Publication

Logarithmic heat kernel estimates without curvature restrictions

2023

Abstract

The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup Pt on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an &#87122 term in the variation allowing greater control.We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time. Furthermore, the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to bypass the lack of continuity of the exit time of the Brownian motions on its initial position.

Official source

https://infoscience.epfl.ch/entities/publication/65ff9770-bae4-4947-bcd7-806893befb79

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/65ff9770-bae4-4947-bcd7-806893befb79

About this result

Related concepts (32)

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

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