Integral Liouville theorem in a complete Riemannian manifold

zbMATH Open1406.53044arXiv1808.00180MaRDI QIDQ4645529

Chandan Kumar Mondal, Absos Ali Shaikh

Publication date: 10 January 2019

Abstract: If the Killing vector field in a Riemannian manifold is the gradient of a smooth real valued function, then it is called Killing potential. In this paper we have deduced a necessary condition for the existence of Killing potential in a complete Riemannian manifold. Yau proved the Liouville theorem of harmonic function in a Riemannian manifold using gradient estimation and after that many authors have generalized this concept and investigated various types of Liouville theorems of harmonic functions. In this article we have also proved a Liouville theorem in integral form of harmonic functions in a complete Riemannian manifold. Finally we have studied the behaviour of harmonic functions in a complete Riemannian manifold that satisfies some gradient Ricci solition and showed that harmonic function is a constant multiple of distance function along some geodesics.

Full work available at URL: https://arxiv.org/abs/1808.00180

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zbMATH Keywords

harmonic function convex function Killing vector field subharmonic function Killing potential potential function Neumann-Poincaré inequality

Mathematics Subject Classification ID

Elliptic equations on manifolds, general theory (58J05) Differential geometric aspects of harmonic maps (53C43) Harmonic maps, etc. (58E20) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21)