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

17 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

Bibliographical note

Funding Information:
The first author was partially supported by NSF grant DMS-1506220. The second author was partially supported by MOST grant 106B2030I2. The third author was partially supported by NSF grant DMS-1606820. The first author was partially supported by NSF grant DMS-1506220. The second author was partially supported by MOST grant 106B2030I2. The third author was partially supported by NSF grant DMS-1606820. We would like to dedicate this paper to our teacher Professor Peter Li. It can not be overstated how much we have benefited from his teaching, encouragement and support over the years. The first author was partially supported by NSF grant DMS-1506220. The second author was partially supported by MOST grant 106B2030I2. The third author was partially supported by NSF grant DMS-1606820.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

Bottom spectrum
Green's function
Poisson equation
Steady Ricci solitons

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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.",

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

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AU - Anna Sung, Chiung Jue

AU - Wang, Jiaping

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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.

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KW - Steady Ricci solitons

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Poisson equation on complete manifolds

Abstract

Bibliographical note

Keywords

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