The maximal principle for properly immersed submanifolds and its applications

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 $K^N\ge -L(1+dis{t}_N(\cdot ,q_0{)}^2{)}^{\frac{\alpha }{2}}$ for some $L>0,2>\alpha \ge 0$ and $q_0\in N$. (i) If $\Delta |\vec{H}{|}^{2p-2}\ge k|\vec{H}{|}^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal. (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $\Delta u\ge k{u}^a$ for some constant $k>0$ and $a>1$. If $|\vec{H}|\le C(1+dis{t}_N(\cdot ,q_0{)}^2{)}^{\frac{\beta }{2}}$ for some $C>0$, $0\le \beta <1$, then $u=0$ on $M$. As applications we get some nonexistence result for $p$-biharmonic submanifolds.

13 pages, all comments are welcome

References in corpus (1)

• Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal