Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature

Alberto Farina; Jesús Ocáriz

doi:10.3934/dcds.2020347

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2021, Volume 41, Issue 4: 1929-1937. Doi: 10.3934/dcds.2020347

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Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature

Alberto Farina^1, and
Jesús Ocáriz^2, ,

1.
LAMFA-CNRS UMR 7352, Université de Picardie Jules Verne, Faculté des Sciences, 33, rue Saint-Leu, 80039 Amiens CEDEX 1, France
2.
Universidad Autónoma de Madrid, Facultad de Ciencias, Ciudad Universitaria de Cantoblanco, Calle Francisco Tomás y Valiente, 7, 28049 Madrid, Spain

^* Corresponding author: Jesús Ocáriz
^* Corresponding author: Jesús Ocáriz

Received: December 31, 2019

Revised: July 31, 2020

Early access: October 2020

Published: April 2021

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In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $ u $ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $ u $.

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Mathematics Subject Classification: Primary: 35J15, 35J61; Secondary: 53B20, 53C20.

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