arXiv research

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

New theorems compare Laplacian on Kähler manifolds.

The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

The paper compares Steklov and Laplacian eigenvalues on graphs.

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.

The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.

We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem.

Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

The paper develops heat kernel comparison theorems and applies them to spectral geometry.

The paper explores inequalities between eigenvalues on Riemannian manifolds.

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the s…

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…

We study comparison formulas for $ζ$-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus $g\geq 2$. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly.

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.

The paper compares eigenvalues of Laplacians on fibred manifolds using symmetrization techniques.

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to …

Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representa…

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet $p$-Laplacian ($1<p<\infty$) obtained by Matei [A.-M. Matei, First eigenvalue for the $p$-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the $p$-Laplacian …

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. …

Paper establishes statistical inference for pairwise comparison models.

The paper studies eigenvalue problems on manifolds and recovers known inequalities.

Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain …

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.

Lower bound found for Kähler manifold eigenvalues.

ELD compares graphs by their embedded Laplacian eigenvectors, resolving ambiguities.

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenböck formula and Laplacian comparison theorem, and study the h…

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The proof is based on a gradient comparison theorem. We will also charachterize the…

Paper develops formulas and theorems in Hermitian geometry.

The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…

Study on second Robin eigenvalue for Laplacian on manifolds.

On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e, Kahler manifolds with constant holomorphic sectional curvature. Moreover, we give a…

The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…