Search

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., Comment: 30 pages. Any comments and suggestions are warmly welcome

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., Comment: 19 pages, 4 figures

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., Comment: 31 pages

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 $\alpha$ and $\beta$ respectively., Comment: 22 pages

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 Poincar\'{e}-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2009.02632

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 Poincar\'{e}-Lichnerowicz inequality. Further, we give a new proof for logarithmic Sobolev inequality. Finally, we obtain an estimate of the volume of geodesic balls., Comment: 14 pages

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.

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.

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.

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 $\beta$ is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

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.