arXiv subjects

Xinyue Cheng

Publications and source records attributed to Xinyue Cheng.

17 recordsLinked to original sources

arXiv2026

The Bonnet-Myers theorem on Finsler manifolds with integral weighted Ricci curvature bounds

In this paper, we derive some new relative volume comparison theorems and Bishop-Gromov volume comparisons on Finsler metric measure manifolds, all of which are controlled by the integral weighted Ricci curvature. In particular, we establish a Bishop-Gromov volume comparison theorem for nonconcentric balls. Based on these, we prove a theorem of Bonnet-Myers type on Finsler metric measure manifolds with integral weighted Ricci curvature bounds.

math.DG↗

arXiv2025

Concentration and relevant properties of Finsler metric measure manifolds

In this paper, we study systematically the concentration properties of Finsler metric measure manifolds. We establish the relationships between the concentration properties and the observable diameter, isoperimetric inequalities and the first eigenvalue. In particular, as an application, we derive a Cheng type upper bound estimate for the first closed eigenvalue via the concentration property. The researches in this paper enrich and extend the concentration theory in Finsler geometry, even in irreversible metric measure spaces.

math.DG↗

arXiv2025

Isoperimetric inequality on Finsler metric measure manifolds with non-negative weighted Ricci curvature

In this paper, we define the volume entropy and the second Cheeger constant and prove a sharp isoperimetric inequality involving the volume entropy on Finsler metric measure manifolds with non-negative weighted Ricci curvature ${\rm Ric}_{\infty}$. As an application, we prove a Cheeger-Buser type inequality for the first eigenvalue of Finsler Laplacian by using the volume entropy and the second Cheeger constant.

math.DG↗

arXiv2025

On Finsler metric measure manifolds with integral weighted Ricci curvature bounds

In this paper, we study deeply geometric and topological properties of Finsler metric measure manifolds with the integral weighted Ricci curvature bounds. We first establish Laplacian comparison theorem, Bishop-Gromov type volume comparison theorem and relative volume comparison theorem on such Finsler manifolds. Then we obtain a volume growth estimate and Gromov pre-compactness under the integral weighted Ricci curvature bounds. Furthermore, we prove the local Dirichlet isoperimetric constant estimate on Finsler metric measure manifolds with integral weighted Ricci curvature bounds. As applications of the Dirichlet isoperimetric constant estimates, we get first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.

math.DG↗

arXiv2025

$(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds

In this paper, we carry out in-depth research centering around the $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds under the condition that ${\rm Ric}_{\infty} \geq -K$ for some $K \geq 0$. We first obtain a global $p$-Poincar\'{e} inequality on such Finsler manifolds. Based on this, we can derive a $(p, q)$-Sobolev inequality. Furthermore, we establish a global optimal $(p, q)$-Sobolev inequality with a sharp Sobolev constant. Finally, as an application of the $p$-Poincar\'{e} inequality, we prove a Nash inequality.

math.DG↗

arXiv2025

Elliptic Harnack inequality and its applications on Finsler metric measure spaces

In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the distortion $\tau$ is of linear growth, $|\tau|\leq ar+b$, where $a,b$ are some non-negative constants, $r=d(x_0,x)$ is the distance function for some point $x_{0} \in M$. We obtain an elliptic $p$-Harnack inequality for positive harmonic functions from a local uniform Poincar\'{e} inequality and a mean value inequality. As applications of the Harnack inequality, we derive the H\"{o}lder continuity estimate and a Liouville theorem for positive harmonic functions. Furthermore, we establish a gradient estimate for positive harmonic functions.

math.DG↗

arXiv2023

Harnack inequality and the relevant theorems on Finsler metric measure manifolds

In this paper, we carry out in-depth research centering around the Harnack inequality for positive solutions to nonlinear heat equation on Finsler metric measure manifolds with weighted Ricci curvature ${\rm Ric}_{\infty}$ bounded below. Aim on this topic, we first give a volume comparison theorem of Bishop-Gromov type. Then we prove a weighted Poincar\'{e} inequality by using Whitney-type coverings technique and give a local uniform Sobolev inequality. Further, we obtain two mean value inequalities for positive subsolutions and supersolutions of a class of parabolic differential equations. From the mean value inequality, we also derive a new local gradient estimate for positive solutions to heat equation. Finally, as the application of the mean value inequalities and weighted Poincar\'{e} inequality, we get the desired Harnack inequality for positive solutions to heat equation.

math.AP↗

arXiv2023

X-ray Spectroscopy of a Rare-Earth Molecular System Measured at the Single Atom Limit in Room Temperature

We investigate the limit of X-ray detection at room temperature on rare-earth molecular films using lanthanum and a pyridine-based dicarboxamide organic linker as a model system. Synchrotron X-ray scanning tunneling microscopy is used to probe the molecules with different coverages on a HOPG substrate. X-ray-induced photocurrent intensities are measured as a function of molecular coverage on the sample allowing a correlation of the amount of La ions with the photocurrent signal strength. X-ray absorption spectroscopy shows cogent M4,5 absorption edges of the lanthanum ion originated by the transitions from the 3d3/2 and 3d5/2 to 4f orbitals. X-ray absorption spectra measured in the tunneling regime further reveal an X-ray excited tunneling current produced at the M4,5 absorption edge of La ion down to the ultimate atomic limit at room temperature.

cond-mat.mtrl-sci↗

arXiv2023

Some inequalities and gradient estimates for harmonic functions on Finsler measure spaces

In this paper, we study functional and geometric inequalities on complete Finsler measure spaces under the condition that the weighted Ricci curvature ${\rm Ric}_\infty$ has a lower bound. We first obtain some local uniform Poincar\'{e} inequalities and Sobolev inequalities. Further, we give a mean value inequality for nonnegative subsolutions of elliptic equations. Finally, we obtain local and global Harnack inequalities, and then, establish a global gradient estimate for positive harmonic functions on forward complete non-compact Finsler measure spaces. Besides, as a by-product of the mean value inequality, we prove a Liouville type theorem.

math.DG↗

arXiv2022

The characterizations on a class of weakly weighted Einstein-Finsler metrics

In this paper, we study the weakly weighted Einstein-Finsler metrics. First, we show that weakly weighted Einstein-Kropina metrics must be of isotropic S-curvature with respect to the Busemann-Hausdorff volume form under a certain condition about the weight constants. Then we characterize weakly weighted Einstein-Kropina metrics completely via their navigation expressions or via $α$ and $β$ respectively.

math.DG↗

arXiv2021

Some inequalities on Finsler manifolds with weighted Ricci curvature bounded below

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.

math.DG↗

arXiv2020

Some important applications of improved Bochner inequality on Finsler manifolds

We establish some important inequalities under the condition that the weighted Ricci curvature $\mathrm{Ric}_{\infty}\geq K$ for some constant $K >0$ by using improved Bochner inequality and its integrated form. Firstly, we obtain a sharp Poincaré-Lichnerowicz inequality. Further, we give a new proof for logarithmic Sobolev inequality. Finally, we obtain an estimate of the volume of geodesic balls.

math.DG↗

arXiv2020

The Randers metrics of weakly isotropic scalar curvature

In this paper, we study the Randers metrics of weakly isotropic scalar curvature. We prove that a Randers metric of weakly isotropic scalar curvature must be of isotropic $S$-curvature. Further, we prove that a conformally flat Randers metric of weakly isotropic scalar curvature is either Minkowskian or Riemannian.

math.DG↗

arXiv2020

The navigation problems and the curvature properties on conic Kropina manifolds

In this paper, we study navigation problems on conic Kropina manifolds. Let $F(x, y)$ be a conic Kropina metric on an $n$-dimensional manifold $M$ and $V$ be a conformal vector field on $(M, F)$ with $F(x, - V_{x})\leq 1$. Let $\widetilde{F}= \widetilde{F} (x,y)$ be the solution of the navigation problem with navigation data $(F, V)$. We prove that $\widetilde{F}$ must be either a Randers metric or a Kropina metric. Then we establish the relationships between some curvature properties of $F$ and the corresponding properties of the new metric $\widetilde{F}$, which involve S-curvature, flag curvature and Ricci curvature.

math.DG↗

arXiv2019

Some fundamental problems in global Finsler geometry

The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of Lie derivatives on Finsler manifolds. We also characterize the gradient vector fields and obtain a gradient estimate for any smooth function on a Randers manifold.

math.DG↗

arXiv2019

A class of Randers metrics of scalar flag curvature

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that $β$ is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

math.DG↗

arXiv2016

Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.

math.DG↗