Heat kernel upper bounds on a complete non-compact manifold.

• Volume: 10, Issue: 2, page 395-452
• ISSN: 0213-2230

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Abstract

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Let M be a smooth connected non-compact geodesically complete Riemannian manifold, Δ denote the Laplace operator associated with the Riemannian metric, n ≥ 2 be the dimension of M. Consider the heat equation on the manifoldut - Δu = 0,where u = u(x,t), x ∈ M, t &gt; 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t → +∞ and r ≡ dist(x,y) → +∞.

How to cite

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Grigor'yan, Alexander. "Heat kernel upper bounds on a complete non-compact manifold.." Revista Matemática Iberoamericana 10.2 (1994): 395-452. <http://eudml.org/doc/39447>.

@article{Grigoryan1994,
abstract = {Let M be a smooth connected non-compact geodesically complete Riemannian manifold, Δ denote the Laplace operator associated with the Riemannian metric, n ≥ 2 be the dimension of M. Consider the heat equation on the manifoldut - Δu = 0,where u = u(x,t), x ∈ M, t &gt; 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t → +∞ and r ≡ dist(x,y) → +∞.},
author = {Grigor'yan, Alexander},
journal = {Revista Matemática Iberoamericana},
keywords = {Kernel; Acotación; Variedades topológicas; Límite superior; heat kernel; isoperimetric inequalities; spectral geometry},
language = {eng},
number = {2},
pages = {395-452},
title = {Heat kernel upper bounds on a complete non-compact manifold.},
url = {http://eudml.org/doc/39447},
volume = {10},
year = {1994},
}

TY - JOUR
AU - Grigor'yan, Alexander
TI - Heat kernel upper bounds on a complete non-compact manifold.
JO - Revista Matemática Iberoamericana
PY - 1994
VL - 10
IS - 2
SP - 395
EP - 452
AB - Let M be a smooth connected non-compact geodesically complete Riemannian manifold, Δ denote the Laplace operator associated with the Riemannian metric, n ≥ 2 be the dimension of M. Consider the heat equation on the manifoldut - Δu = 0,where u = u(x,t), x ∈ M, t &gt; 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t → +∞ and r ≡ dist(x,y) → +∞.
LA - eng
KW - Kernel; Acotación; Variedades topológicas; Límite superior; heat kernel; isoperimetric inequalities; spectral geometry
UR - http://eudml.org/doc/39447
ER -

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