L² harmonic forms and finiteness of ends

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) 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) generalized this result to the manifold satisfying a weighted Poincaré inequality. Recalling that in Li and Wang (2006), 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) -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) generalized this result and applied to minimal hypersurfaces. Lam (2010) also obtained a result similar to Theorem 5.2 in Li and Wang (2006) 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 2010):

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 r_p(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 2006):

(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), respectively. We should also point out that Carron (2002) 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 ∈ T_pM, we can choose an orthonormal frame {e ₁, e ₂, ..., e_n } 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 2002). 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) 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), it was proved that a complete oriented weakly minimal hypersurface in must have only one end. It was shown that in Cheng (2008) 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. 2008):

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).

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) with some changes of technique, we obtain that:

Theorem 3.1.Let M be a complete n-dimensional manifold with property (P_ρ ) (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).

Now we prove Theorem 3.1. A theory of Li-Tam (Li and Tam 1991) 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 f_R as R → +∞, satisfying Δ f_R = 0 in B(R) with boundary conditions that and f_R = 0 on ∂B(R) \ E ₁. It follows from the Maximum Principle that 0 ≤ f_R ≤ 1 for all R and hence 0 ≤ f ₁ ≤ 1. For each non-parabolic ends E_i , we can construct a corresponding fi by the above method. Let be the linear space containing all f_i 's constructed as above. Thus, the number of non-parabolic ends of M is dim . The following lemma is due to Li (Li 1993):

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 B_ρ (R) \ B_ρ (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 (P_ρ ), the growth estimate for |∇f| (Li and Wang 2006, Corollary 2.3):

(3.9)

and the decay estimate for f (Li and Wang 2006, (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) 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 ∈ B_ρ (R) \ B(R ₀ − 1), . Therefore,

for x ∈ B_ρ (R) \ B(R ₀−1). Replacing f by 1 − f, we obtain that

(3.16)

for x ∈ B_ρ (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) 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 2006), 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), 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):

(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) -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 N^{n +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)

As an application of Theorem 3.6, we obtain that

Corollary 3.7.Let N^{n +1}be a complete simply connected manifold with non-positive sectional curvature. Let Mⁿ be a complete minimal finite index hypersurface in N^{n +1} (n ≥ 3). If

for all orthonormal tangent vectors X, Y in T_pN, 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 1974):

for all . Set

Obviously, τ ≥ 0. Thus,

Combining Theorem 3.6 with Proposition 2.3 in Cheng et al. (2008), we obtain the result.

From Corollary 3.7, we obtain the following results directly:

Corollary 3.8.Li and Wang (2002) 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.

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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