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On reduced L2 cohomology of hypersurfaces in spheres with finite total curvature

Abstract

In this paper, we prove that the dimension of the second space of reduced L2 cohomology of M is finite if is a complete noncompact hypersurface in a sphere 𝕊n+1and has finite total curvature (n≥3).

total curvature; reduced L2 cohomology; hypersurface in sphere; L2 harmonic 2-form

Introduction

For a complete manifold Mn, the p-th space of reduced L2-cohomology is defined, for 0pn in Carron (2007r2 CARRON G. 1999. L2-Cohomologie et inégalités de Sobolev. Math Ann 314: 613-639.). It is interesting and importantto discuss the finiteness of the dimension of these spaces.Carron (1999r1 CARRON G. 2007. L2 harmonic forms on non compact manifolds. arXiv:0704.3194v1.) proved that if Mn (n3) is a complete noncompact submanifold of n+pwith finite total curvature and finite mean curvature (i. e., the Ln-norm of the mean curvature vector is finite),then each p-th space of reduced L2-cohomology on M has finite dimension, for 0pn.The reduced L2 cohomology is related with the L2 harmonic forms (Carron 2007r1 CARRON G. 2007. L2 harmonic forms on non compact manifolds. arXiv:0704.3194v1.). In fact, several mathematicians studied the space of L2harmonic p-forms for p=1,2.If Mn (n3) is a complete minimal hypersurface in n+1 with finite index, Li and Wang (2002r11 LI P AND WANG JP. 2002. Minimal hypersurfaces with finite index. Math Res Lett 9: 95-103.) proved that the dimension of the space of the L2 harmonic 1-forms M is finite and M has finitely many ends.More generally, Zhu (2013r15 ZHU P. 2013. L2-harmonic forms and finiteness of ends. An Acad Bras Cienc 85: 457-471.) showed that:suppose that Nn+1 (n3) is a complete simply connected manifold with non-positive sectional curvature andMn is a complete minimal hypersurface in N with finite index. If the bi-Ricci curvature satisfies

b - R i c ¯ ( X , Y ) + 1 n | A | 2 0 ,

for all orthonormal tangent vectors X,Y in TpN for pM, then the dimension of the space of the L2 harmonic 1-forms M is finite. Furthermore, following the idea of Cheng and Zhou (2009r4 CHENG X AND ZHOU DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154.); Zhu (2013r15 ZHU P. 2013. L2-harmonic forms and finiteness of ends. An Acad Bras Cienc 85: 457-471.) gave a result on finitely many ends of complete manifolds with a weighted Poincaréinequality by use of the space of L2 harmonic functions.Cavalcante et al. (2014r3 CAVALCANTE MP, MIRANDOLA H AND VITÓRIO F. 2014. L2-harmonic 1-forms on submanifolds with finite total curvature. J Geom Anal 24: 205-222.) discussed a complete noncompact submanifold Mn (n3) isometrically immersed in a Hadamard manifoldNn+p with sectional curvature satisfying -k2KN0 for some constant k and showed that if the total curvature is finiteand the first eigenvalue of the Laplacian operator of M is bounded from below by a suitable constant, then the dimension of the space of the L2 harmonic 1-forms on M is finite.Fu and Xu (2010r7 FU HP AND XU HW. 2010. Total curvature and L2 harmonic 1-forms on complete submanifolds in space forms. Geom Dedicata 144: 129-140.) studied a complete submanifold Mn in a sphere 𝕊n+p with finite total curvature and bounded mean curvature and proved that the dimension of the space of the L2 harmonic 1-forms on M is finite. Zhu and Sw. (2014r16 ZHU P AND FANG SW. 2014. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Global Anal Geom 46: 187-196.) proved Fu-Xu’s result without the restriction on the mean curvature vector and therefore obtained that the first space of reduced L2-cohomology on M has finite dimension.Zhu (2011r14 ZHU P. 2011. Harmonic two-forms on manifolds with nonnegative isotropic curvature. Ann Global Anal Geom 40: 427-434.) studied the existence of the symplectic structure and L2 harmonic 2-forms on complete noncompact manifolds by use of a special version of Bochner formula.

Motivated by above results, we discuss a complete noncompact hypersurface Mn in a sphere 𝕊n+1 with finite total curvature in this paper.We obtain the following finiteness results on the space of all L2 harmonic 2-forms and the second space of reduced L2cohomology:

Theorem 1. Let Mn (n3) be an n-dimensional complete noncompact oriented manifold isometrically immersed in an (n+1)-dimensional sphere𝕊n+1. If the total curvature is finite, then the space of all L2 harmonic 2-forms has finite dimension.

Corollary 2. Let Mn (n3) be an n-dimensional complete noncompact oriented manifold isometrically immersed in𝕊n+1. If the total curvature is finite, then the dimension of the second space of reduced L2cohomology of M is finite.

Remark 3.Under the same condition of Corollary 2, we conjecture thatthe p-th space of reduced L2 cohomology of M has finite dimension for 3pn-3.

PRELIMINARIES

In this section, we recall some relevant definitions and results. Suppose that Mn is an n-dimensional complete Riemannian manifold.The Hodge operator*:p(M)n-p(M) is defined by

* e i 1 e i p = sgn σ ( i 1 , i 2 , , i n ) e i p + 1 e i n ,

where σ(i1,i2,,in) denotes a permutation ofthe set (i1,i2,,in) and sgnσ is the sign ofσ. The operator d*:p(M)p-1(M)is given by

d * ω = ( - 1 ) ( n k + k + 1 ) * d * ω .

The Laplacian operator is defined by

ω = - d d * ω - d * d ω .

A p-form ω is called L2 harmonic if ω=0and

M ω * ω < + .

We denote by Hp(L2(M))the space of all L2 harmonic p-forms on M.Let

Z 2 p ( M ) = { α L 2 ( p ( T * M ) ) : d α = 0 }

and

D p ( d ) = { α L 2 ( p ( T * M ) ) : d α L 2 ( p + 1 ( T * M ) ) } .

We define the p-th space of reduced L2 cohomology by

H 2 p ( M ) = Z 2 p ( M ) D p - 1 ( d ) ¯ .

Suppose that x:Mn𝕊n+1 is an isometric immersion of an n-dimensional manifold M in an (n+1)-dimensionalsphere.Let A denote the second fundamental form and H the mean curvature of the immersion x.Let

Φ ( X , Y ) = A ( X , Y ) - H X , Y ,

for all vector fields X and Y, where , is the induced metric of M.We say the immersion x has finite total curvature if

Φ L n ( M ) < + .

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

Proposition 4. (Carron 2007r1 CARRON G. 2007. L2 harmonic forms on non compact manifolds. arXiv:0704.3194v1.)Let (M,g) is a complete Riemannian manifold, then the space of L2 harmonic p-forms Hp(L2(M)) is isomorphic tothe p-th space of reduced L2 cohomology H2p(M).

Lemma 5. (Li 1993r9 LI P. 1993. Lecture notes on geometric analysis. Lecture Notes Series n. 6. Seoul: Seoul National University, Research Institute of Mathematics, Global Analysis Reseach Center, p. 47-48.) If (Mn,g) is a Riemannianmanifold and ω=aIωIp(M), then

| ω | 2 = 2 ω , ω + 2 | ω | 2 + 2 E ( ω ) , ω ,

where E(ω)=Rkβiβjαiαai1kβipeipejαei1.

Proposition 6. (Hoffman and Spruck 1974r6 HOFFMAN D AND SPRUCK J. 1974. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm Pure Appl Math 27: 715-727., Zhu and Fang 2014r16 ZHU P AND FANG SW. 2014. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Global Anal Geom 46: 187-196.)Let Mn be a complete noncompact oriented manifold isometrically immersed in a sphere𝕊n+1. Then

( M | f | 2 n n - 2 ) n - 2 n C 0 ( M | f | 2 + n 2 M ( H 2 + 1 ) f 2 )

for each fC01(M), where C0 depends only on n and H is the mean curvature of M in 𝕊n+1.

AN INEQUALITY FOR L2 HARMONIC 2-FORMS

In this section, we show an inequality for L2 harmonic 2-forms on hypersurfaces in a sphere 𝕊n+1,which plays an important role in the proof of main results.

Proposition 7. Let Mn (n3) be an n-dimensional complete noncompact hypersurface isometrically immersed in an (n+1)-dimensional sphere 𝕊n+1. If ωH2(L2(M)), then

h h | h | 2 + 2 h 2 - | Φ | 2 h 2 + 3 2 H 2 h 2 ,

for n=3 and

h h 1 n - 2 | h | 2 + 2 ( n - 2 ) h 2 - n - 2 2 | Φ | 2 h 2 + n H 2 h 2 ,

for n4, where h=|ω|.

Proof. Suppose that ωH2(L2(M)). Then we have

| ω | 2 = 2 | | ω | | 2 + 2 | ω | | ω | . (1)

By Lemma 5, we get that:

| ω | 2
= 2 ω , ω + 2 | ω | 2 + 2 E ( ω ) , ω
= 2 | ω | 2 + 2 E ( ω ) , ω . (2)

Combining (1) with (AN INEQUALITY FOR L2 HARMONIC 2-FORMS), we obtain that

| ω | | ω | = | ω | 2 - | | ω | | 2 + E ( ω ) , ω . (3)

There exists the Kato inequality for L2 harmonic 2-forms as follows (Cibotaru and Zhu 2012r5 CIBOTARU D AND ZHU P. 2012. Refined Kato inequalities for harmonic fields on Kähler manifolds. Pacific J Math 256: 51-66., Wang 2002r12 WANG XD. 2002. On the L2-cohomology of a convex cocompact hyperbolic manifold. Duke Math J 115: 311-327.):

n - 1 n - 2 | | ω | | 2 | ω | 2 . (4)

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

| ω | | ω | 1 n - 2 | | ω | | 2 + E ( ω ) , ω . (5)

Now, we give the estimate of the term E(ω),ω. Let ω1=bi1i2ei2ei12(M) andω2=ci1i2ei2ei12(M), wherebi1i2=-bi2i1 and ci1i2=-ci2i1. By Lemma 5, we obtain that

E ( ω 1 )
= R k 1 i 1 j 1 i 1 b k 1 i 2 e i 2 e j 1 + R k 2 i 2 j 2 i 2 b i 1 k 2 e j 2 e i 1
+ R k 2 i 2 j 1 i 1 b i 1 k 2 e i 2 e j 1 + R k 1 i 1 j 2 i 2 b k 1 i 2 e j 2 e i 1
= R i c k 1 j 1 b k 1 i 2 e i 2 e j 1 + R i c k 2 j 2 b i 1 k 2 e j 2 e i 1
+ R k 2 i 2 j 1 i 1 b i 1 k 2 e i 2 e j 1 + R k 1 i 1 j 2 i 2 b k 1 i 2 e j 2 e i 1 .

So, we get that

E ( ω 1 ) , ω 2 =
R i c k 1 j 1 b k 1 i 2 c j 1 i 2 + R i c k 2 j 2 b i 1 k 2 c i 1 j 2
+ R k 2 i 2 j 1 i 1 b i 1 k 2 c j 1 i 2 + R k 1 i 1 j 2 i 2 b k 1 i 2 c i 1 j 2 ,

which implies that

E ( ω ) , ω =
R i c k 1 j 1 a k 1 i 2 a j 1 i 2 + R i c k 2 j 2 a i 1 k 2 a i 1 j 2
+ R k 2 i 2 j 1 i 1 a i 1 k 2 a j 1 i 2 + R k 1 i 1 j 2 i 2 a k 1 i 2 a i 1 j 2 . (6)

By Gauss equation, we have that

R i j k l = ( δ i k δ j l - δ i l δ j k ) + h i k h j l - h i l h j k .

A direct computation shows that

R i c k 1 j 1 = ( n - 1 ) δ k 1 j 1 + n H h k 1 j 1 - h k 1 i h i j 1 ; (7)
R i c k 2 j 2 = ( n - 1 ) δ k 2 j 2 + n H h k 2 j 2 - h k 2 i h i j 2 ; (8)
R k 2 i 2 j 1 i 1 = ( δ k 2 j 1 δ i 2 i 1 - δ k 2 i 1 δ i 2 j 1 ) + h k 2 j 1 h i 2 i 1 - h k 2 i 1 h i 2 j 1 (9)

and

R k 1 i 1 j 2 i 2 = ( δ k 1 j 2 δ i 1 i 2 - δ k 1 i 2 δ i 1 j 2 ) + h k 1 j 2 h i 1 i 2 - h k 1 i 2 h i 1 j 2 . (10)

Since the curvature operator E is linear andzero order, and hence tensorial, it is sufficient to compute E(ω),ω at a point p.We can choose an orthonormal frame {ei} such thathij=λiδij at p. Obviously,

n H = λ 1 + + λ n .

By (6)-(10), we have

E ( ω ) , ω =
( n - 1 ) ( a j 1 i 2 ) 2 + n H λ k 1 ( a k 1 i 2 ) 2 - λ k 1 2 ( a k 1 i 2 ) 2
+
( n - 1 ) ( a i 1 j 2 ) 2 + n H λ k 2 ( a i 1 k 2 ) 2 - λ k 2 2 ( a i 1 k 2 ) 2
+
a i 1 j 1 a j 1 i 1 - λ k 2 λ i 2 ( a k 2 i 2 ) 2
+
a j 2 i 2 a i 2 j 2 - λ j 2 λ i 2 ( a j 2 i 2 ) 2
=
2 i j ( ( n - 2 ) + ( λ 1 + + λ n ) λ i - λ i 2 - λ i λ j ) ( a i j ) 2 .

Note that

| A | 2 = | Φ | 2 + n H 2 .

For n=3, we have that

E ( ω ) , ω = 2 i j ( 1 + ( λ 1 + λ 2 + λ 3 ) λ i - λ i 2 - λ i λ j ) ( a i j ) 2
= i j ( 2 + ( λ 1 + λ 2 + λ 3 ) ( λ i + λ j ) - ( λ i 2 + λ j 2 ) - 2 λ i λ j ) ( a i j ) 2
= i j ( 2 + 1 2 ( 3 H ) 2 - 1 2 k = 1 , k i , j 3 λ k 2 - 1 2 ( λ i + λ j ) 2 ) ( a i j ) 2
i j ( 2 + 1 2 ( 3 H ) 2 - 1 2 k = 1 , k i , j 3 λ k 2 - ( λ i 2 + λ j 2 ) ) ( a i j ) 2
i j ( 2 + 9 2 H 2 - | A | 2 ) ( a i j ) 2
= ( 2 + 3 2 H 2 - | Φ | 2 ) | ω | 2 .

For n4, we obtain that

E ( ω ) , ω = 2 i j ( ( n - 2 ) + ( λ 1 + + λ n ) λ i - λ i 2 - λ i λ j ) ( a i j ) 2
= i j ( 2 ( n - 2 ) + ( λ 1 + + λ n ) ( λ i + λ j ) - ( λ i 2 + λ j 2 ) - 2 λ i λ j ) ( a i j ) 2
= i j ( 2 ( n - 2 ) + ( λ 1 + + λ i ^ + + λ j ^ + + λ n ) ( λ i + λ j ) ) ( a i j ) 2
= i j ( 2 ( n - 2 ) + 1 2 ( n H ) 2 - 1 2 ( k = 1 , k i , j n λ k ) 2 - 1 2 ( λ i + λ j ) 2 ) ( a i j ) 2
i j ( 2 ( n - 2 ) + 1 2 ( n H ) 2 - n - 2 2 ( k = 1 , k i , j n λ k 2 ) - ( λ i 2 + λ j 2 ) ) ( a i j ) 2
i j ( 2 ( n - 2 ) + 1 2 ( n H ) 2 - n - 2 2 | A | 2 ) ( a i j ) 2
= ( 2 ( n - 2 ) + 1 2 ( n H ) 2 - n - 2 2 | A | 2 ) | ω | 2
= ( 2 ( n - 2 ) + n H 2 - n - 2 2 | Φ | 2 ) | ω | 2 .

By (5), we have that:

h h | h | 2 + 2 h 2 - | Φ | 2 h 2 + 3 2 H 2 h 2 ,

for n=3 and

h h 1 n - 2 | h | 2 + 2 ( n - 2 ) h 2 - n - 2 2 | Φ | 2 h 2 + n H 2 h 2 ,

for n4.∎

Remark 8. If ω is 1-form , then the term E(ω,ω) is equal to Ric(ω,ω).The corresponding estimate for this term was given by Leung (1992r8 LEUNG PF. 1992. An estimate on the Ricci curvature of a submanifold and some applications. Proc Amer Math Soc 114: 1051-1061.).

PROOF OF MAIN RESULTS

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

If η is a compactly supported piecewise smooth function on M, then

d i v ( η 2 h h )
= η 2 h h + ( η 2 h ) , h
= η 2 h h + η 2 | h | 2 + 2 η h η , h .

Integrating by parts on M, we obtain that

M η 2 h h + M η 2 | h | 2 + 2 M η h η , h = 0. (11)

Case I: n=3. By Proposition 7 and (11), we obtain that

-
2 M η h η , h - 2 M η 2 | h | 2 - 2 M η 2 h 2
+ M | Φ | 2 η 2 h 2 - 3 2 M H 2 h 2 η 2 0. (12)

Note that

- 2 M η h η , h a 1 M η 2 | h | 2 + 1 a 1 M h 2 | η | 2 , (13)

for any positive real number a1.Now we give an estimate of the term M|Φ|2η2h2 as follows:set ϕ1(η)=(Suppη|Φ|3)13. Then there exists

M | Φ | 2 η 2 h 2 ( S u p p η ( | Φ | 2 ) 3 2 ) 2 3 ( M ( η 2 h 2 ) 3 ) 1 3
= ϕ 1 ( η ) 2 ( M ( η h ) 6 ) 1 3
C 0 ϕ 1 ( η ) 2 ( M | ( η h ) | 2 + 9 M ( H 2 + 1 ) ( η h ) 2 )
C 0 ϕ 1 ( η ) 2 ( ( 1 + 1 b 1 ) M h 2 | η | 2 + ( 1 + b 1 ) M η 2 | h | 2 + 9 M ( H 2 + 1 ) ( η h ) 2 ) , (14)

for any positive real number b1, where the second inequality holds because of Proposition 6.By (12)-(14), we obtain that

𝒜 1 M η 2 | h | 2 + 1 M H 2 η 2 h 2 + 𝒞 1 M η 2 h 2 𝒟 1 M h 2 | η | 2 , (15)

where

𝒜 1 :
= ( 2 - C 0 ϕ 1 ( η ) 2 ) - ( a 1 + b 1 C 0 ϕ 1 ( η ) 2 ) ,
1 :
= 3 2 - 9 C 0 ϕ 1 ( η ) 2 ,
𝒞 1 :
= 2 - 9 C 0 ϕ 1 ( η ) 2

and

𝒟 1 := 1 a 1 + C 0 ϕ 1 ( η ) 2 ( 1 + 1 b 1 ) .

Since the total curvature ΦL3(M) is finite, we can choose a fixedr0 such that

Φ L 3 ( M - B r 0 ) < δ 1 = 1 12 C 0 .

Set

𝒜 ~ 1 :
= ( 2 - C 0 δ 1 2 ) - ( a 1 + b 1 C 0 δ 1 2 ) ,
~ 1 :
= 3 2 - 9 C 0 δ 1 2 ,
𝒞 ~ 1 :
= 2 - 9 C 0 δ 1 2

and

𝒟 ~ 1 := 1 a 1 + C 0 δ 1 2 ( 1 + 1 b 1 ) .

Thus,

𝒜 ~ 1 M η 2 | h | 2 + ~ 1 M H 2 η 2 h 2 + 𝒞 ~ 1 M η 2 h 2 𝒟 ~ 1 M h 2 | η | 2 , (16)

for any ηC0(M-Br0).By Proposition 6, we have

1 C 0 ( M ( η h ) 6 ) 1 3 M | ( η h ) | 2 + 9 M ( H 2 + 1 ) ( η h ) 2
( 1 + 1 c 1 ) M h 2 | η | 2 + ( 1 + c 1 ) M η 2 | h | 2 + 9 M ( H 2 + 1 ) ( η h ) 2 , (17)

for any positive real number c1.By (16) and (17), we have

1 C 0 ( M ( η h ) 6 ) 1 3
( 1 + 1 c 1 ) M h 2 | η | 2 + ( 1 + c 1 ) M η 2 | h | 2 + 9 M ( H 2 + 1 ) ( η h ) 2
( 1 + 1 c 1 + ( 1 + c 1 ) 𝒟 ~ 1 𝒜 ~ 1 ) M h 2 | η | 2 + ( 9 - ( 1 + c 1 ) ~ 1 𝒜 ~ 1 ) M H 2 η 2 h 2
+ ( 9 - ( 1 + c 1 ) 𝒞 ~ 1 𝒜 ~ 1 ) M η 2 h 2 . (18)

Choose a sufficient large c1 such that

9 - ( 1 + c 1 ) ~ 1 𝒜 ~ 1 < 0

and

9 - ( 1 + c 1 ) 𝒞 ~ 1 𝒜 ~ 1 < 0.

Then (18) implies that

( M ( η h ) 6 ) 1 3 A ~ M h 2 | η | 2 , (19)

for any ηC0(M-Br0). where A~ is a positive constant.

Case II: n4. By Proposition 7 and (11), we obtain that

-
2 M η h η , h - n - 1 n - 2 M η 2 | h | 2 - 2 ( n - 2 ) M η 2 h 2
+ n - 2 2 M | Φ | 2 η 2 h 2 - n M H 2 h 2 η 2 0. (20)

Note that

- 2 M η h η , h a 2 M η 2 | h | 2 + 1 a 2 M h 2 | η | 2 , (21)

for any positive real number a2. Weset ϕ2(η)=(Suppη|Φ|n)1n and obtain that

M | Φ | 2 η 2 h 2 ( S u p p η ( | Φ | 2 ) n 2 ) 2 n ( M ( η 2 h 2 ) n n - 2 ) n - 2 n
= ϕ 2 ( η ) 2 ( M ( η h ) 2 n n - 2 ) n - 2 n
C 0 ϕ 2 ( η ) 2 ( M | ( η h ) | 2 + n 2 M ( H 2 + 1 ) ( η h ) 2 )
C 0 ϕ 2 ( η ) 2 ( M ( 1 + 1 b 2 ) h 2 | η | 2 + ( 1 + b 2 ) η 2 | h | 2 + n 2 M ( H 2 + 1 ) ( η h ) 2 ) , (22)

for any positive real number b2, where the second inequality holds because of Proposition 6.By (20)-(22), there exists

𝒜 2 M η 2 | h | 2 + 2 M H 2 η 2 h 2 + 𝒞 2 M η 2 h 2 𝒟 2 M h 2 | η | 2 , (23)

where

𝒜 2 :
= ( n - 1 n - 2 - n - 2 2 C 0 ϕ 2 ( η ) 2 ) - ( a 2 + n - 2 2 b 2 C 0 ϕ 2 ( η ) 2 ) ,
2 :
= n - n 2 ( n - 2 ) 2 C 0 ϕ 2 ( η ) 2 ,
𝒞 2 :
= 2 ( n - 2 ) - n 2 ( n - 2 ) 2 C 0 ϕ 2 ( η ) 2

and

𝒟 2 := 1 a 2 + n - 2 2 ( 1 + 1 b 2 ) C 0 ϕ 2 ( η ) 2 .

Since the total curvature ΦLn(M) is finite, we can choose a fixedr0 such that

Φ L n ( M - B r 0 ) < δ 2 = 1 n ( n - 2 ) C 0 .
𝒜 ~ 2 :
= ( n - 1 n - 2 - n - 2 2 C 0 δ 2 2 ) - ( a 2 + n - 2 2 b 2 C 0 δ 2 2 ) ,
~ 2 :
= n - n 2 ( n - 2 ) 2 C 0 δ 2 2 ,
𝒞 ~ 2 :
= 2 ( n - 2 ) - n 2 ( n - 2 ) 2 C 0 δ 2 2

and

𝒟 ~ 2 := 1 a 2 + n - 2 2 ( 1 + 1 b 2 ) C 0 δ 2 2 .

Obviously, 𝒜~2, ~2, 𝒞~2 and 𝒟~2 are positive.Thus,

𝒜 ~ 2 M η 2 | h | 2 + ~ 2 M H 2 η 2 h 2 + 𝒞 ~ 2 M η 2 h 2 𝒟 ~ 2 M h 2 | η | 2 , (24)

for any ηC0(M-Br0).Combining with Proposition 6, we get that

1 C 0
( M | η h | 2 n n - 2 ) n - 2 n M | ( η h ) | 2 + n 2 M ( H 2 + 1 ) ( η h ) 2
( 1 + c 2 ) M η 2 | h | 2 + ( 1 + 1 c 2 ) M h 2 | η | 2 + n 2 M ( H 2 + 1 ) η 2 h 2 , (25)

for any positive real number c2.By (24) and (25), we have

1 C 0 ( M | η h | 2 n n - 2 ) n - 2 n
( 1 + 1 c 2 + ( 1 + c 2 ) 𝒟 ~ 2 𝒜 ~ 2 ) M h 2 | η | 2 + ( n 2 - ( 1 + c 2 ) ~ 2 𝒜 ~ 2 ) M H 2 η 2 h 2
+ ( n 2 - ( 1 + c 2 ) 𝒞 ~ 2 𝒜 ~ 2 ) M η 2 h 2 . (26)

We choose a sufficient large c2 such that

n 2 - ( 1 + c 2 ) ~ 2 𝒜 ~ 2 < 0

and

n 2 - ( 1 + c 2 ) 𝒞 ~ 2 𝒜 ~ 2 < 0.

Then (26) implies that

( M ( η h ) 2 n n - 2 ) n - 2 n A ~ M h 2 | η | 2 , (27)

for any ηC0(M-Br0), where A~ is a positive constant depending only on n.

By Case I and Case II, we have that

( M ( η h ) 2 n n - 2 ) n - 2 n A ~ M h 2 | η | 2 , (28)

for any ηC0(M-Br0), where A~ is a positive constant depending only on n (n3).

Next, the proof follows standard techniques (after inequality (33) in Cavalcante et al. (2014r3 CAVALCANTE MP, MIRANDOLA H AND VITÓRIO F. 2014. L2-harmonic 1-forms on submanifolds with finite total curvature. J Geom Anal 24: 205-222. and uses a Moser iteration argument (lemma 11 in Li (1980r10 LI P. 1980. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann Sci Éc Norm Super 13: 451-468.)).We include a concise proof here for the sake of completeness.Choose r>r0+1 and ηC0(M-Br0) such that

{ η = 0 o n B r 0 ( M - B 2 r ) , η = 1 o n B r - B r 0 + 1 , | η | < c ~ o n B r 0 + 1 - B r 0 , | η | c ~ r - 1 o n B 2 r - B r ,

for some positive constant c~. Then (28) becomes that

( B r - B r 0 + 1 h 2 n n - 2 ) n - 2 n A ~ B r 0 + 1 - B r 0 h 2 + A ~ r 2 B 2 r - B r h 2 .

Letting r and noting that hL2(M), we obtain that

( M - B r 0 + 1 h 2 n n - 2 ) n - 2 n A ~ B r 0 + 1 - B r 0 h 2 . (29)

By Hölder inequality

B r 0 + 2 - B r 0 + 1 h 2 ( B r 0 + 2 - B r 0 + 1 h 2 n n - 2 ) n - 2 n ( B r 0 + 2 - B r 0 + 1 1 n 2 ) 2 n ,

we get that

B r 0 + 2 h 2 ( 1 + A ~ V o l ( B r 0 + 2 ) 2 n ) B r 0 + 1 h 2 . (30)

Set

Ψ = { | 2 - | Φ | 2 + 3 2 H 2 | , f o r n = 3 , | 2 ( n - 2 ) - n - 2 2 | Φ | 2 + n H 2 | , f o r n 4.

Fix xM and take τC01(B1(x)). Proposition 7 implies that

h h α | h | 2 - Ψ h 2 ,

where

α = { 1 2 , f o r n = 3 , 1 n - 2 , f o r n 4.

Then, for p>2, there exists

M τ 2 h p - 1 h α M τ 2 h p - 2 | h | 2 - M τ 2 Ψ h p .

That is,

- 2 B 1 ( x ) τ h p - 1 τ , h
( α + ( p - 1 ) ) B 1 ( x ) τ 2 h p - 2 | h | 2
- B 1 ( x ) τ 2 Ψ h p . (31)

Note that

- 2 τ h p - 1 τ , h
= - 2 h p 2 τ , τ h p 2 - 1 h
1 α h p | τ | 2 + α τ 2 h p - 2 | h | 2 .

Combining with (31), we obtain that

( p - 1 ) B 1 ( x ) τ 2 h p - 2 | h | 2 B 1 ( x ) Ψ τ 2 h p + 1 α B 1 ( x ) | τ | 2 h p . (32)

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

B 1 ( x ) | ( τ h p 2 ) | 2 B 1 ( x ) 𝒜 Ψ τ 2 h p + | τ | 2 h p , (33)

where 𝒜=1p-1(p24+p2) and =(1+p2)+1α(p-1)(p24+p2).Choose f=τhp2 in Proposition 6. Combining with (33), we obtain that

( B 1 ( x ) ( τ h p 2 ) 2 n n - 2 ) n - 2 2 p 𝒞 B 1 ( x ) ( τ 2 + | τ | 2 ) h p , (34)

where 𝒞 depends on n and supB1(x)Ψ.Set pk=2nk(n-2)k and ρk=12+12k+1 for k=0,1,2,.Take a function τkC0(Bρk(x)) satisfying:

{ 0 τ k 1 , τ k = 1 o n B ρ k + 1 ( x ) , | τ k | 2 k + 3 .

Choosing p=pk and τ=τk in (34), we obtain that

( B ρ k + 1 ( x ) h p k + 1 ) 1 p k + 1 ( 𝒞 p k 4 k + 4 ) 1 p k ( B ρ k ( x ) h p k ) 1 p k . (35)

By recurrence, we have

h L p k + 1 ( B 1 2 ( x ) ) i = 0 k p i 1 p i 4 i p i ( 𝒞 4 4 ) 1 p i h L 2 ( B 1 ( x ) ) 𝒟 h L 2 ( B 1 ( x ) ) , (36)

where 𝒟 is a positive constant depending only on n, Vol(Br0+2) and supBr0+2Ψ. Letting k,we get

h L ( B 1 2 ( x ) ) 𝒟 h L 2 ( B 1 ( x ) ) . (37)

Now, choose yB¯r0+1 such that supBr0+1h2=h(y)2. Note that B1(y)Br0+2.(37) implies that

sup B r 0 + 1 h 2 𝒟 h L 2 ( B 1 ( y ) ) 2 𝒟 h L 2 ( B r 0 + 2 ) 2 . (38)

By (30), we have

sup B r 0 + 1 h 2 h L 2 ( B r 0 + 1 ) 2 , (39)

where depends only on n, Vol(Br0+2) and supBr0+2Ψ.In order to show the finiteness of the dimension of H2(L2(M)), it suffices to provethat the dimension of any finite dimensional subspaces of H2(L2(M)) is bounded above by a fixed constant. Combining (39)with Lemma 11 in Li (1980r10 LI P. 1980. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann Sci Éc Norm Super 13: 451-468.), we show that dimH2(L2(M))<+.By Proposition 4, we obtain that the dimension of the second space of reduced L2cohomology of M is finite.

Remark 9.For the case of n=3, Theorem 1 can also be obtained by a different method. In fact, Yau (1976r13 YAU ST. 1976. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ Math J 25: 659-670.) proved that if ωH2(L2(M)), thenω is closed and coclosed. By use of the Hodge-* operator, we obtain the dimensions of H2(L2(M)) and H1(L2(M)) are equal.By Theorem 1.1 in Zhu and Fang (2014r16 ZHU P AND FANG SW. 2014. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Global Anal Geom 46: 187-196.), we obtain the desired result.

ACKNOWLEDGMENTS

The author would like to thank professor Detang Zhou for useful suggestions.The work was partially supported by NSFC Grants 11471145, 11371309 and Qing Lan Project.

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Publication Dates

  • Publication in this collection
    Dec 2016

History

  • Received
    6 Feb 2015
  • Accepted
    17 June 2016