Generalized $1$-harmonic Equation and The Inverse Mean Curvature Flow

Details Generalized $1$-harmonic Equation and The Inverse Mean Curvature Flow Generalized $1$-harmonic Equation and The Inverse Mean Curvature Flow

2010-11-13

arxiv.org/abs/1011.3138v1

Mathematics

Differential Geometry

14 pages, to appear in Journal of Geometry and Physics

Scientific paper

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), we find an analytic quantity $w$ in the generalized $1$-harmonic equations (1.1) on a domain in a Riemannian $n$-manifold that affects the behavior of weak solutions of (1.1), and establish its link with the geometry of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for $p$-subharmonic functions of constant $p$-tension field, $p \ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.

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