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 , the -th space of reduced -cohomology is defined, for 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 is a complete noncompact submanifold of with finite total curvature and finite mean curvature (i. e., the -norm of the mean curvature vector is finite),then each -th space of reduced -cohomology on has finite dimension, for .The reduced cohomology is related with the 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 harmonic -forms for .If is a complete minimal hypersurface in 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 harmonic -forms is finite and 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 is a complete simply connected manifold with non-positive sectional curvature and is a complete minimal hypersurface in with finite index. If the bi-Ricci curvature satisfies
for all orthonormal tangent vectors in for , then the dimension of the space of the harmonic -forms 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 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 isometrically immersed in a Hadamard manifold with sectional curvature satisfying for some constant and showed that if the total curvature is finiteand the first eigenvalue of the Laplacian operator of is bounded from below by a suitable constant, then the dimension of the space of the harmonic -forms on 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 in a sphere with finite total curvature and bounded mean curvature and proved that the dimension of the space of the harmonic -forms on 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 -cohomology on 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 harmonic -forms on complete noncompact manifolds by use of a special version of Bochner formula.
Motivated by above results, we discuss a complete noncompact hypersurface in a sphere with finite total curvature in this paper.We obtain the following finiteness results on the space of all harmonic -forms and the second space of reduced cohomology:
Theorem 1. Let be an -dimensional complete noncompact oriented manifold isometrically immersed in an -dimensional sphere. If the total curvature is finite, then the space of all harmonic -forms has finite dimension.
Corollary 2. Let be an -dimensional complete noncompact oriented manifold isometrically immersed in. If the total curvature is finite, then the dimension of the second space of reduced cohomology of is finite.
Remark 3.Under the same condition of Corollary 2, we conjecture thatthe p-th space of reduced cohomology of has finite dimension for .
PRELIMINARIES
In this section, we recall some relevant definitions and results. Suppose that is an -dimensional complete Riemannian manifold.The Hodge operator is defined by
where denotes a permutation ofthe set and is the sign of. The operator is given by
The Laplacian operator is defined by
A -form is called harmonic if and
We denote by the space of all harmonic -forms on .Let
and
We define the -th space of reduced cohomology by
Suppose that is an isometric immersion of an -dimensional manifold in an -dimensionalsphere.Let denote the second fundamental form and the mean curvature of the immersion .Let
for all vector fields and , where is the induced metric of .We say the immersion has finite total curvature if
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 is a complete Riemannian manifold, then the space of harmonic -forms is isomorphic tothe -th space of reduced cohomology .
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 is a Riemannianmanifold and , then
where
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 be a complete noncompact oriented manifold isometrically immersed in a sphere. Then
for each , where depends only on and is the mean curvature of in .
AN INEQUALITY FOR HARMONIC -FORMS
In this section, we show an inequality for harmonic -forms on hypersurfaces in a sphere ,which plays an important role in the proof of main results.
Proposition 7. Let be an -dimensional complete noncompact hypersurface isometrically immersed in an -dimensional sphere . If , then
for and
for , where .
Proof. Suppose that . Then we have
By Lemma 5, we get that:
Combining (1) with (AN INEQUALITY FOR HARMONIC -FORMS), we obtain that
There exists the Kato inequality for harmonic -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.):
By (3) and (4), we get that
Now, we give the estimate of the term . Let and, where and . By Lemma 5, we obtain that
So, we get that
which implies that
By Gauss equation, we have that
A direct computation shows that
and
Since the curvature operator is linear andzero order, and hence tensorial, it is sufficient to compute at a point .We can choose an orthonormal frame such that at . Obviously,
By (6)-(10), we have
Note that
For , we have that
For , we obtain that
By (5), we have that:
for and
for .∎
Remark 8. If is -form , then the term is equal to .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 is a compactly supported piecewise smooth function on , then
Integrating by parts on , we obtain that
Case I: . By Proposition 7 and (11), we obtain that
Note that
for any positive real number .Now we give an estimate of the term as follows:set . Then there exists
for any positive real number , where the second inequality holds because of Proposition 6.By (12)-(14), we obtain that
where
and
Since the total curvature is finite, we can choose a fixed such that
Set
and
Thus,
for any .By Proposition 6, we have
for any positive real number .By (16) and (17), we have
Choose a sufficient large such that
and
Then (18) implies that
for any . where is a positive constant.
Case II: . By Proposition 7 and (11), we obtain that
Note that
for any positive real number . Weset and obtain that
for any positive real number , where the second inequality holds because of Proposition 6.By (20)-(22), there exists
where
and
Since the total curvature is finite, we can choose a fixed such that
and
Obviously, , , and are positive.Thus,
for any .Combining with Proposition 6, we get that
for any positive real number .By (24) and (25), we have
We choose a sufficient large such that
and
Then (26) implies that
for any , where is a positive constant depending only on .
By Case I and Case II, we have that
for any , where is a positive constant depending only on .
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 and such that
for some positive constant . Then (28) becomes that
Letting and noting that , we obtain that
By Hölder inequality
we get that
Set
Fix and take . Proposition 7 implies that
where
Then, for , there exists
That is,
Note that
Combining with (31), we obtain that
Combining Cauchy-Schwarz inequality with (32), we obtain that
where and Choose in Proposition 6. Combining with (33), we obtain that
where depends on and .Set and for .Take a function satisfying:
Choosing and in (34), we obtain that
By recurrence, we have
where is a positive constant depending only on , and . Letting ,we get
Now, choose such that . Note that .(37) implies that
By (30), we have
where depends only on , and .In order to show the finiteness of the dimension of , it suffices to provethat the dimension of any finite dimensional subspaces of 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 .By Proposition 4, we obtain that the dimension of the second space of reduced cohomology of is finite.
Remark 9.For the case of , 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 , then is closed and coclosed. By use of the Hodge- operator, we obtain the dimensions of and 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.
REFERENCES
-
r2CARRON G. 1999. L2-Cohomologie et inégalités de Sobolev. Math Ann 314: 613-639.
-
r1CARRON G. 2007. L2 harmonic forms on non compact manifolds. arXiv:0704.3194v1.
-
r3CAVALCANTE MP, MIRANDOLA H AND VITÓRIO F. 2014. L2-harmonic 1-forms on submanifolds with finite total curvature. J Geom Anal 24: 205-222.
-
r4CHENG X AND ZHOU DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154.
-
r5CIBOTARU D AND ZHU P. 2012. Refined Kato inequalities for harmonic fields on Kähler manifolds. Pacific J Math 256: 51-66.
-
r7FU HP AND XU HW. 2010. Total curvature and L2 harmonic 1-forms on complete submanifolds in space forms. Geom Dedicata 144: 129-140.
-
r6HOFFMAN D AND SPRUCK J. 1974. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm Pure Appl Math 27: 715-727.
-
r8LEUNG PF. 1992. An estimate on the Ricci curvature of a submanifold and some applications. Proc Amer Math Soc 114: 1051-1061.
-
r10LI P. 1980. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann Sci Éc Norm Super 13: 451-468.
-
r9LI 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.
-
r11LI P AND WANG JP. 2002. Minimal hypersurfaces with finite index. Math Res Lett 9: 95-103.
-
r12WANG XD. 2002. On the L2-cohomology of a convex cocompact hyperbolic manifold. Duke Math J 115: 311-327.
-
r13YAU ST. 1976. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ Math J 25: 659-670.
-
r14ZHU P. 2011. Harmonic two-forms on manifolds with nonnegative isotropic curvature. Ann Global Anal Geom 40: 427-434.
-
r15ZHU P. 2013. L2-harmonic forms and finiteness of ends. An Acad Bras Cienc 85: 457-471.
-
r16ZHU P AND FANG SW. 2014. Finiteness of non-parabolic ends on submanifolds in spheres. Ann Global Anal Geom 46: 187-196.
Publication Dates
-
Publication in this collection
Dec 2016
History
-
Received
6 Feb 2015 -
Accepted
17 June 2016