Abstracts
In this paper, we obtain vanishing theorems and finitely many ends theorems of complete Riemannian manifolds with weighted Poincaré inequality, applying them to minimal hypersurfaces.
end; minimal hypersurface; weighted Poincaré inequality; L2 harmonic forms; complete manifolds
Neste artigo, obtemos teoremas de anulamento e de número finito de extremidades sobre variedades Riemannianas completas com desigualdade de Poincaré ponderada, aplicando-os a superfícies mínimas.
extremidade; superfície mínima; desigualdade de Poincaré ponderada; formas L 2 harmônicas; variedades completas
INTRODUCTION
Li and Wang (2001)Li P and Wang JP. 2001. Complete manifolds with positive spectrum I. J Diff Geom 58: 501-534. proved a vanishing theorem for L 2 harmonic one-forms on M under the assumption of the Ricci curvature bounded from below in terms of λ 1(M), which is the greatest lower bound of the spectrum of the Laplacian acting on L 2 functions. Lam (2010)Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062. generalized this result to the manifold satisfying a weighted Poincaré inequality. Recalling that in Li and Wang (2006)Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982., a complete Riemannian manifold (M, ds 2) is said to satisfy a weighted Poincaré inequality with a non-negative weighted function ρ if the inequality
(1.1)holds for all compactly supported piecewise smooth function . (M, ds 2) is is said to satisfy the property () if M satisfies a weighted Poincaré inequality with ρ and ρds2 is complete. If M is a complete manifold with property (), Li and Wang (2006)Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982. -Theorem 5.2- gave a rigidity theorem under the assumption of the Ricci curvature bounded from below in terms of ρ. Cheng and Zhou (2009)Cheng X and Zhou DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154. generalized this result and applied to minimal hypersurfaces. Lam (2010)Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062. also obtained a result similar to Theorem 5.2 in Li and Wang (2006)Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982. by relaxing Ricci curvature to be only satisfied outside a compact set of M.
In this paper, on the one hand, we obtain vanishing theorems of two classes of complete manifolds that satisfy weighted Poincaré inequalities (Theorems 2.2 and 2.7). At the same time, we apply Theorem 2.2 to minimal hypersurfaces (Corollaries 2.4 and 2.5). On the other hand, we also obtain two results on finitely many ends of complete manifolds (Theorems 3.1 and 3.6). As an application, we obtain that finitely many ends of minimal hypersurfaces in non-positively curved manifolds (Corollaries 3.7 and 3.9).
2 VANISHING THEOREMS
In this section, we study H 1(L 2(M)), the space of L 2 integrable harmonic 1-forms. The following lemma is due to Lam (Lam 2010Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062.):
Lemma 2.1.Suppose that ω ∈ H 1(L 2(M)) and h = |ω| satisfies that
for a function a on M and some positive constant b. Then, for anyand, the following inequality holds
Theorem 2.2.Suppose that M is an n-dimensional complete non-compact Riemannian manifold satisfying the weighted Poincaré inequality with a non-negative weight function ρ (n ≥ 2). Assume the Ricci curvature satisfies
(2.1)for a non-negative continuous function σ (σ ≠ 0). If ρ(x)=O(r 2−α), where rp(x) is the distance function from x to some fixed point p, for some 0 ≪ α ≪ 2, then H 1(L 2(M)) = {0}.
Proof. If ω ∈ H 1(L 2(M)), then h = |ω| satisfies a formula (Li and Wang 2006Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982.):
(2.2)By Lemma 2.1 with b = (n − 1)−1 and a = (b + 1)ρ − σ, we obtain that
that is,
(2.3)for any and . Choose (R sufficient large) and
such that
on B(2R) \ B(R). By (2.3) and , we obtain that
Let R→ +∞. We obtain that
since h ∈ L 2(M). By assumption, there exist p 0 ∈ M and r 0 > 0 such that σ > 0 on B(p 0, r 0). Thus,
It implies that ω = 0 on B(p 0, r 0). Therefore, ω = 0 on M, by uniqueness of expanding theorem of harmonic forms.
Remark 2.3.When σ is equal to a positive constant and cρ for some positive constant c and a positive weight function ρ, respectively, Theorem 2.2 is just Theorem 3.4 and Theorem 3.5 in Lam (2010)Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062., respectively. We should also point out that Carron (2002)Carron G. 2002. L2 harmonic forms on non-compact Riemannian manifold. In: Surveys in analysis and operator theory (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ. n. 40. Canberra: Austral Nat Univ, p. 49-59. gave a very nice survey about L2 harmonic k-forms on non-compact Riemannian manifolds.
Let Mn be a minimal hypersurface of . v denotes the unit normal vector field of M. |A| is the normal of the second fundamental form A. A minimal hypersurface is called δ-stable if, for each ,
(2.4)Corollary 2.4. Suppose that Mn (n ≥ 2) is a complete minimal δ-stable () hypersurface in . If , for some 0 ≪ α ≪ 2, then H 1(L 2(M)) = {0}.
Proof. First, a complete minimal hypersurface in is non-compact. For any point p ∈ M and any unit tangent vector v ∈ TpM, we can choose an orthonormal frame {e 1, e 2, ..., en } on M at p such that e 1 = v. Since M is a minimal hypersurface, there has the following inequality:
(2.5)The Gauss equation implies that
(2.6)From (2.5) and (2.6), we obtain that (2.1) holds, where
and
Since M is δ-stable, (1.1) holds. If |A| = 0 then H 1(L 2(M)) = {0}. Otherwise, by Theorem 2.2, we have H 1(L 2(M)) = {0}.
If Mn is a minimal hypersurface in , then each end of M is non-parabolic and the number of non-parabolic end of M is bounded from above by dim H 1(L 2(M)) +1 (Li and Wang 2002Li P and Wang JP. 2002. Minimal hypersurfaces with finite index. Math Res Lett 9: 95-103.). Thus, Corollary 2.4 implies that:
Corollary 2.5.Suppose that Mn (n ≥ 3) is a complete minimal δ-stable () hypersurface in . If it has the bounded norm of the second fundamental form, then M has only one end.
Remark 2.6.Cheng and Zhou (2009)Cheng X and Zhou DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154. proved that: let M be an -stable complete minimal hypersurface . If it has bounded norm of its second fundamental form, then M either has only one end or is a catenoid. In Cheng et al. (2008)Cheng X, Cheung LF and Zhou DT. 2008. The structure of weakly stable constant mean curature hypersurfaces. Tohoku Math J 60: 101-121., it was proved that a complete oriented weakly minimal hypersurface in must have only one end. It was shown that in Cheng (2008)Cheng X. 2008. On end theorem and application to stable minimal hypersurfaces. Arch Math 90: 461-470. one end theorem holds for complete noncompact stable minimal hypersurfaces in general manifolds and the method is different from above.
If M is a quaternionic manifold and ω ∈ H 1(L 2(M)), then h = |ω| satisfies (Kong et al. 2008Kong SL, Li P and Zhou DT. 2008. Spectrum of the Laplacian on quaternionic Kähler manifolds. J Diff Geom 78: 295-332.):
Using the similar method as proof of Theorem 2.2, we also can obtain:
Theorem 2.7.Suppose that M is a 4n-dimensional complete non-compact quaternionic manifold satisfying the weighted Poincaré inequality with a nonnegative weight function ρ(x). Assume the Ricci curvature satisfies
for a positive function σ. If
for some 0 ≪ α ≪ 2, then H 1(L 2(M)) = {0}.
Remark 2.8.If we choose σ is equal to positive constant and ρ = λ 1(M) > 0, Theorems 2.7 is Theorem 4.3 in Lam (2010)Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062..
3 FINITENESS OF ENDS
In this section, we study complete Riemannian manifolds with a weighted Poincaré inequality, giving two results on finitely many ends of complete manifolds. Then, we apply one of the results to hypersurfaces in manifolds with non-positive sectional curvature. First, following the idea of Cheng and Zhou (2009)Cheng X and Zhou DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154. with some changes of technique, we obtain that:
Theorem 3.1.Let M be a complete n-dimensional manifold with property (Pρ ) (n ≥ 3). Suppose that the Ricci curvature of M satisfies that
for a constant ϵ > 0 and Ω is a compact subset of M, where the non-negative function τ (x) satisfies Poincaré inequality
for all. Set
If ρ and τ satisfy the growth estimates
then M has only finitely many non-parabolic ends.
Remark 3.2.Ifthen Theorem 3.1 is just Theorem 0.2 in Lam (2010)Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062..
Now we prove Theorem 3.1. A theory of Li-Tam (Li and Tam 1991Li P and Tam LF. 1991. The heat equation and harmonic maps of complete manifolds. Invent Math 105: 1-46.) gives the number of non-parabolic ends of M. More precisely, suppose that M has at least two non-parabolic ends E 1 and E2. There exists a non-constant harmonic function f 1 with finite Dirichlet integral by taking a convergent subsequence of harmonic functions fR as R → +∞, satisfying Δ fR = 0 in B(R) with boundary conditions that and fR = 0 on ∂B(R) \ E 1. It follows from the Maximum Principle that 0 ≤ fR ≤ 1 for all R and hence 0 ≤ f 1 ≤ 1. For each non-parabolic ends Ei , we can construct a corresponding fi by the above method. Let be the linear space containing all fi 's constructed as above. Thus, the number of non-parabolic ends of M is dim . The following lemma is due to Li (Li 1993Li 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.):
Lemma 3.3.Suppose thatis a finite dimensional subspace of L2 1-forms defined over a set . Then there exists such that
where V (D) is the volume of the set D.
Let
Obviously,
In order to estimate dim , by Lemma 3.3, it is suffice to choose a suitable subset D = B(R 0) of M and prove that
(3.1)for each non-constant bounded harmonic function . Next, we will prove (3.1) by Lemmas 3.4 and 3.5. For convenience, we replace ϵ by . By (2.2) and the assumption of the Ricci curvature in Theorem 3.1, we obtain that, on M \ Ω,
(3.2)Set . Inequality (3.2) becomes
(3.3)on M \ Ω. For each , we have
Combining with (3.3), we obtain that
(3.4)Let B(R) be the geodesic ball of radius R for some fixed point p. Since Ω is compact, we can choose a positive R 0 (to be fixed in Lemma 3.4) such that
Let R > 0, such that
Lemma 3.4. Under the assumption of Theorem 3.1, there exists a constant C depending on n and ρ such that
Proof. From (3.4), we have
(3.5)for all . Choose φ = ψχ, where ψ and χ will be chosen later. Then
(3.6)where
and
For n ≥ 4. Choose ψ, χ as follows,
For σ ∈ (0, 1) and , we define χ on the levels sets of f:
where,
From the definition of ψ, we obtain
(3.7)where
and
First, consider
where
and
Denote the set
Note that
So
on Bρ (R) \ Bρ (R − 1). Thus, by the definition of χ and the HÖlder inequality, we obtain that
(3.8)Recall that, under the assumption of complete manifold M with the property (Pρ ), the growth estimate for |∇f| (Li and Wang 2006Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982., Corollary 2.3):
(3.9)and the decay estimate for f (Li and Wang 2006Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982., (2.10)):
(3.10)From the definition of S(R), we obtain that
(3.11)where the second inequality holds because on Ω1 and the last one holds because of (3.10). From (3.8), (3.9) and (3.11), we obtain
(3.12)Now, we consider . Let
and
then
Now, we only consider . Suppose that
where l is a non-negative function on M. Set
x ∈ M. Then
Under the assumption of
Cheng and Yau (1975)Cheng SY and Yau ST. 1975. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math 28: 333-354. gives a local gradient estimate for positive harmonic functions:
(3.13)for any R > 0. Fix x ∈ M and consider the function
Obviously, η(r) tends to a negative number as r → 0 and tends to +∞ as r → +∞. Thus, there exists a r 0 depending on x such that . Combining with (3.13), we obtain that
(3.14)For any y ∈ B(x, r 0), let γ(s) s ∈ [0, b] be a minimizing geodesic with respect to the background metric ds 2 jointing x and y. Then
which implies that . Hence, for any x ∈ M, there is
(3.15)Since the Ricci cuvature has low bound only on M \ Ω, we can choose R 0 such that, for every x ∈ Bρ (R) \ B(R 0 − 1), . Therefore,
for x ∈ Bρ (R) \ B(R 0−1). Replacing f by 1 − f, we obtain that
(3.16)for x ∈ Bρ (R 0−1). Thus, from the definition of χ, we have
where the second inequality holds because, on , and the last one holds because of (3.16). Lemma 5.1 in Li and Wang (2006)Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982. implies the integral of ∇f on the level set , 0 ≤ t ≤ 1 is independent of t and bounded. By the co-area formula, we obtain
for any level b. Thus,
Replacing f by 1 − f, similar to the above argument, we have that
Set and . We obtain that and , as R→ +∞. Combining with (3.5)-(3.7), we have
For n = 3, we may choose ψ as above and χ to be
By an argument similar to the above one for n ≥ 4 and the corresponding estimate in reference (Li and Wang 2006Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982.), we have
and
where C is independent of R. Choose
with
As R → +∞, we also have . Thus, for n ≥ 3, the desired result is obtained.
Lemma 3.5.
Proof. Since the function h satisfies the differential inequality
where
By the mean value inequality of Li and Tam (1991)Li P and Tam LF. 1991. The heat equation and harmonic maps of complete manifolds. Invent Math 105: 1-46., for any x ∈ B(p,R 0), there is
where . From Lemma 3.4, we obtain
(3.17)Thus,
(3.18)By the Schwarz's inequality, we have
Together with (3.18), we obtain the desired result.
Now, we give finite ends theorem when a weighted Poincaré inequality and the Sobolev inequality hold outside a compact subset.
Theorem 3.6. Let Mn (n ≥ 3) be a complete Riemannian manifold. Suppose that
and τ is a non-negative function satisfying
(3.19)for all , where Ω is a compact subset of M. If M satisfies the Sobolev inequality
(3.20)for all, then M has finitely many non-parabolic ends. Proof. Suppose that , where B(R0) is a geodesic ball center at p ∈ M of radius R 0. Let ω ∈ H 1(L 2(M)). Then h = |ω| satisfies Li and Wang (2006)Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982.:
(3.21)Thus, for ,
(3.22)From (3.20) and (3.22), we have
(3.23)Choose φ as follows (R > R 0 + 1):
such that
on B(R 0 + 1) \ B(R 0) and
on B(2R) \ B(R), where C > 0. Thus, by (3.23), we have
Let R → +∞. Then
(3.24)The Schwarz inequality implies that
Combining with (3.24), we obtain that
(3.25)By (3.21) and Li (1993)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. -Lemma 11.1-, we obtain that
For each x ∈ B(R 0 + 1). Thus, Combining with (3.25), we obtain that
Choose D = B(R 0 + 1), by Lemma 3.3, we obtain the desired result.
Let Nn +1 be a Riemannian manifold and X, Y two orthonormal tangent vectors. Then the bi-Ricci curvature in the directions X, Y is defined as Cheng (2000)Cheng X. 2000. L2 harmonic forms and stablity of hypersurfaces with constant mean curvature. Bol Soc Bras Mat 31: 225-239.
As an application of Theorem 3.6, we obtain that
Corollary 3.7.Let Nn +1be a complete simply connected manifold with non-positive sectional curvature. Let Mn be a complete minimal finite index hypersurface in Nn +1 (n ≥ 3). If
for all orthonormal tangent vectors X, Y in TpN, p ∈ M, then M must has finitely many ends.
Proof. Fix a point p ∈ M and a local orthonormal frame field such that are tangent fields and v is unit normal vector at p on M. The Gauss equation implies that
Since M has finite index, it implies that there exists a compact set Ω such that
for all . Since N is a complete simply connected manifold with non-positive sectional curvature, there is the following Sobolev inequality (Hoftman and Spruck 1974Hoftman D and Spruck J. 1974. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm Pure Appl Math 27: 715-727.):
for all . Set
Obviously, τ ≥ 0. Thus,
Combining Theorem 3.6 with Proposition 2.3 in Cheng et al. (2008)Cheng X, Cheung LF and Zhou DT. 2008. The structure of weakly stable constant mean curature hypersurfaces. Tohoku Math J 60: 101-121., we obtain the result.
From Corollary 3.7, we obtain the following results directly:
Corollary 3.8.Li and Wang (2002)Li P and Wang JP. 2002. Minimal hypersurfaces with finite index. Math Res Lett 9: 95-103. Let Mn be a complete minimal finite index hypersurface in . Then M must has finitely many ends.
Corollary 3.9.Let Mn be a complete minimal finite index hypersurface in. If |A|2 ≥ 2n 2 − n, then M must has finitely many ends.
The author would like to express his sincere gratitude toward the referee for having pointed out several improvements in the writing the paper. This work was partially supported by NSF Grant (China) 11101352 and New Century Talent Project of Yangzhou University.
REFERENCES
- Carron G. 2002. L2 harmonic forms on non-compact Riemannian manifold. In: Surveys in analysis and operator theory (Canberra, 2001), Proc. Centre Math. Appl. Austral. Nat. Univ. n. 40. Canberra: Austral Nat Univ, p. 49-59.
- Cheng SY and Yau ST. 1975. Differential equations on Riemannian manifolds and their geometric applications. Comm Pure Appl Math 28: 333-354.
- Cheng X. 2000. L2 harmonic forms and stablity of hypersurfaces with constant mean curvature. Bol Soc Bras Mat 31: 225-239.
- Cheng X. 2008. On end theorem and application to stable minimal hypersurfaces. Arch Math 90: 461-470.
- Cheng X, Cheung LF and Zhou DT. 2008. The structure of weakly stable constant mean curature hypersurfaces. Tohoku Math J 60: 101-121.
- Cheng X and Zhou DT. 2009. Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces. Comm Anal Geom 17: 135-154.
- Hoftman D and Spruck J. 1974. Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm Pure Appl Math 27: 715-727.
- Kong SL, Li P and Zhou DT. 2008. Spectrum of the Laplacian on quaternionic Kähler manifolds. J Diff Geom 78: 295-332.
- Lam KH. 2010. Results on a weighted Poincaré inequality of complete manifolds. Trans Amer Math Soc 362: 5043-5062.
- 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.
- Li P and Tam LF. 1991. The heat equation and harmonic maps of complete manifolds. Invent Math 105: 1-46.
- Li P and Wang JP. 2001. Complete manifolds with positive spectrum I. J Diff Geom 58: 501-534.
- Li P and Wang JP. 2002. Minimal hypersurfaces with finite index. Math Res Lett 9: 95-103.
- Li P and Wang JP. 2006. Weighted Poincaré inequality and rigidity of complete manifolds. Ann Sci École Norm Sup 39: 921-982.
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AMS Classication: 53C21, 54C42.
Publication Dates
-
Publication in this collection
June 2013
History
-
Received
24 Feb 2011 -
Accepted
1 Oct 2012