On reduced L 2 cohomology of hypersurfaces in spheres with finite total curvature

For a complete manifold Mn, the p-th space of reduced L2-cohomology is defined, for 0 ≤ p ≤ n in Carron (2007). It is interesting and important to discuss the finiteness of the dimension of these spaces. Carron (1999) proved that if Mn (n ≥ 3) is a complete noncompact submanifold of Rn+p with 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 onM has finite dimension, for 0 ≤ p ≤ n. The reduced L2 cohomology is related with the L2 harmonic forms (Carron 2007). In fact, several mathematicians studied the space of L2 harmonic p-forms for p = 1, 2. If Mn (n ≥ 3) is a complete minimal hypersurface in Rn+1 with finite index, Li and Wang (2002) 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 (2013) showed that: suppose that Nn+1 (n ≥ 3) is a complete simply connected manifold with non-positive sectional curvature and Mn is a complete minimal hypersurface in N with finite index. If the bi-Ricci curvature satisfies

PENG ZHU space of L 2 harmonic functions.Cavalcante et al. (2014) discussed a complete noncompact submanifold M n (n ≥ 3) isometrically immersed in a Hadamard manifold N n+p with sectional curvature satisfying −k 2 ≤ K N ≤ 0 for some constant k and showed that if the total curvature is finite and 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 L 2 harmonic 1-forms on M is finite.Fu and Xu (2010) studied a complete submanifold M n in a sphere S n+p with finite total curvature and bounded mean curvature and proved that the dimension of the space of the L 2 harmonic 1-forms on M is finite.Zhu and Fang (2014) proved Fu-Xu's result without the restriction on the mean curvature vector and therefore obtained that the first space of reduced L 2 -cohomology on M has finite dimension.Zhu (2011) studied the existence of the symplectic structure and L 2 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 M n in a sphere S n+1 with finite total curvature in this paper.We obtain the following finiteness results on the space of all L 2 harmonic 2-forms and the second space of reduced L 2 cohomology: Theorem 1.Let M n (n ≥ 3) be an n-dimensional complete noncompact oriented manifold isometrically immersed in an (n + 1)-dimensional sphere S n+1 .If the total curvature is finite, then the space of all L 2 harmonic 2-forms has finite dimension.
Corollary 2. Let M n (n ≥ 3) be an n-dimensional complete noncompact oriented manifold isometrically immersed in S n+1 .If the total curvature is finite, then the dimension of the second space of reduced L 2 cohomology of M is finite.
Remark 3.Under the same condition of Corollary 2, we conjecture that the p-th space of reduced L 2 cohomology of M has finite dimension for 3 ≤ p ≤ n − 3.

PRELIMINARIES
In this section, we recall some relevant definitions and results.Suppose that M n is an n-dimensional complete Riemannian manifold.The Hodge operator * : The Laplacian operator is defined by We denote by H p (L 2 (M )) the space of all L 2 harmonic p-forms on M .Let We define the p-th space of reduced L 2 cohomology by .
Suppose that x : M n → S n+1 is an isometric immersion of an n-dimensional manifold M in an (n + 1)-dimensional sphere.Let A denote the second fundamental form and H the mean curvature of the immersion for all vector fields X and Y , where , is the induced metric of M .We say the immersion x has finite total curvature if We state several results which will be used to prove Theorem 1.
Proposition 4. (Carron 2007) Let (M, g) is a complete Riemannian manifold, then the space of L 2 harmonic p-forms H p (L 2 (M )) is isomorphic to the p-th space of reduced L 2 cohomology H p 2 (M ).
Lemma 5. (Li 1993) If (M n , g) is a Riemannian manifold and ω = a I ω I ∈ ∧ p (M ), then where Proposition 6. (Hoffman andSpruck 1974, Zhu andFang 2014) Let M n be a complete noncompact oriented manifold isometrically immersed in a sphere S n+1 .Then for each f ∈ C 1 0 (M ), where C 0 depends only on n and H is the mean curvature of M in S n+1 .

AN INEQUALITY FOR L 2 HARMONIC 2-FORMS
In this section, we show an inequality for L 2 harmonic 2-forms on hypersurfaces in a sphere S n+1 , which plays an important role in the proof of main results.

PENG ZHU
Proof.Suppose that ω ∈ H 2 (L 2 (M )).Then we have By Lemma 5, we get that: Combining ( 1) with ( 2), we obtain that There exists the Kato inequality for L 2 harmonic 2-forms as follows (Cibotaru and Zhu 2012, Wang 2002): By ( 3) and ( 4), we get that Now, we give the estimate of the term By Lemma 5, we obtain that So, we get that By Gauss equation, we have that A direct computation shows that and Since the curvature operator E is linear and zero order, and hence tensorial, it is sufficient to compute E(ω), ω at a point p.We can choose an orthonormal frame {e i } such that h ij = λ i δ ij at p. Obviously, By ( 6)-( 10), we have Note that For n = 3, we have that An Acad Bras Cienc (2016) 88 (4)

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PENG ZHU For n ≥ 4, we obtain that By (5), we have that: for n ≥ 4.

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 Integrating by parts on M , we obtain that An Acad Bras Cienc (2016) 88 (4) ON REDUCED L 2 COHOMOLOGY

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Case I: n = 3.By Proposition 7 and (11), we obtain that for any positive real number a 1 .Now we give an estimate of the term M |Φ| 2 η 2 h 2 as follows: set φ 1 (η) = Suppη |Φ| 3 1 3 .Then there exists and Since the total curvature Φ L 3 (M ) is finite, we can choose a fixed r 0 such that Choose a sufficient large c 1 such that Then (18) implies that where Ã is a positive constant.Case II: n ≥ 4. By Proposition 7 and (11), we obtain that for any positive real number a 2 .We set φ 2 (η) = Suppη |Φ| n 1 n and obtain that for any positive real number b 2 , where the second inequality holds because of Proposition 6.By ( 20)-( 22), there exists where and Since the total curvature Φ L n (M ) is finite, we can choose a fixed r 0 such that We choose a sufficient large c 2 such that and Then (26) implies that for any η ∈ C ∞ 0 (M − B r0 ), where Ã is a positive constant depending only on n.By Case I and Case II, we have that , where Ã is a positive constant depending only on n (n ≥ 3).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 r > r 0 + 1 and η for some positive constant c.Then (28) becomes that Letting r → ∞ and noting that h ∈ L 2 (M ), we obtain that By Hölder inequality we get that An Acad Bras Cienc (2016) 88 (4)

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PENG ZHU Combining with (31), we obtain that Combining Cauchy-Schwarz inequality with (32), we obtain that where in Proposition 6. Combining with (33), we obtain that where C depends on n and sup B1(x) Ψ. Set Choosing p = p k and τ = τ k in (34), we obtain that By recurrence, we have where D is a positive constant depending only on n, V ol(B r0+2 ) and sup Br 0 +2 Ψ. Letting k → ∞, we get Now, choose y ∈ B r0+1 such that sup Br 0 +1 h 2 = h(y) 2 .Note that B 1 (y) ⊂ B r0+2 .(37) implies that sup By (30), we have sup where F depends only on n, V ol(B r0+2 ) and sup Br 0 +2 Ψ.In order to show the finiteness of the dimension of H 2 (L 2 (M )), it suffices to prove that the dimension of any finite dimensional subspaces of H 2 (L 2 (M )) is bounded above by a fixed constant.Combining (39) with Lemma 11 in Li (1980), we show that dim H 2 (L 2 (M )) < +∞.By Proposition 4, we obtain that the dimension of the second space of reduced L 2 cohomology of M is finite.
An Acad Bras Cienc (2016) 88 (4) ON REDUCED L 2 COHOMOLOGY 2065 Remark 9.For the case of n = 3, Theorem 1 can also be obtained by a different method.In fact, Yau (1976) proved that if ω ∈ H 2 (L 2 (M )), then ω is closed and coclosed.By use of the Hodge- * operator, we obtain the dimensions of H 2 (L 2 (M )) and H 1 (L 2 (M )) are equal.By Theorem 1.1 in Zhu and Fang (2014), we obtain the desired result.