Poisson equation on complete manifolds

Ovidiu Munteanu; Chiung Jue Anna Sung; Jiaping Wang

doi:10.1016/j.aim.2019.03.019

Poisson equation on complete manifolds

Ovidiu Munteanu, Chiung Jue Anna Sung, Jiaping Wang

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

Original language	English (US)
Pages (from-to)	81-145
Number of pages	65
Journal	Advances in Mathematics
Volume	348
DOIs	https://doi.org/10.1016/j.aim.2019.03.019
State	Published - May 25 2019

Keywords

Bottom spectrum
Green's function
Poisson equation
Steady Ricci solitons

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10.1016/j.aim.2019.03.019

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title = "Poisson equation on complete manifolds",

abstract = "We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.",

keywords = "Bottom spectrum, Green's function, Poisson equation, Steady Ricci solitons",

author = "Ovidiu Munteanu and {Anna Sung}, {Chiung Jue} and Jiaping Wang",

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month = may,

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AU - Munteanu, Ovidiu

AU - Anna Sung, Chiung Jue

AU - Wang, Jiaping

PY - 2019/5/25

Y1 - 2019/5/25

N2 - We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

AB - We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

KW - Bottom spectrum

KW - Green's function

KW - Poisson equation

KW - Steady Ricci solitons

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