arXiv research

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

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

New theorems compare Laplacian on Kähler manifolds.

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

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

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

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

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

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…

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.

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…

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

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 …

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

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…

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

Paper develops formulas and theorems in Hermitian geometry.

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

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

Paper develops methods for estimating gradients of Finslerian Schrödinger equations.

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

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

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

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

The paper explores dualities in differential equations and their applications in Riemannian geometry.

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…

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…

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…

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 …

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…

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…

Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model…

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

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…

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

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.

Study on second Robin eigenvalue for Laplacian on manifolds.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of…

We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under p…

The paper compares Steklov and Laplacian eigenvalues on graphs.

Develops a Barta theorem for p-Laplacian on manifolds.

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …

In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsi…