Introduction

For a complete manifold ^{2007}). It is interesting and importantto discuss the finiteness of the dimension of these spaces.Carron (^{1999}) proved that if ^{Carron 2007}). In fact, several mathematicians studied the space of ^{2002}) proved that the dimension of the space of the ^{2013}) showed that:suppose that

for all orthonormal tangent vectors ^{2009}); Zhu (^{2013}) gave a result on finitely many ends of complete manifolds with a weighted Poincaréinequality by use of the space of ^{2014}) discussed a complete noncompact submanifold ^{2010}) studied a complete submanifold ^{2014}) proved Fu-Xu’s result without the restriction on the mean curvature vector and therefore obtained that the first space of reduced ^{2011}) studied the existence of the symplectic structure and

Motivated by above results, we discuss a complete noncompact hypersurface

**Theorem 1.**
*Let *

**Corollary 2.**
*Let *

**Remark 3.**
*Under the same condition of Corollary 2*, we conjecture thatthe p-th space of reduced

PRELIMINARIES

In this section, we recall some relevant definitions and results. Suppose that

where

The Laplacian operator is defined by

A

We denote by

and

We define the

Suppose that

for all vector fields

We state several results which will be used to prove Theorem 1.

**Proposition 4.**
*( ^{Carron 2007})Let *

**Lemma 5.**
*( ^{Li 1993}) If *

*where *

**Proposition 6.**
*( ^{Hoffman and Spruck 1974}, ^{Zhu and Fang 2014})Let *

*for each *

AN INEQUALITY FOR

In this section, we show an inequality for

**Proposition 7.**
*Let *

*for *

*for *

*Proof.* Suppose that

By Lemma 5, we get that:

Combining (1) with (AN INEQUALITY FOR

There exists the Kato inequality for ^{Cibotaru and Zhu 2012}, ^{Wang 2002}):

By (3) and (4), we get that

Now, we give the estimate of the term

So, we get that

which implies that

By Gauss equation, we have that

A direct computation shows that

and

Since the curvature operator

By (6)-(10), we have

Note that

For

For

By (5), we have that:

for

for

**Remark 8.**
*If *

^{1992}).

PROOF OF MAIN RESULTS

In this section, we prove Theorem 1 and Corollary 2.

If

Integrating by parts on

Case I:

Note that

for any positive real number

for any positive real number

where

and

Since the total curvature

Set

and

Thus,

for any

for any positive real number

Choose a sufficient large

and

Then (18) implies that

for any

Case II:

Note that

for any positive real number

for any positive real number

where

and

Since the total curvature

and

Obviously,

for any

for any positive real number

We choose a sufficient large

and

Then (26) implies that

for any

By Case I and Case II, we have that

for any

Next, the proof follows standard techniques (after inequality (33) in Cavalcante et al. (^{2014} and uses a Moser iteration argument (lemma 11 in Li (^{1980})).We include a concise proof here for the sake of completeness.Choose

for some positive constant

Letting

By Hölder inequality

we get that

Set

Fix

where

Then, for

That is,

Note that

Combining with (31), we obtain that

Combining Cauchy-Schwarz inequality with (32), we obtain that

where

where

Choosing

By recurrence, we have

where

Now, choose

By (30), we have

where ^{1980}), we show that

**Remark 9.**
*For the case of * can also be obtained by a different method. In fact, Yau (

^{1976}) proved that if

^{2014}), we obtain the desired result.