Biharmonic hypersurfaces in a sphere

In this paper, we study the existence of p-harmonic maps into Riemannian manifolds admitting a conformal vector field. We also prove a Liouville type theorem for p-biharmonic maps.

On the p -harmonic and p -biharmonic maps

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 $K^N\ge -L(1+dist_N(\cdot ,q_0)^2)^\frac{\alpha }{2}$ 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 $\Delta u\ge ku^a$ for some constant $k>0$ and $a>1$. If $|\vec {H}|\le C(1+dist_N(\cdot ,q_0)^2)^\frac{\beta }{2}$ for some $C>0$, $0\le \beta <1$, then $u=0$ on M. As applications we get some nonexistence result for p-biharmonic submanifolds.

The maximal principle for properly immersed submanifolds and its applications

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.

Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and $(N, h)$. The $p$-bienergy of $u$ is defined by $E_p(u)=\int_M|\tau(u)|^pd\nu_g$, where $\tau(u)$ is the tension field of $u$ and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. In this paper we will prove nonexistence result of proper $p$-biharmonic maps when $p\geq2$. In particular when $M=\mathbb{R}^m$, we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas \cite{BFO}.

Nonexistence of proper $p$-biharmonic maps and Liouville type theorems I: case of $p\geq2$

Let u: (M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). The p-bienergy of u is τp(u) = ∫M|τ(u)|p dνg, where τ(u) is the tension field of u and p > 1. Critical points of τp are called p-biharmonic maps and isometric p-biharmonic maps are called p-biharmonic submanifolds. When p = 2, p-biharmonic submanifolds are biharmonic submanifolds and in recent years many nonexistence results are found for biharmonic submanifolds in nonpositively curved manifolds. In this paper we will study the nonexistence result for general p-biharmonic submanifolds.

Yong Luo

Papers

On p-biharmonic submanifolds in nonpositively curved manifolds