho$, we can obtain the following relative volume
comparison of Bishop-Gromov type for the geodesic balls with respect to $m$.

###### Theorem 1.1

Let $(M,F,m)$ be an $n$-dimensional forward complete Finsler manifold equipped
with a smooth volume form $dm=\phi(x)dm_{BH}$ and $p\in M$. Suppose that the
distance function $\rho(x)=d_{F}(p,x)$ satisfies

$\Delta\rho(x)\leq\frac{d}{d\rho}\Big{[}\ln\chi(\rho)\Big{]}|_{\rho=\rho(x)},\
\ x\in\Omega_{p}\setminus B_{p}(\rho_{o})$ (1.1)

for a function $\chi(t)$ as above, where $\Omega_{p}$ denotes the cut-domain
at $p$. Then for $\rho_{o}<r<R<t_{o}$,

$\frac{{\rm Vol}(B_{p}(R)\setminus B_{p}(\rho_{o}))}{{\rm
Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}\leq\frac{\int_{\rho_{o}}^{R}\chi(t)dt}{\int_{\rho_{o}}^{r}\chi(t)dt}.$
(1.2)

Moreover, if $\rho_{o}=0$ and $\chi(t)=t^{n-1}\\{1+O(t)\\}$, then for
$0<r<R<t_{o}$

${\rm Vol}(B_{p}(R)\setminus
B_{p}(r))\leq\phi(p)\omega_{n-1}\int_{r}^{R}\chi(t)dt,$ (1.3)

where $B_{p}(R)$ denotes the geodesic ball of radius $R$ at the center $p$ and
$\omega_{n-1}={\rm Vol}(S^{n-1}(1))$ denotes the volume of the standard unit
sphere in $R^{n}$.

We must point out that, under certain curvature bounds on the Ricci curvature
and the S-curvature, one can find $\chi(t)$ such that (1.1) holds (e.g.
[8][14][15]). However, in order to get an absolute upper bound on the volume
of geodesic ball, one must get a function $\chi(t)$ with the property
$\chi(t)=t^{n-1}\\{1+O(t)\\}$.

###### Theorem 1.2

Let $(M,F,m)$ be an $n$-dimensional forward complete Finsler manifold equipped
with a smooth volume form $dm=\phi(x)dm_{BH}$ and $p\in M$. Suppose that there
are constants $K>0$ and $\delta\geq 0$ such that

${\rm Ric}^{\infty}\geq K>0,\ \ \ \ {\bf
S}|_{B_{p}(\frac{\pi}{2}\sqrt{\frac{n-1}{K}})}\geq-\delta,$ (1.4)

where ${\bf S}$ denotes the S-curvature of $F$. Then the volume of $M$ is
bounded by a constant depending on $n$, $K$ and $\delta$,

${\rm Vol}(M)\leq\phi(p)K^{-n/2}c\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)},$

where $c\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}$ is a constant depending only
on $n$ and $\frac{\delta}{\sqrt{K}}$.

It is known that for a complete Riemannian manifold with a volume form, if
${\rm Ric}^{\infty}$ is strictly positive, then the volume is finite. Theorem
1.2 shows that this is true for Finsler manifolds and gives an estimate of
upper bound of the volume. However, the Ricci curvature bound ${\rm
Ric}^{\infty}\geq K>0$ does not control the size of the Finsler manifold and
the volume of metric balls. Thus the lower bound on the S-curvature cannot
dropped. Further, if one assumes that $|{\bf S}|\leq\delta$ on the whole
manifold, then $M$ must be compact with ${\rm Diam}(M)\leq C(n,K,\delta)$ and
the volume is bounded too, ${\rm
Vol}(M)\leq\phi(p)K^{-n/2}C(n,\frac{\delta}{\sqrt{K}})$. See Theorem 4.1 below
for details.

Finally, let us consider an analytic problem on Finsler manifolds with
weighted Ricci curvature ${\rm Ric}^{\infty}$ bounded below. Starting from
(5.3), we can get the following inequality.

###### Theorem 1.3

(Poincaré-Lichnerowicz inequality) Suppose that $(M,F,m)$ is closed and
satisfies $m(M)=1$ and $\operatorname{Ric}^{\infty}\geq K>0$. Then, for any
$f\in H^{1}(M)$, we have the following

$\operatorname{Var}_{m}(f)\leq\frac{1}{K}\int_{M}F^{2}({\nabla}f)d{m}-\frac{2}{K}\int_{M}\left(\int_{0}^{\infty}g(t)dt\right)dm,$
(1.5)

where ${\rm Var}_{m}(f)$ denotes the variance of $f$ and

$g(t):=g_{\nabla u_{t}}\left(\nabla^{{\nabla u_{t}}}F({\nabla
u_{t}}),\nabla^{{\nabla u_{t}}}F({\nabla u_{t}})\right),$

$\left(u_{t}\right)_{t\geq 0}$ is the global solution to the heat equation
with $u_{0}=f$.

In [6], Ohta obtained Poincaré-Lichnerowicz inequality under the condition
that ${\rm Ric}^{\infty}\geq K>0$ as follows

${\rm Var}_{m}(f)\leq\frac{1}{K}\int_{M}F^{2}({\nabla}f)d{m}.$ (1.6)

Note that (1.5) is stronger than (1.6).

For a closed Finsler manifold $(M,F,m)$ with $m(M)=1$, the first eigenvalue
$\lambda_{1}$ is defined by ([2][12])

$\lambda_{1}:=\inf_{f}\frac{\int_{M}F^{2}({\nabla}f)d{m}}{\operatorname{Var}_{m}(f)}.$

Under the assumption that ${\rm Ric}^{\infty}\geq K>0$, we get from (1.6) that

$\lambda_{1}\geq K.$

However, according to Theorem 1.3, the above estimate is not sharp, namely, we
can get from (1.5) that $\lambda_{1}\geq K+\delta$, where

$\delta:=\inf_{f}\frac{2}{{\rm
Var}_{m}(f)}\int_{M}\left(\int_{0}^{\infty}g(t)dt\right)dm.$

The paper is organized as follows. In Section 2, we give some necessary
definitions and notations. Then we derive a relative volume comparison of
Bishop-Gromov type and give the proof of Theorem 1.2 based on Theorem 1.1 in
Section 3. Further, when the S-curvature is bounded on the whole manifold, we
obtain a theorem of Bonnet-Myers type on Finsler manifolds (that is, Theorem
4.1) in Section 4. Finally, the proof of a sharp Poincaré-Lichnerowicz
inequality and a sharp lower bound for the first eigenvalue on Finsler
manifolds are given in Section 5.

##  2 Preliminaries

Let $M$ be an $n$-dimensional manifold. A Finsler metric $F$ on $M$ is a non-
negative function on $TM$ such that $F$ is ${\cal C}^{\infty}$ on
$TM\backslash\\{0\\}$ and the restriction $F_{x}:=F|_{T_{x}M}$ is a Minkowski
function on $T_{x}M$ for all $x\in M$. For Finsler metric $F$ on $M$, there is
a Finsler co-metric $F^{*}$ on $M$ which is non-negative function on the
cotangent bundle $T^{*}M$ given by

$F^{*}(x,\xi):=\sup\limits_{y\in
T_{x}M\setminus\\{0\\}}\frac{\xi(y)}{F(x,y)},\ \ \forall\xi\in T^{*}_{x}M.$
(2.1)

We call $F^{*}$ the dual Finsler metric of $F$. Finsler metric $F$ and its
dual Finsler metric $F^{*}$ satisfy the following relation.

###### Lemma 2.1

(Lemma 3.1.1, [12]) Let $F$ be a Finsler metric on $M$ and $F^{*}$ its dual
Finsler metric. For any vector $y\in T_{x}M\setminus\\{0\\}$, $x\in M$, the
covector $\xi=g_{y}(y,\cdot)\in T^{*}_{x}M$ satisfies

$F(x,y)=F^{*}(x,\xi)=\frac{\xi(y)}{F(x,y)}.$ (2.2)

Conversely, for any covector $\xi\in T_{x}^{*}M\setminus\\{0\\}$, there exists
a unique vector $y\in T_{x}M\setminus\\{0\\}$ such that $\xi=g_{y}(y,\cdot)\in
T^{*}_{x}M$.

Naturally, by Lemma 2.1, we define a map ${\cal L}:TM\rightarrow T^{*}M$ by

${\cal L}(y):=\left\\{\begin{array}[]{ll}g_{y}(y,\cdot),&y\neq 0,\\\
0,&y=0.\end{array}\right.$

It follows from (2.2) that

$F(x,y)=F^{*}(x,{\cal L}(y)).$ (2.3)

Thus ${\cal L}$ is a norm-preserving transformation. We call ${\cal L}$ the
Legendre transformation on Finsler manifold $(M,F)$.

Take a basis $\\{{\bf b}_{i}\\}^{n}_{i=1}$ for $TM$ and its dual basis
$\\{\theta^{i}\\}_{i=1}^{n}$ for $T^{*}M$. Express $\xi={\cal
L}(y)=\xi_{i}\theta^{i}$. Then

$\xi_{i}=g_{ij}(x,y)y^{j},$ (2.4)

where $g_{ij}(x,y):=\frac{1}{2}\left[F^{2}\right]_{y^{i}y^{j}}(x,y)$. Let

$g^{*kl}(x,\xi):=\frac{1}{2}\left[F^{*2}\right]_{\xi_{k}\xi_{l}}(x,\xi).$
(2.5)

For any $\xi={\cal L}(y)$, differentiating $F^{2}(x,y)=F^{*2}(x,{\cal L}(y))$
with respect to $y^{i}$ yields

$\frac{1}{2}\left[F^{2}\right]_{y^{i}}(x,y)=\frac{1}{2}\left[F^{*2}\right]_{\xi_{k}}(x,\xi)g_{ik}(x,y),$

which implies

$g^{*kl}(x,\xi)\xi_{l}=\frac{1}{2}\left[F^{*2}\right]_{\xi_{k}}(x,\xi)=\frac{1}{2}g^{ik}(x,y)\left[F^{2}\right]_{y^{i}}(x,y)=y^{k}.$
(2.6)

Then, we can get (see [2][12])

$g^{*kl}(x,\xi)=g^{kl}(x,y),$ (2.7)

where $(g^{kl}(x,y))=(g_{kl}(x,y))^{-1}.$

Given a smooth function $u$ on $M$, the differential $du_{x}$ at any point
$x\in M$,

$du_{x}=\frac{{\partial}u}{{\partial}x^{i}}(x)dx^{i}$

is a linear function on $T_{x}M$. We define the gradient vector $\nabla u(x)$
of $u$ at $x\in M$ by $\nabla u(x):={\cal L}^{-1}\left(du(x)\right)\in
T_{x}M$. In a local coordinate system, by (2.6), we can express $\nabla u$ as

$\nabla
u(x)=\left\\{\begin{array}[]{ll}g^{*ij}(x,du)\frac{{\partial}u}{{\partial}x^{i}}\frac{{\partial}}{{\partial}x^{j}},&x\in
M_{u},\\\ 0,&x\in M\setminus M_{u},\end{array}\right.$ (2.8)

where $M_{u}=\\{x\in M\mid du(x)\neq 0\\}.$ Further, by Lemma 2.1, we have the
following

$du_{x}(v)=g_{\nabla u_{x}}(\nabla u_{x},v),\ \ \ \forall v\in T_{x}M$ (2.9)

and

$F(x,\nabla u_{x})=F^{*}(x,du_{x})=\frac{du_{x}(\nabla u_{x})}{F(x,\nabla
u_{x})}.$ (2.10)

We must be careful when $du(x)=0,$ because $g^{*ij}(du(x))$ is not defined and
the Legendre transform $\mathcal{L}^{-1}$ is only continuous at the zero
section. Besides, if $u\in\mathcal{C}^{l}(M),$ then ${\nabla}u$ is
$\mathcal{C}^{l-1}$ on $M_{u}$ whereas only continuous on $M\backslash M_{u}$.

Let $(M,F,{m})$ be a Finsler manifold equipped with a measure ${m}$ on $M.$
Given an open set $\Omega\subset M,$ let $H_{\rm loc}^{1}(\Omega)$ be the
space of weakly differentiable functions $u$ on $\Omega$ such that both $u$
and $F^{*}(du)$ belong to $L_{\rm loc}^{2}(\Omega)$. Define the energy
functional $\mathcal{E}_{\Omega}:H_{\rm
loc}^{1}(\Omega)\longrightarrow[0,\infty]$ by

$\mathcal{E}_{\Omega}(u):=\frac{\int_{\Omega}F^{*}(x,du)^{2}d{m}}{\int_{\Omega}u^{2}dm}=\frac{\int_{\Omega}F(\nabla
u)^{2}d{m}}{\int_{\Omega}u^{2}dm}.$ (2.11)

Further, define the Sobolev space $H^{1}(\Omega):=W^{1,2}(\Omega)$ as a space
of functions $u\in L^{2}(M)$ with $\int_{M}[F(\nabla u)]^{2}dm<\infty$, which
is a Banach space with respect to the (absolutely homogeneous) Sobolev norm

$\|u\|_{H^{1}(\Omega)}:=\left\\{\|u\|_{L^{2}(\Omega)}+\frac{1}{2}\|F(\nabla
u)\|_{L^{2}(\Omega)}+\frac{1}{2}\|F\left(\nabla(-u)\right)\|_{L^{2}(\Omega)}\right\\}^{1/2}.$

Let $H_{0}^{1}(\Omega)$ be the closure of $\mathcal{C}_{c}^{\infty}(\Omega)$
in $H^{1}(\Omega)$, where $\mathcal{C}_{c}^{\infty}(\Omega)$ denotes the set
of $\mathcal{C}^{\infty}$-functions on $\Omega$ with compact support.

Associated with the measure $m$ on $M$, we first decomposed the volume form
$dm$ of $m$ as $dm={\rm e}^{\Phi}dx^{1}dx^{2}\cdots dx^{n}$. Then the
divergence of a differentiable vector field $V$ on $M$ is defined by

${\rm div}_{m}V:=\frac{\partial V^{i}}{\partial
x^{i}}+V^{i}\frac{\partial\Phi}{\partial x^{i}},\quad
V=V^{i}\frac{\partial}{\partial x^{i}}.$ (2.12)

One can also define ${\rm div}_{m}V$ in the weak form by following divergence
formula:

$\int_{M}\phi\ {\rm div}_{m}V\ d{m}=-\int_{M}d\phi(V)\ d{m}$ (2.13)

for all $\phi\in\mathcal{C}_{c}^{\infty}(M)$.

Now we define the Finsler Laplacian $\Delta u$ of $u\in H_{\rm loc}^{1}(M)$ by

$\Delta u:={\rm div}_{m}(\nabla u).$ (2.14)

Equivalently, we can define Laplacian $\Delta u$ on the whole $M$ in the weak
sense by

$\int_{M}\phi\ {\Delta}u\ d{m}:=-\int_{M}d\phi({\nabla}u)d{m}$ (2.15)

for all $\phi\in\mathcal{C}_{c}^{\infty}(M)$.

From (2.14), Finsler Laplacian is a nonlinear elliptic differential operator
of the second order. Moreover, since the gradient vector field $\nabla u$ is
merely continuous on $M\backslash M_{u}$, even when
$u\in\mathcal{C}^{\infty}(M),$ it is necessary to introduce the Laplacian in
the weak form as (2.15).

By the definitions, we have the following for any smooth function $\varphi$ on
$M$,

${\rm div}_{m}(\varphi\nabla u)=\varphi\Delta u+d\varphi(\nabla u).$ (2.16)

Let $V=V^{i}\frac{\partial}{\partial x^{k}}$ be a nonzero measurable vector
field on $M.$ One can introduce a weighted Riemannian metric $g_{V}$ on $M$
via

$g_{V}(X,Y)=g_{ij}(x,V)X^{i}Y^{j},\quad\text{ for }X,Y\in T_{x}M.$

In particular, $g_{V}(V,V)=F^{2}(x,V)$. For $u\in H_{\rm loc}^{1}(M)$ such
that $du=0$ almost everywhere on $\\{x\in M\mid V(x)=0\\}$ (in other words,
$V\neq 0$ almost everywhere on $M_{u}$ ), we can define the linearized
gradient vector and the linearized Laplacian on the weighted Riemannian
manifold $\left(M,g_{V},{m}\right)$ by

$\nabla^{V}u:=\left\\{\begin{array}[]{ll}g^{ij}(x,V)\frac{\partial u}{\partial
x^{j}}\frac{\partial}{\partial x^{i}}&\text{on }M_{u},\\\ 0&\text{on
}M\backslash M_{u},\end{array}\quad\Delta^{V}u:={\rm
div}_{m}\left(\nabla^{V}u\right)\right..$ (2.17)

It is easy to get the following by (2.17): for any $u\in H_{\rm loc}^{1}(M)$,
we have

$\nabla^{\nabla u}u=\nabla u,\ \ \ \Delta^{\nabla u}u=\Delta u.$ (2.18)

Further, for any $f_{1},f_{2}\in H_{\mathrm{loc}}^{1}(M)$ satisfying
$df_{1}=df_{2}=0$ almost everywhere on $M\backslash M_{u}$, we have

$df_{2}\left(\nabla^{\nabla u}f_{1}\right)=df_{1}\left(\nabla^{\nabla
u}f_{2}\right).$ (2.19)

Finsler geometry is just Riemannian geometry without the quadratic restriction
([3]). The Ricci curvature in Finsler geometry is just a natural extension of
the Ricci curvature in Riemann geometry. However, we have a difficulty on the
choice of a measure in Finsler geometry, because it is impossible to choose a
unique canonical measure like the volume measure in the Riemannian setting.
Naturally, by choosing an arbitrary measure $m$ on a Finsler manifold, Ohta
modified the Ricci curvature and defined the weighted Ricci curvature in
Finsler geometry ([5]). Concretely, for an $n$-dimensional Finsler manifold
$(M,F,m)$ equipped with a smooth measure $m$ and for any $v\in
T_{x}M\backslash\\{0\\},$ let $\eta:(-\varepsilon,\varepsilon)\longrightarrow
M$ be the geodesic with $\dot{\eta}(0)=v$ and decompose the volume form $dm$
of measure ${m}$ along $\eta$ as

$dm={\rm e}^{-\psi_{\eta}}\sqrt{{\rm
det}\left(g_{ij}(\eta,\dot{\eta})\right)}\ dx^{1}dx^{2}\cdots dx^{n},$

where
$\psi_{\eta}=\psi_{\eta}\left(\eta(t),\dot{\eta}(t)\right):(-\varepsilon,\varepsilon)\longrightarrow
R$ is a $\mathcal{C}^{\infty}$-function. Then, for $N\in R\backslash\\{n\\},$
define the weighted Ricci curvature

${\rm Ric}^{N}(v):={\rm
Ric}(v)+\psi_{\eta}^{\prime\prime}(0)-\frac{\psi_{\eta}^{\prime}(0)^{2}}{N-n}.$
(2.20)

As the limits of $N\rightarrow\infty$ and $N\downarrow n,$ we define the
weighted Ricci curvatures as follows.

$\displaystyle{\rm Ric}^{\infty}(v)$ $\displaystyle:={\rm
Ric}(v)+\psi_{\eta}^{\prime\prime}(0),$ (2.21) $\displaystyle{\rm Ric}^{n}(v)$
$\displaystyle:=\left\\{\begin{array}[]{ll}{\rm
Ric}(v)+\psi_{\eta}^{\prime\prime}(0)&\text{ if }\psi_{\eta}^{\prime}(0)=0,\\\
-\infty&\text{ if }\psi_{\eta}^{\prime}(0)\neq 0.\end{array}\right.$

Note that the quantity $\psi_{\eta}^{\prime}(0)={\bf S}(x,v)$ is just the
S-curvature with respect to the measure $m$ and
$\psi_{\eta}^{\prime\prime}(0)=\dot{\bf S}(x,v)={\bf S}_{|m}(x,v)v^{m}$ in
Finsler geometry, where $``|"$ denotes the horizontal covariant derivative
with respect to the Chern connection ([4][5]). Hence we can rewrite ${\rm
Ric}^{N}(v)$ as

${\rm Ric}^{N}(v):={\rm Ric}(v)+\dot{\bf S}(x,v)-\frac{{\bf
S}(x,v)^{2}}{N-n}.$ (2.22)

From (2.22), we can get the following further description about the weighted
Ricci curvature in Finsler geometry. Let $(M,F,m)$ be an $n$-dimensional
Finsler manifold with $dm=\sigma(x)dx^{1}\cdots dx^{n}$. Let $Y$ be a
$C^{\infty}$ geodesic field on an open subset $U\subset M$ and
$\hat{g}=g_{Y}.$ Let

$dm:=e^{-\psi}{\rm Vol}_{\hat{g}},\ \ \ {\rm
Vol}_{\hat{g}}=\sqrt{{det}\left(g_{ij}\left(x,Y_{x}\right)\right)}dx^{1}\cdots
dx^{n}.$

It is easy to see that $\psi$ is given by

$\psi(x)=\ln\frac{\sqrt{\operatorname{det}\left(g_{ij}\left(x,Y_{x}\right)\right)}}{\sigma(x)}=\tau\left(x,Y_{x}\right),$

which is just the distortion along $Y_{x}$ at $x\in M$ ([4]). Let $y:=Y_{x}\in
T_{x}M$ (that is, $Y$ is a geodesic extension of $y\in T_{x}M$). Then, by the
definitions of the S-curvature and the Hessian ([11][12]), we have

$\displaystyle{\bf S}(x,y)=y[\tau(x,Y_{x})]=d\psi(y),$ $\displaystyle\dot{\bf
S}(x,y)=y[{\bf S}(x,Y)]=y[Y(\psi)]={\rm Hess}\psi(y).$

Hence,

${\rm Ric}^{N}(y)={\rm Ric}(y)+{\rm Hess}\psi(y)-\frac{d\psi(y)^{2}}{N-n}.$
(2.23)

Obviously, (2.23) is an analogue of the weighted Ricci curvature in Riemannian
geometry.

##  3 A relative volume comparison of Bishop-Gromov type and its applications

In this section, firstly we will derive a relative volume comparison of
Bishop-Gromov type and prove Theorem 1.1.

Let $(M,F,m)$ be an $n$-dimensional forward complete Finsler manifold with a
smooth measure $m$. Let $dm_{BH}$ denote the volume form of the Busemann-
Hausdorff measure $m_{BH}$ of $F$. Then the volume form $dm$ of $m$ is a
multiple of $dm_{BH}$,

$dm=\phi(x)dm_{BH}.$

These measures induce measures on hypersurfaces if a normal vector field is
chosen. Let $p\in M$. The volumes of a small geodesic sphere $S_{p}(r)$ and a
small geodesic ball $B_{p}(r)$ with respect to $m$ are given by

$\displaystyle{\rm Vol}(S_{p}(r))$ $\displaystyle=$
$\displaystyle\phi(p)\omega_{n-1}r^{n-1}\Big{\\{}1+O(r)\Big{\\}},$ (3.1)
$\displaystyle{\rm Vol}(B_{p}(r))$ $\displaystyle=$
$\displaystyle\phi(p)\frac{\omega_{n-1}}{n}r^{n}\Big{\\{}1+O(r)\Big{\\}},$
(3.2)

where $\omega_{n-1}={\rm Vol}(S^{n-1}(1))$ and $S_{p}(r)$ denotes the geodesic
sphere of radius $r$ at center $p$. See [12].

Let $\Omega_{p}$ be the cut-domain – the open domain enclosed by the cut-locus
${\rm Cut}(p)$ at $p$. The distance function $\rho(x)=d_{F}(p,x)$ is
$C^{\infty}$ on $\Omega_{p}\setminus\\{p\\}$. Then
$\tilde{S}_{p}(t):=S_{p}(t)\cap\Omega_{p}$ is a $C^{\infty}$ hypersurface in
$M$ for $t>0$. The volume form $dm$ induces a volume form $dA_{t}$ on
$\tilde{S}_{p}(t)$. On the other hand, the volume form $dm$ determines the
volume form $dm_{p}$ on $T_{p}M$. $dm_{p}$ induces the volume form $dA_{p}$ on
$S_{p}M:=\\{y\in T_{p}M~{}|\ F(p,y)=1\\}$. Define a map $\varphi_{t}:S_{p}M\to
M$ by

$\varphi_{t}(y):=\exp_{p}(ty),\ \ \ \ \ y\in S_{p}M.$ (3.3)

Set

$(\varphi_{t})^{*}dA_{t}=\eta_{t}(y)dA_{p}.$ (3.4)

The function $\eta_{t}(y)$ is positive for $0<t<i_{y}$ (the cut-value of $y$).
For a small $s>0$,

${\rm Vol}(S_{p}(s))=\int_{S_{p}(s)}dA_{s}=\int_{S_{p}M}\eta_{s}(y)dA_{p}.$

Further, the Laplace of $\rho(x)$ with respect to $F$ and $dm$ can be
expressed by

$\Delta\rho|_{\tilde{S}_{p}(t)}=\frac{d}{dt}\Big{[}\ln\eta_{t}(y)\Big{]}.$
(3.5)

This is the link between the volume of geodesic spheres $S_{p}(r)$ and the
Laplace of the distance function $\Delta\rho(x)$ (see (16.6) and Proposition
14.3.1 in [12]).

Let $0\leq\rho_{o}<t_{o}\leq\infty$ and $\chi:(\rho_{o},t_{o})\to(0,\infty)$
be a $C^{\infty}$ function such that $\lim_{t\to 0^{+}}\chi(t)=0$ if
$\rho_{o}=0$ and $\lim_{t\to t_{o}^{+}}\chi(t)=0$ if $t_{o}<\infty$. Assume
that $d_{p}:=\sup_{x\in M}d_{F}(p,x)\leq t_{o}$ and

$\Delta\rho(x)\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},\ \ \ \
\\\ \forall x\in\Omega_{p}\setminus B_{p}(\rho_{o}).$ (3.6)

By (3.5) and (3.6), we get

$\frac{d}{dt}\Big{[}\ln\eta_{t}(y)\Big{]}\leq\frac{\chi^{\prime}(t)}{\chi(t)},\
\ \ \ \ \rho_{o}<t<i_{y}.$

This implies that

$\frac{d}{dt}\Big{[}\ln\frac{\eta_{t}(y)}{\chi(t)}\Big{]}\leq 0,\ \ \ \ \
\rho_{o}<t<i_{y}.$

Thus $\eta_{t}(y)/\chi(t)$ is non-increasing monotonically. Let
$\tilde{\eta}_{t}(y):=\eta_{t}(y)$ for $\rho_{o}\leq t<i_{y}$ and
$\tilde{\eta}_{t}(y)=0$ for $t\geq i_{y}$. Then $\tilde{\eta}_{t}(y)/\chi(t)$
is non-increasing monotonically for $\rho_{o}<t<t_{o}$, that is,

$\frac{\tilde{\eta}_{t}(y)}{\chi(t)}\leq\frac{\tilde{\eta}_{s}(y)}{\chi(s)},\
\ \ \ \ \rho_{o}<s<t<t_{o}.$

Integrating it over $S_{p}M$ with respect to $dA_{p}$, we get

$\frac{{\rm Vol}(\tilde{S}_{p}(t))}{\chi(t)}\leq\frac{{\rm
Vol}(\tilde{S}_{p}(s))}{\chi(s)},\ \ \ \ \ \rho_{o}<s<t<t_{o}.$ (3.7)

Rewrite (3.7) as

${\rm Vol}(\tilde{S}_{p}(t))\chi(s)\leq{\rm Vol}(\tilde{S}_{p}(s))\chi(t),\ \
\ \ \ \ \rho_{o}<s<t<t_{o}.$ (3.8)

Let $\rho_{o}<r<R<t_{o}$. Fix $t\in[r,R]$. Integrating (3.8) with respect to
$s\in[\rho_{o},r]$, we get

${\rm Vol}(\tilde{S}_{p}(t))\int_{\rho_{o}}^{r}\chi(s)ds\leq{\rm
Vol}(B_{p}(r)\setminus B_{p}(\rho_{o}))\chi(t),\ \ \ \ \ \ r\leq t\leq R.$

Then integrating it with respect to $t\in[r,R]$, we get

${\rm Vol}(B_{p}(R)\setminus B_{p}(r))\int_{\rho_{o}}^{r}\chi(s)ds\leq{\rm
Vol}(B_{p}(r)\setminus B_{p}(\rho_{o}))\int_{r}^{R}\chi(t)dt.$ (3.9)

By adding ${\rm Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))\int_{\rho_{o}}^{r}\chi(t)dt$ to both sides of (3.9), we get

${\rm Vol}(B_{p}(R)\setminus
B_{p}(\rho_{o}))\int_{\rho_{o}}^{r}\chi(t)dt\leq{\rm Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))\int_{\rho_{o}}^{R}\chi(t)dt.$

Then, we obtain

$\frac{{\rm Vol}(B_{p}(R)\setminus B_{p}(\rho_{o}))}{{\rm
Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}\leq\frac{\int_{\rho_{0}}^{R}\chi(t)dt}{\int_{\rho_{0}}^{r}\chi(t)dt}.$

It is just (1.2).

From (3.9), we also get

$\frac{{\rm Vol}(B_{p}(R)\setminus
B_{p}(r))}{\int_{r}^{R}\chi(t)dt}\leq\frac{{\rm Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}{\int_{\rho_{o}}^{r}\chi(t)dt}.$ (3.10)

Now, assume that $\rho_{o}=0$. It follows from (3.10) that

$\frac{{\rm Vol}(B_{p}(R)\setminus
B_{p}(r))}{\int_{r}^{R}\chi(t)dt}\leq\frac{{\rm
Vol}(B_{p}(r))}{\int_{0}^{r}\chi(t)dt}.$ (3.11)

Further, assume that

$\chi(t)=t^{n-1}\\{1+O(t)\Big{\\}}.$

By (3.2), we have

$\lim_{r\to 0}\frac{{\rm
Vol}(B_{p}(r))}{\int_{0}^{r}\chi(t)dt}=\phi(p)\omega_{n-1}.$

Letting $r\to 0^{+}$ in (3.11) yields

$\frac{{\rm Vol}(B_{p}(R))}{\int_{0}^{R}\chi(t)dt}\leq\phi(p)\omega_{n-1}.$
(3.12)

Replacing $R$ by $r$ in (3.12) and applying the result to (3.11), we get

$\frac{{\rm Vol}(B_{p}(R)\setminus
B_{p}(r))}{\int_{r}^{R}\chi(t)dt}\leq\phi(p)\omega_{n-1}.$ (3.13)

This gives rise to (1.3). This completes the proof of Theorem 1.1.

In the following, we will show that we can find some $\chi(t)$ satisfying the
conditions in Theorem 1.1. Firstly, define

${s}_{c}(t):=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{c}}\sin(\sqrt{c}t)&\text{
for }c>0,\\\ t&\text{ for }c=0,\\\ \frac{1}{\sqrt{-c}}\sinh(\sqrt{-c}t)&\text{
for }c<0.\end{array}\right.$ (3.14)

Note that ${s}_{c}(t)$ is the solution to the differential equation

$f^{\prime\prime}+cf=0,\quad f(0)=0,\quad f^{\prime}(0)=1.$

Further, define

$ct_{c}(t):=\frac{{s}^{\prime}_{c}(t)}{{s}_{c}(t)}=\left\\{\begin{array}[]{ll}\sqrt{c}\cdot\operatorname{cotan}(\sqrt{c}t)&\text{
for }c>0,\\\ \frac{1}{t}&\text{ for }c=0,\\\
\sqrt{-c}\cdot\operatorname{cotanh}(\sqrt{-c}t)&\text{ for
}c<0.\end{array}\right.$ (3.15)

Let $\hat{g}:=g_{\nabla\rho}$ denote the induced Riemannian metric on
$\Omega_{p}$ and $\hat{\Delta}$ denote the Laplacian on $(M,\hat{g},{\rm
Vol}_{\hat{g}})$. Let $dm=e^{-f}{\rm Vol}_{\hat{g}}$. It is proved in [12]
that $\nabla\rho=\hat{\nabla}\rho$ is a geodesic field of $F$ and $\hat{g}$
and

${\bf S}(x,\nabla\rho_{x})=df_{x}(\hat{\nabla}\rho_{x}),\ \ {\rm
Ric}(x,\nabla\rho_{x})=\widehat{\rm Ric}(x,\hat{\nabla}\rho_{x}),\ \ {\rm
Hess}f(\nabla\rho_{x})=\widehat{\rm Hess}f(\hat{\nabla}\rho_{x}).$ (3.16)

Then

$\displaystyle{\rm Ric}^{\infty}(x,\nabla\rho_{x})$ $\displaystyle=$
$\displaystyle\widehat{\rm Ric}^{\infty}(x,\hat{\nabla}\rho_{x}):=\widehat{\rm
Ric}(x,\hat{\nabla}\rho_{x})+{\rm Hess}f(\nabla\rho_{x})$ $\displaystyle{\rm
Ric}^{N}(x,\nabla\rho_{x})$ $\displaystyle=$ $\displaystyle\widehat{\rm
Ric}^{N}(x,\hat{\nabla}\rho_{x}):=\widehat{\rm
Ric}(x,\hat{\nabla}\rho_{x})+{\rm
Hess}f(\nabla\rho_{x})-\frac{1}{N-n}[df_{x}(\hat{\nabla}\rho_{x})]^{2}.$

Further,

$\Delta\rho=\hat{\Delta}_{f}\rho:=\hat{\Delta}\rho-
df(\hat{\nabla}\rho)=\hat{\Delta}\rho-df_{x}(\hat{\nabla}\rho_{x}).$ (3.17)

Under the lower Ricci curvature bound ${\rm Ric}\geq(n-1)c$, we have
$\widehat{\rm Ric}(x,\hat{\nabla}\rho_{x})\geq(n-)c$. Then

$\hat{\Delta}\rho\leq(n-1)\operatorname{ct}_{c}(\rho).$

Under the lower S-curvature bound ${\bf S}\geq-\delta$, we have
$df_{x}(\hat{\nabla}\rho_{x})\geq-\delta$. Then

$\Delta\rho\leq(n-1)\operatorname{ct}_{c}(\rho)+\delta.$

This leads to the following

###### Lemma 3.1

(Laplacian comparison, [12][15]) Let $(M,F,dm)$ be an $n$-dimensional Finsler
manifold with Ricci curvature satisfying ${\rm Ric}\geq(n-1)c$ and ${\bf
S}\geq-\delta$. Then the following holds whenever the distance function
$\rho=\rho(x)$ is smooth

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},$ (3.18)

where $\chi(t)=s_{c}(t)^{n-1}e^{\delta t}$, $0<t<d_{p}$.

Then, by Theorem 1.1 and Lemma 3.1, we can obtain a volume comparison theorem
of Bishop-Gromov type given in [10].

Assume the lower Ricci curvature bound ${\rm Ric}^{\infty}\geq(n-1)c$, then
$\widehat{\rm Ric}^{\infty}(x,\hat{\nabla}\rho_{x})\geq(n-1)c$. Then we obtain
the following lemma by applying a result in [14].

###### Lemma 3.2

(Laplacian comparison, [14][17]) Let $(M,F,m)$ be an $n$-dimensional Finsler
manifold with weighted Ricci curvature satisfying ${\rm
Ric}^{\infty}\geq(n-1)c.$ Then the following bound on $\Delta\rho$ holds.

  1. (a)

If ${\bf S}\geq-\delta$, then the following holds on $\Omega_{p}\cap
B_{p}(r_{o})$,

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},$

where $\chi(t)=[s_{c}(t)]^{n-1}e^{\delta t}$, $0<t<r_{o}$. Here
$r_{o}=+\infty$ when $c\leq 0$ and $r_{o}=\frac{\pi}{2\sqrt{c}}$ when $c>0$.

  2. (b)

If the distortion $\tau$ satisfies that $|\tau|\leq k$, then the following
holds on $\Omega_{p}\cap B_{p}(r_{o})$,

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},$

where $\chi(t):=[s_{c}(t)]^{n+4k-1}$, $0<t<r_{o}$. Here $r_{o}:=+\infty$ when
$c\leq 0$ and $r_{o}=\pi/(4\sqrt{c})$ when $c>0$.

###### Remark 3.3

Lemma 3.2(a) is similar to Lemma 3.1. However, in the case when $c>0$, it
might be possible that $r_{o}<d_{p}$. Thus Theorem 1.1 holds only for
$0\leq\rho_{o}<r<R<r_{o}$. By Theorem 1.1 and Lemma 3.2, we can obtain a
volume comparison theorem of Bishop-Gromov type for metric balls of radius
less than $r_{o}$. See [17].

###### Lemma 3.4

(Laplacian comparison, [8]) Let $(M,F,{m})$ be forward or backward complete
and assume that ${\rm Ric}^{N}\geq(N-1)K$ for some $K\in{R}$ and
$N\in[n,\infty).$ Then, for any $p\in M,$ the distance function
$\rho(x):=d_{F}(p,x)$ satisfies the following on $\Omega_{p}$

$\Delta\rho(x)\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)}$ (3.19)

where $\chi(t):=[s_{K}(t)]^{N-1}$, $0<t<r_{o}$. Here $r_{o}=\pi/\sqrt{K}$ when
$K>0$.

By Theorem 1.1 and Lemma 3.4, we can obtain a volume comparison theorem of
Bishop-Gromov type given in [5].

Assume that

${\rm Ric}^{\infty}\geq K.$

This implies that $\widehat{\rm Ric}^{\infty}(x,\hat{\nabla}\rho_{x})\geq K$.
By Theorem 3.1 in [14], along a minimal geodesic $\gamma(t)$, $0\leq t\leq
t_{o}$ from $\gamma(0)=p$, for any $0<\rho_{o}<r<t_{o}$,

$\hat{\Delta}_{f}\rho|_{\gamma(r)}\leq\hat{\Delta}_{f}\rho|_{\gamma(\rho_{o})}-K(r-\rho_{o}).$
(3.20)

Then

$\Delta\rho(x)\leq\Delta\rho(x^{\prime})-K\\{\rho(x)-\rho_{o}\\},\ \ \ \
\rho(x)>\rho_{o},$

where $x^{\prime}\in S_{p}(\rho_{o})$ is on the minimal geodesic from $p$ to
$x$. We assume that $0<\rho_{o}<i_{p}$. Then $S_{p}(\rho_{o})$ is a smooth
hypersurface in $\Omega_{p}$. Let

$m_{o}:=\sup_{\rho(x^{\prime})=\rho_{o}}\Delta\rho(x^{\prime})<+\infty.$

Then

$\Delta\rho(x)\leq m_{o}-K\\{\rho(x)-\rho_{o}\\},\ \ \ \ \rho(x)>\rho_{o}.$

This leads to the following

###### Lemma 3.5

(Laplacian comparison, [14][17]) Let $(M,F,m)$ be an $n$-dimensional Finsler
manifold with weighted Ricci curvature satisfying ${\rm Ric}^{\infty}\geq K$.
Then, for any $p\in M$, the following holds whenever the distance function
$\rho(x):=d_{F}(p,x)$ is smooth and $\rho(x)>\rho_{o}$

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},$ (3.21)

where $\chi(t)=e^{m_{o}(t-\rho_{o})-\frac{1}{2}K(t-\rho_{o})^{2}}$,
$\rho_{o}<t<\infty$.

Proof of Theorem 1.2: We assume that

${\rm Ric}^{\infty}\geq K>0,\ \ \ \ {\bf
S}|_{B_{p}(\frac{\pi}{2}\sqrt{\frac{n-1}{K}})}\geq-\delta.$

Then ${\rm Ric}^{\infty}\geq(n-1)c$ with $c=K/(n-1)$.

Let

$r:=\frac{\pi}{2}\sqrt{\frac{n-1}{K}}.$

By Lemma 3.2 (a), the following holds on $B_{p}(r)$,

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi_{o}(t)\Big{]}|_{t=\rho(x)},\ \ \ \
\rho(x)<r$ (3.22)

where $\chi_{o}(t):=[s_{c}(t)]^{n-1}e^{\delta t}$ where $c=\frac{K}{n-1}$.
Note that $\chi_{o}(t)=t^{n-1}\\{1+O(t)\\}$. By (3.12),

$\displaystyle{\rm Vol}(B_{p}(r))$ $\displaystyle\leq$
$\displaystyle\phi(p)\omega_{n-1}\int_{0}^{r}[s_{c}(t)]^{n-1}e^{\delta t}dt$
(3.23) $\displaystyle=$
$\displaystyle\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\int_{0}^{\pi/2}\sin^{n-1}(s)e^{\delta\sqrt{\frac{n-1}{K}}s}ds,$

Suppose that

$d_{p}\leq r.$

Then

${\rm Vol}(M)={\rm
Vol}(B_{p}(r))\leq\phi(p)K^{-n/2}c\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}.$

We now assume that

$d_{p}>r=\frac{\pi}{2}\sqrt{\frac{n-1}{K}}.$

In this case, the manifold might noncompact, i.e. $d_{p}=+\infty$. Let
$\rho_{o}:=r/2$. By (3.22), we have

$m_{o}=\sup_{\rho(x)=\rho_{o}}\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi_{o}(t)\Big{]}_{t=\rho_{o}}=\sqrt{(n-1)K}+\delta.$

Let

$\chi(t):=e^{(\sqrt{(n-1)K}+\delta)(t-\rho_{o})-\frac{K}{2}(t-\rho_{o})^{2}}.$
(3.24)

Then by Lemma 3.5, the following holds on $M\setminus B_{p}(\rho_{o})$,

$\Delta\rho\leq\frac{d}{dt}\Big{[}\ln\chi(t)\Big{]}|_{t=\rho(x)},\ \ \ \
\rho(x)>\rho_{o}.$ (3.25)

It follows from Theorem 1.1 that for $\rho_{0}<r<R\leq d_{p}$,

${\rm Vol}(B_{p}(R))\leq{\rm Vol}(B_{p}(\rho_{o}))+\frac{{\rm
Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}{\int_{\rho_{o}}^{r}\chi(t)dt}{\int_{\rho_{o}}^{R}\chi(t)dt}.$
(3.26)

It follows from (3.26) that

${\rm Vol}(M)\leq{\rm Vol}(B_{p}(\rho_{o}))+\frac{{\rm Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}{\int_{\rho_{o}}^{r}\chi(t)dt}{\int_{\rho_{o}}^{\infty}\chi(t)dt}.$
(3.27)

We now estimate each term in (3.27). Similar to (3.23), we have

$\displaystyle{\rm Vol}(B_{p}(\rho_{o}))$ $\displaystyle\leq$
$\displaystyle\phi(p)\omega_{n-1}\int_{0}^{\rho_{o}}[s_{c}(t)]^{n-1}e^{\delta
t}dt$ (3.28) $\displaystyle=$
$\displaystyle\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\int_{0}^{\pi/4}\sin^{n-1}(s)e^{\delta\frac{\sqrt{n-1}}{\sqrt{K}}s}ds.$

Further, by (1.3), we have

$\displaystyle{\rm Vol}(B_{p}(r)\setminus B_{p}(\rho_{o}))$
$\displaystyle\leq$
$\displaystyle\phi(p)\omega_{n-1}\int_{\rho_{0}}^{r}[s_{c}(t)]^{n-1}e^{\delta
t}dt$ (3.29) $\displaystyle=$
$\displaystyle\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\int_{\pi/4}^{\pi/2}\sin^{n-1}(s)e^{\delta\frac{\sqrt{n-1}}{\sqrt{K}}s}ds.$

By substitution, we have

$\int_{\rho_{o}}^{r}\chi(t)dt=\frac{1}{\sqrt{(n-1)K}+\delta}\int_{0}^{h}e^{s-\frac{1}{2}K_{o}s^{2}}ds,$
$\int_{\rho_{o}}^{\infty}\chi(t)t=\frac{1}{\sqrt{(n-1)K}+\delta}\int_{0}^{\infty}e^{s-\frac{1}{2}K_{o}s^{2}}ds,$

where $h=\frac{\pi}{4}\sqrt{n-1}\left(\sqrt{(n-1)}+\delta/\sqrt{K}\right)$ and
$K_{o}:=1/(\sqrt{n-1}+\delta/\sqrt{K})^{2}$. Then by (3.27), we get

$\displaystyle{\rm Vol}(M)$ $\displaystyle\leq$
$\displaystyle\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\Big{\\{}\int_{0}^{\pi/4}\sin^{n-1}(s)e^{\delta{\frac{\sqrt{n-1}}{\sqrt{K}}}s}ds$
$\displaystyle+\frac{\int_{\pi/4}^{\pi/2}\sin^{n-1}(s)e^{\delta{\frac{\sqrt{n-1}}{\sqrt{K}}}s}ds}{\int_{0}^{h}e^{s-\frac{1}{2}K_{o}s^{2}}ds}\int_{0}^{\infty}e^{s-\frac{1}{2}K_{o}s^{2}}ds\Big{\\}}$
$\displaystyle=:$
$\displaystyle\phi(p)K^{-n/2}c\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}.$

This completes the proof of Theorem 1.2.

##  4 A Theorem of Bonnet-Myers type

Let $(M,F,m)$ be an $n$-dimensional forward complete Finsler manifold with a
smooth volume form $dm=\phi(x)dm_{BH}$. The classical Bonnet-Myers Theorem in
Finsler geometry says that, if ${\rm Ric}\geq(n-1)K>0,$ then $M$ is in fact
compact and ${\rm Diam}(M)\leq\frac{\pi}{\sqrt{K}}$ ([1], Theorem 7.7.1). If
the weighted Ricci curvature satisfies ${\rm
Ric}_{N}\geq(N-1)K>0,N\in[n,\infty),$ then ${\rm
Diam}(M)\leq\frac{\pi}{\sqrt{K}}$ ([5]).

In this section, we will consider the case when the S-curvature is bounded on
the whole manifold and give a theorem of Bonnet-Myers type.

###### Theorem 4.1

Let $(M,F,m)$ be an $n$-dimensional forward complete Finsler manifold with a
smooth volume form $dm=\phi(x)dm_{BH}$ and $p\in M$. Suppose that

${\rm Ric}^{\infty}\geq K>0,\ \ \ \ \ |{\bf S}|\leq\delta.$

Then the diameter and the volume is bounded.

${\rm
Diam}(M)\leq\frac{\pi}{\sqrt{K}}\Big{(}\frac{\delta}{\sqrt{K}}+\sqrt{\frac{\delta^{2}}{K}+n-1}\Big{)},$
(4.1) ${\rm
Vol}(M)\leq\phi(p)K^{-n/2}C\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)},$ (4.2)

where $C\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}$ is a constant depending only
on $n$ and $\frac{\delta}{\sqrt{K}}$.

Proof: Firstly let $N$ be an arbitrary number with

$N>n+\frac{\delta^{2}}{K}.$

Let

$H:=\frac{K(N-n)-\delta^{2}}{(N-1)(N-n)}>0.$

Then

${\rm Ric}^{N}={\rm Ric}^{\infty}-\frac{1}{N-n}{\bf S}^{2}\geq
K-\frac{1}{N-n}\delta^{2}=(N-1)H>0.$ (4.3)

By a theorem of Bonnet-Myers type proved by S. Ohta in [5],

${\rm Diam}(M)\leq\frac{\pi}{\sqrt{H}}.$

Then

${\rm Diam}(M)\leq
f_{K,\delta}(N):=\pi\sqrt{\frac{(N-1)(N-n)}{K(N-n)-\delta^{2}}}.$

The minimum value of $f_{K,\delta}(N)$ over $n+\delta^{2}/K<N<+\infty$ is

$\min f_{K,\delta}=\frac{\pi\gamma}{\sqrt{K}},$

where

$\gamma:=\frac{\delta}{\sqrt{K}}+\sqrt{\frac{\delta^{2}}{K}+n-1}.$

The minimum value is achieved when

$N=n+\frac{\delta\gamma}{\sqrt{K}}.$

This gives (4.1). In this case,

$H=\frac{K}{\gamma^{2}}.$

It is tricky to estimate the volume of the manifolds since there is no upper
bound on ${\rm Vol}(B(p,R))$ only under the weighted Ricci curvature bound
${\rm Ric}^{N}\geq(N-1)H$. We shall use (3.26) to get an upper bound on ${\rm
Vol}(M)={\rm Vol}(B_{p}(R))$ with $R=\pi/\sqrt{H}$.

Let

$r:=\frac{\pi}{2}\sqrt{\frac{n-1}{K}}.$

Firstly suppose that

$d_{p}\leq r.$

Then by (3.23), we get

${\rm Vol}(M)={\rm
Vol}(B_{p}(r))\leq\phi(p)K^{-n/2}C\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)},$

where
$C\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}:=(n-1)^{n/2}\int_{0}^{\pi/2}\sin^{n-1}(s)e^{\delta\sqrt{\frac{n-1}{K}}s}ds$.

We now assume that $r<d_{p}\leq\pi\gamma/\sqrt{K}$. Let $\rho_{o}=r/2$. By
Lemma 3.2 (a), we have got two bounds (3.28) and (3.29) on ${\rm
Vol}(B_{p}(\rho_{o}))$ and ${\rm Vol}(B_{p}(r)\setminus B_{p}(\rho_{o}))$,
respectively. On the other hand, from (4.3) and by Lemma 3.4 and Theorem 1.1,
we can get an upper bound on ${\rm Vol}(B_{p}(R))$ for
$\chi(t):=[s_{H}(t)]^{N-1}$ and $R=\pi/\sqrt{H}$ by (3.26). Then we obtain an
upper bound on ${\rm Vol}(M)={\rm Vol}(B_{p}(R))$ as following

$\displaystyle{\rm Vol}(M)$ $\displaystyle\leq$ $\displaystyle{\rm
Vol}(B_{p}(\rho_{o}))+\frac{{\rm Vol}(B_{p}(r)\setminus
B_{p}(\rho_{o}))}{\int_{\rho_{o}}^{r}\chi(t)dt}\int_{\rho_{o}}^{R}\chi(t)dt$
$\displaystyle\leq$
$\displaystyle\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\Big{\\{}\int_{0}^{\pi/4}\sin^{n-1}(s)e^{\delta\sqrt{\frac{n-1}{K}}s}ds$
$\displaystyle+\frac{\int_{\pi/4}^{\pi/2}\sin^{n-1}(s)e^{\delta\sqrt{\frac{n-1}{K}}s}ds}{\int_{h}^{2h}\sin^{N-1}(s)ds}\int_{h}^{\pi}\sin^{N-1}(s)ds\Big{\\}},$

where

$h:=\frac{\pi}{4}\frac{\sqrt{n-1}}{\gamma}.$

Thus there is a constant $C(n,\frac{\delta}{\sqrt{K}})$ depending only on $n$
and $\frac{\delta}{\sqrt{K}}$ such that

${\rm Vol}(M)\leq\phi(p)K^{-n/2}C\Big{(}n,\frac{\delta}{\sqrt{K}}\Big{)}.$

Note that when $\delta=0$, $\gamma=\sqrt{n-1}$, $N=n$ and $H=K/(n-1)$. Then

${\rm
Vol}(M)\leq\phi(p)\omega_{n-1}\Big{(}\frac{n-1}{K}\Big{)}^{n/2}\int_{0}^{\pi}\sin^{n-1}(s)ds.$

Clearly, this is a sharp bound. This completes the proof of Theorem 4.1.
Q.E.D.

##  5 Poincaré-Lichnerowicz inequality

Let $(M,F,m)$ be a Finsler manifold equipped with a measure $m$ on $M$. In
order to prove Theorem 1.3, we need some necessary lemmas. Firstly, we have
the following lemma.

###### Lemma 5.1

([7]) If $(M,F,m)$ satisfies that $\Lambda_{F}<\infty,\ {m}(M)<\infty$ and is
complete, then the constant function $1$ belongs to $H_{0}^{1}(M)$, where
$\Lambda_{F}$ denotes the reversibility constant of $F$.

In our discussions, $(M,F,m)$ always satisfies that $\Lambda_{F}<\infty$.
Further, we have the following result.

###### Lemma 5.2

For any $f,u\in H_{\mathrm{loc}}^{1}(M)$ satisfying $df=0$ almost everywhere
on $M\backslash M_{u}$, we have

$\Delta^{\nabla u}f^{2}=2f\Delta^{\nabla u}f+2g_{\nabla u}\left(\nabla^{\nabla
u}f~{},\nabla^{\nabla u}f\right),$ (5.1)

equivalently,

$f\Delta^{\nabla u}f=\frac{1}{2}\Delta^{\nabla u}f^{2}-g_{\nabla
u}\left(\nabla^{\nabla u}f~{},\nabla^{\nabla u}f\right).$ (5.2)

Proof. By (2.17), we have

$\nabla^{\nabla u}f^{2}=2f\nabla^{\nabla u}f.$

Further, by (2.16) and (2.17), we have

$\displaystyle\Delta^{\nabla u}f^{2}$ $\displaystyle=$ $\displaystyle{\rm
div}_{m}\left(\nabla^{\nabla u}f^{2}\right)={\rm div}_{m}(2f\nabla^{\nabla
u}f)$ $\displaystyle=$ $\displaystyle 2f\Delta^{\nabla u}f+2df(\nabla^{\nabla
u}f).$

It is easy to see that

$df(\nabla^{\nabla u}f)=g_{\nabla u}\left(\nabla^{\nabla u}f~{},\nabla^{\nabla
u}f\right).$

Thus we get (5.1). Q.E.D.

In the following, we say that ${\rm Ric}^{\infty}\geq K$ for some $K\in{R}$ if
${\rm Ric}^{\infty}(v)\geq KF^{2}(x,v)$ for all $x\in M$ and $v\in T_{x}M$.
The following integrated form of the improved Bochner inequality is important
for our discussion in this section.

###### Lemma 5.3

(Integrated form, [7]) Assume that ${\rm Ric}^{\infty}\geq K$ for some
$K\in{R}.$ Then, given $u\in H_{\mathrm{loc}}^{2}(M)\cap{\cal C}^{1}(M)$ such
that $\Delta u\in H_{\mathrm{loc}}^{1}(M),$ we have

$\displaystyle-\int_{M}d\phi\left(\nabla^{\nabla
u}\left[\frac{F^{2}({\nabla}u)}{2}\right]\right)d{m}$
$\displaystyle\geq\int_{M}\phi\left\\{d({\Delta}u)({\nabla}u)+KF^{2}({\nabla}u)+d[F({\nabla}u)]\left(\nabla^{\nabla
u}[F({\nabla}u)]\right)\right\\}d{m}$ (5.3)

for all nonnegative functions $\phi\in H_{0}^{1}(M)\cap L^{\infty}(M)$.

The following lemma is derived from Lemma 5.3 and is necessary for the proof
of Theorem 1.3.

###### Lemma 5.4

Assume that $(M,F,{m})$ is compact and satisfies ${\rm Ric}^{\infty}\geq K>0$.
Then , for any $u\in H^{2}(M)\cap\mathcal{C}^{1}(M)$ such that $\Delta u\in
H^{1}(M)$, we have

$\int_{M}F^{2}({\nabla}u)d{m}\leq\frac{1}{K}\left\\{\int_{M}({\Delta}u)^{2}d{m}-\int_{M}g_{\nabla
u}\left(\nabla^{\nabla u}F(\nabla u),\nabla^{\nabla u}F(\nabla
u)\right)dm\right\\}.$ (5.4)

Proof. From (5.3) and by taking test function $\phi\equiv 1$ according to
Lemma 5.1, we have the following

$\int_{M}\left\\{d({\Delta}u)({\nabla}u)+KF^{2}({\nabla}u)+d[F({\nabla}u)]\left(\nabla^{\nabla
u}[F({\nabla}u)]\right)\right\\}d{m}\leq 0.$ (5.5)

Noticing that

$\displaystyle\int_{M}d({\Delta}u)({\nabla}u)dm=-\int_{M}(\Delta
u)^{2}dm=-\|\Delta u\|^{2}_{L^{2}},$
$\displaystyle\int_{M}d[F({\nabla}u)]\left(\nabla^{\nabla
u}[F({\nabla}u)]\right)=-\int_{M}F({\nabla}u)\Delta^{\nabla
u}\left[F({\nabla}u)\right]dm,$

(5.5) becomes

$K\int_{M}F^{2}({\nabla}u)dm\leq\int_{M}(\Delta
u)^{2}dm+\int_{M}F({\nabla}u)\Delta^{\nabla u}\left[F({\nabla}u)\right]dm.$

Further, by Lemma 5.2, we have

$\displaystyle K\int_{M}F^{2}({\nabla}u)dm$ $\displaystyle\leq$
$\displaystyle\int_{M}(\Delta u)^{2}dm+\int_{M}\Delta^{\nabla
u}\left[\frac{F^{2}({\nabla}u)}{2}\right]dm$ $\displaystyle-\int_{M}g_{\nabla
u}\left(\nabla^{\nabla u}F(\nabla u),\nabla^{\nabla u}F(\nabla u)\right)dm$
$\displaystyle=$ $\displaystyle\int_{M}(\Delta u)^{2}dm-\int_{M}g_{\nabla
u}\left(\nabla^{\nabla u}F(\nabla u),\nabla^{\nabla u}F(\nabla u)\right)dm.$

From this, we obtain (5.4). Q.E.D.

In the following, normalizing $m$ as $m(M)=1,$ we define the variance of $f\in
L^{2}(M)$ as

$\operatorname{Var}_{m}(f):=\int_{M}\left(f-\int_{M}fdm\right)^{2}dm=\int_{M}f^{2}dm-\left(\int_{M}fdm\right)^{2}.$
(5.6)

Further, for any $f\in H^{1}(M)$, let $(u_{t})_{t\geq 0}$ be the global
solution to the heat equation ${\partial}_{t}u_{t}=\Delta u_{t}$ with
$u_{0}=f$. By the properties of the linearized heat semigroups, we have the
mass conservation

$\int_{M}u_{t}dm=\int_{M}fdm$ (5.7)

and ergodicity

$\lim\limits_{t\rightarrow\infty}u_{t}=\int_{M}fd{m}\ \ \ $ (5.8)

in $L^{2}(M)$ (see [6][7]).

Now we are in the position to prove Theorem 1.3.

Proof of Theorem 1.3. Let $(u_{t})_{t\geq 0}$ be the global solution to the
heat equation with $u_{0}=f$. Put
$\Phi(t):=\|u_{t}\|^{2}_{L^{2}}=\int_{M}u^{2}_{t}dm$. Then the ergodicity
(5.8) implies that

$\lim\limits_{t\rightarrow\infty}\Phi(t)=\lim\limits_{t\rightarrow\infty}\int_{M}u^{2}_{t}dm=\left(\int_{M}fdm\right)^{2}$

and

$\displaystyle{\rm Var}_{m}(f)$ $\displaystyle=$
$\displaystyle\int_{M}f^{2}dm-\left(\int_{M}fdm\right)^{2}$ $\displaystyle=$
$\displaystyle\Phi(0)-\lim\limits_{t\rightarrow\infty}\Phi(t)=-\int_{0}^{\infty}\Phi^{\prime}(t)dt.$

On the other hand, by the definition of $\Phi(t)$ , we know that

$\displaystyle\Phi^{\prime}(t)$ $\displaystyle=$ $\displaystyle
2\int_{M}u_{t}\Delta u_{t}d{m}=-2\int_{M}du_{t}(\nabla u_{t})dm$ (5.9)
$\displaystyle=$
$\displaystyle-2\int_{M}F^{2}\left(\boldsymbol{\nabla}u_{t}\right)d{m}.$

By (4.2) in [9], we have the following

$\frac{\partial}{\partial
t}\left[F^{2}\left({\nabla}u_{t}\right)\right]=2d\left({\Delta}u_{t}\right)\left({\nabla}u_{t}\right)$
(5.10)

for all $t>0$. Hence, from (5.9) and by (5.10), we have

$\Phi^{\prime\prime}(t)=-4\int_{M}d\left(\Delta u_{t}\right)\left(\nabla
u_{t}\right)d{m}=4\int_{M}\left(\Delta u_{t}\right)^{2}dm=4\left\|\Delta
u_{t}\right\|_{L^{2}}^{2}$

for all $t>0$. Thus, from (5.4) for $u_{t}$ and by (5.9), we get the following

$-\frac{1}{2}\Phi^{\prime}(t)\leq\frac{1}{4K}\Phi^{\prime\prime}(t)-\frac{1}{K}\int_{M}g_{\nabla
u_{t}}\left(\nabla^{{\nabla u_{t}}}F({\nabla u_{t}}),\nabla^{{\nabla
u_{t}}}F({\nabla u_{t}})\right)dm,$

that is,

$-\Phi^{\prime}(t)\leq\frac{1}{2K}\Phi^{\prime\prime}(t)-\frac{2}{K}\int_{M}g_{\nabla
u_{t}}\left(\nabla^{{\nabla u_{t}}}F({\nabla u_{t}}),\nabla^{{\nabla
u_{t}}}F({\nabla u_{t}})\right)dm.$

Then,

$\displaystyle{\rm Var}_{m}(f)=-\int_{0}^{\infty}\Phi^{\prime}(t)dt$
$\displaystyle\leq\frac{1}{2K}\left(\lim\limits_{t\rightarrow\infty}\Phi^{\prime}(t)-\Phi^{\prime}(0)\right)-\frac{2}{K}\int_{M}\left(\int_{0}^{\infty}g(t)dt\right)dm,$

where

$g(t):=g_{\nabla u_{t}}\left(\nabla^{{\nabla u_{t}}}F({\nabla
u_{t}}),\nabla^{{\nabla u_{t}}}F({\nabla u_{t}})\right).$

From (5.9), $\Phi^{\prime}(0)=-2\int_{M}F^{2}(\nabla f)dm$. Further, it is not
difficult to prove that $\lim\limits_{t\rightarrow\infty}{\cal E}(u_{t})=0$
(e.g. see [6][8]), and then,
$\lim\limits_{t\rightarrow\infty}\Phi^{\prime}(t)=0$. Thus, we get the
following

${\rm Var}_{m}(f)\leq\frac{1}{K}\int_{M}F^{2}(\nabla
f)dm-\frac{2}{K}\int_{M}\left(\int_{0}^{\infty}g(t)dt\right)dm.$ (5.11)

This completes the proof of Theorem 1.3. Q.E.D.

As an application of Theorem 1.3, the following result is natural.

###### Corollary 5.5

Suppose that $(M,F,m)$ is closed and satisfies $m(M)=1$ and
$\operatorname{Ric}^{\infty}\geq K>0$. Let $\lambda_{1}$ be the first
(nonzero) eigenvalue of the Finsler Laplacian. Then

$\lambda_{1}\geq K+\delta,$ (5.12)

where

$\delta:=\inf_{f}\frac{2}{{\rm
Var}_{m}(f)}\int_{M}\left(\int_{0}^{\infty}g(t)dt\right)dm.$

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Xinyue Cheng
School of Mathematical Sciences
Chongqing Normal University
Chongqing 401331, P. R. of China
E-mail<EMAIL_ADDRESS>
Zhongmin Shen
Department of Mathematical Sciences
Indiana University-Purdue University Indianapolis
IN 46202-3216, USA
E-mail<EMAIL_ADDRESS>