The maximal principle for properly immersed submanifolds and its applications

TL;DR

This paper establishes Liouville type theorems for properly immersed submanifolds in certain Riemannian manifolds, showing conditions under which the submanifold must be minimal or certain functions must vanish, with applications to p-biharmonic submanifolds.

Contribution

It introduces new Liouville theorems for immersed submanifolds under curvature bounds, extending previous results and applying them to nonexistence of p-biharmonic submanifolds.

Findings

Submanifolds are minimal if certain Laplacian inequalities hold.

Nonnegative functions satisfying specific Laplacian inequalities must be zero under curvature conditions.

New nonexistence results for p-biharmonic submanifolds are derived.

Abstract

In this note we consider the Liouville type theorem for a properly immersed submanifold $M$ in a complete Riemmanian manifold $N$. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\geq-L(1+dist_N(\cdot,q_0)^2)^\frac{\alpha}{2}$ for some $L>0, 2>\alpha\geq 0$ and $q_0\in N$. (i) If $\Delta|\vec{H}|^{2p-2}\geq k|\vec{H}|^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal. (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $\Delta u\geq ku^a$ for some constant $k>0$ and $a>1$. If $|\vec{H}|\leq C(1+dist_N(\cdot,q_0)^2)^\frac{\beta}{2}$ for some $C>0$, $0\leq\beta<1$, then $u=0$ on $M$. As applications we get some nonexistence result for $p$-biharmonic submanifolds.