Jiuru Zhou

School of Mathematical Sciences,Yangzhou University,Yangzhou 225002,China.

Abstract.In general,the space of Lpharmonic forms Hk(Lp(M))and reduced Lp cohomology Hk(Lp(M))might be not isomorphic on a complete Riemannian manifold M,except for p=2.Nevertheless,one can consider whether dim Hk(Lp(M))<+∞are equivalent to dim Hk(Lp(M))<+∞.In order to study such kind of problems,this paper obtains that dimension of space of Lpharmonic forms on a hypersurface in unit sphere with finite total curvature is finite,which is also a generalization of the previous work by Zhu.The next step will be the investigation of dimension of the reduced Lp cohomology on such hypersurfaces.

Key words:Lpharmonic k-form,hypersurface in sphere,total curvature.

1 Introduction

For compact manifolds,Hodge Theorem says that de Rham cohomology can be computed by harmonic forms.This motivates us to study Lpharmonic forms.Notice that on a compact Riemannian manifold a harmonic form α satisfies Δα=0,where Δ:=?(δd+dδ)is the Hodge Laplacian,and this is equivalent to dα =0,δα =0.By the Gaffney cut-off trick,the same equivalence holds for L2harmonic forms on a complete Riemannian manifold[18],and Hk(L2(M))is isomorphic to Hk(L2(M));however,in general,it is not the case for Lp(p/=2).Actually,Alexandru-Rugina[1]found Lpintegrable k-forms α satisfying Δα=0,which are neither closed nor co-closed,on hyperbolic space Hnfor n≥3.From this we see that the Lp(p/=2)and L2harmonic theory are much different.In this paper,similar to[14,3622],we define the space of Lpharmonic k-forms to be

Besides,Alexandru-Rugina[1]observed that for manifolds with bounded geometry,there is a continuous embedding

Hence,if the dimension of Hk(Lp(M))is finite,then so is the dimension of Hk(Lp(M)).However,the opposite implication is unknown.Hence,one can ask the following question:

(?)Whether”dim Hk(Lp(M))<+∞??dim Hk(Lp(M))<+∞”holds in general.

Furthermore,

(??)Whether”dim Hk(Lp(M))<+∞??dim Hk(Lp(M))<+∞”is true.

This is the motivation of the current work.It’s also interesting to consider similar questions between Hk(Lp(M))and Hk(Lp(M)),and under which conditions,Hk(Lp(M))and Hk(Lp(M))will equal.Recently,Li(Theorem 2.1 in[14])has given sufficient conditions to obtain strong Lp-Hodge decompositions on k-forms,and moreover,the isomorphism

Since there are affluent finiteness results on reduced L2cohomology and L2harmonic forms both on manifolds and submanifolds,such as[2,5,6,8,12,13,16,17,19,22],and references therein.We want to investigate which kind of finiteness results can be generalized to Lpcase,and here we will concentrate on hypersurfaces in Sn+1.

Let x:Mn→Sn+1be an isometric immersion, and denote by A the second fundamental form,H the mean curvature of the immersion x.The traceless second fundamental form is give by

for all vector fields X and Y,where 〈·,·〉is the induced metric of M,and the immersion x is said to be of finite total curvature if

Fu and Xu[8]proved that if a complete noncompact submanifold Mn(n≥3)isometrically immersed in Sn+thas finite total curvature and bounded mean curvature,then the space of L2harmonic 1-forms on M has finite dimension.Later,Fang and Zhu[21]obtained the same result by removing the restriction on the mean curvature.Recently,Zhu[20]extended their result to L2harmonic k-forms(0≤k≤n)on a complete noncompact hypersurface M in a sphere Sn+1,so obtained that the reduced L2cohomology on M has finite dimension.In this paper,we obtain the following finiteness theorem for Hk(Lp(M)),

Theorem 1.1.Let Mnbe a complete connected noncompact hypersurface isometrically immersed in Sn+1with n≥3.If the total curvature of M is finite,then the space of Lpharmonic k-forms Hk(Lp(M))on M has finite dimension,for p>1 and 0≤k≤n.

Our next step is to investigate whether the dimension of the reduced Lpcohomology Hk(Lp(M))is finite or not.If yes,this might show some light on validity of question(??).Otherwise,we would like to find counterexamples to question(??).

2 Preliminaries

In this section,we recall some Lemmas which will be used in the proof of Theorem 1.1.The first is the famous Hoffman-Spruck Sobolev inequality,

3 Proof of Theorem 1.1

Acknowledgments

The author would like to thank Prof.Xingwang Xu and Prof.Hongyu Wang for continual support and Prof.Peng Zhu for useful discussion.The author is partially supported by National Natural Science Foundation of China(Grant No.11771377)and the Natural Science Foundation of Jiangsu Province(Grant No.BK20191435).