Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite Lq-Energy?

Xiangzhi CAO

1 Introduction

Liouville type theorem is very important in geometry and topology.In[6],Wang considered complete sub manifolds in manifolds with partially non-negative curvature and studied harmonic map from complete submanifolds in manifolds of partially non-negative curvature to non-positively curved manifold.She obtained the Liouville type theorem for harmonic map.Notice that this curvature condition on the ambient manifold is interesting.

In[2],Feng and Han derived a Liouville type theorem for p-harmonic function on minimal submanifold in.About the operator?+Ak(x)with constant A on manifold M withmany results about the p-harmonic maps with finite energy from M to non-positively curved manifold has been obtained,such as[3],where they required thatChanging the condition that p-harmonic map is of finite energy into the condition which is of finite Lq-energy,Pigola,Rigoli and Setti[5]investigated p-harmonic maps of finite Lq-energy into non-positively curved manifolds.

In this paper,we mainly consider the case that A=1.We always assume p-harmonic map has finite Lq-energy,because for p-harmonic map with finite energy,we find that it is hard to get better estimates since the Kato inequaltiy for p-harmonic maps is not nicer than that for harmonic map,but luckily for p-harmonic function,we have a better Kato inequality than p-harmonic map.The purpose of this paper is to study p-harmonic function on submanifold in a manifold with partially non-negative curvature and p-harmonic maps from a submanifold in a manifold with partially non-negative curvature to non-positively curved manifold respectively,but with different conditions.For example,we can give the following:

(1)The following Sobolev inequality holds on M for any compactly supported function ψ

2 Preliminary

The energy functional of p-harmonic map is defined by

where q(x)is any continuous function onis a piecewise smooth function with compact support,△ is the Laplacian,is the gradient of φ.The index of the operator L is the supremum of the number of the negative eigenvalue of L with Dirichlet boundary condition.

We say a map u is of Lq- finite energy if R

3 The Proof of the Main Theorems

Theorem 3.1 Let M be an n(≥3)-dimensional complete non-compact submanifold isometrically immersed in an(n+k)-dimensional complete Riemannian manifoldof nonnegative(n?1)-th Ricci curvature.If the index of the operatoris zero andthen any p-harmonic function with finiteon M is a constant.

So we haveFrom(3.3),we have RicM≥0,thus M has infinity volume(see[7]).This is a contradiction since RConsequently,f=0 and u is a constant.

Theorem 3.2 Let M be an n(≥3)-dimensional complete non-compact isometrically immersed in an(n+k)-dimensional complete Riemannian manifoldof non-negative(n?1)-th Ricci curvature,and let the Sobolev inequality(1.1)hold on M.Furthermore,we suppose

where the constant c appearing in(3.10)is the same as that from the Sobolev inequality(1.1).Then any p-harmonic map with finite L2p?2-energy from M to a complete manifold with non-positive curvature is a constant.

Proof For p-harmonic maps,we have the Bochner formula(see Lemma 12 in[5])

Then any p-harmonic map with finite L2p?2-energy from M to a complete manifold with non-positive curvature is a constant.

Proof From the proof of Corollary 3.3 in[6],we see that the Sobolev inequality holds on M.So it follows from Theorem 3.2.

Remark 3.1 Combining the skills used in the proof of Theorem 2 in[3]and Theorem 1.1 in[6],we can also prove the Liouville theorem about p-harmonic map under the similar curvature condition on domain manifold,codomain manifold and index condition imposed on some operator.We should note that the similarity between the index condition and the stable condition for submanifold.

AcknowledgmentThe authors would like to thank his tutor Professor Qun Chen for his useful comments and suggestions for pointing out some mistakes in the former version of this manuscript.

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