The Journal of Geometric Analysis

, Volume 12, Issue 3, pp 425–436

The fundamental solution on manifolds with time-dependent metrics

Article

DOI: 10.1007/BF02922048

Cite this article as:
Guenther, C.M. J Geom Anal (2002) 12: 425. doi:10.1007/BF02922048

Abstract

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.

Math Subject Classifications

5835

Key Words and Phrases

Fundamental solutionHarnack inequalitygeometric heat flow

Copyright information

© Mathematica Josephina, Inc. 2002

Authors and Affiliations

  1. 1.Department of MathematicsPacific UniversityForest Grove