arXiv research

The paper compares heat kernels on manifolds with Robin boundary conditions.

The Liouville theorem is proven for V T-harmonic map heat flow.

Develops a local Fokker--Planck geometric framework for more accurate score estimation.

Optimal Poincaré constant estimates on manifolds with ends.

Study spectral properties of sub-Laplacians in Carnot groups.

Global existence and convergence of heat flow for p-harmonic maps.

Study on biharmonic heat equation on manifolds with curvature constraints.

The paper solves numerical integration on graphs by optimizing vertex sampling and weights.

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

We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman's recent results o…

One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations…

Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley …

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…

New approach to heat flow for half-harmonic maps, related to minimal surfaces.

In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…

Proves gradient estimate for heat equation on curved manifolds.

In this paper, by applying a linear trace Li-Yau-Hamilton inequality for a positive (1,1)-form solution of the CR Hodge-Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudocovex CR manifold with nonnegative pseudohermitian bisectional c…

This paper proves energy convexity for bi-harmonic maps into spheres, with applications to heat flow and uniqueness.

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. …

Upper bounds on Laplace-Beltrami operator found via volume growth.

Let $(M^m,g)$ be a m-dimensional complete Riemannian manifold which satisfies the n-Sobolev inequality and on which the volume growth is comparable to the one of $\R^n$ for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in $L^{\frac{n}{2}\pm ε}$ for an $ε>0$, then we prove a G…

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 …

Let $M$ be a Riemannian manifold and $Ω$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $Ω$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the nor…

Study shows uniform doubling property for SU(2) geometries.

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to $p$-exponential …

New formulae for minimal submanifolds improve area bounds.

Analyzes constant heat flow properties on Riemannian manifolds.

The paper proves stability of curvature bounds in geometric analysis.

The paper connects fractional Laplacians on spheres to number theory and differential geometry.

Study local invariants and geometry of sub-Laplacian on H-type foliations.

We develop a novel Gaussian process method for manifold data.

The paper establishes bounds for Ricci flows using entropy and heat kernel methods.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.

The Riesz transform is characterized on non-compact manifolds with specific conditions.

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough,…

By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for Kähler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius $r$ centered at any f…

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed e…

Smooth Ricci flows from metric spaces can be extended to smooth solutions.

Data-driven method clusters and analyzes heat load patterns in district heating networks.

The main goal in this paper is to point out that quantity $||\nabla R||^2(p)$ on a harmonic space can not be determined by the spectra of local geodesic spheres or balls, therefore the main results of [AM-S] (quoted in the title) are wrong. My strong interest in the above theorem is motivated by the fact that it contra…

Study Kleinian groups using orbital functions and heat kernels.

Maps from spheres and disks to convex shapes via curvature flow.

Quantum heat traces study new invariants from elliptic operators.

We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this…