Rigidity of Complete Minimal Submanifolds in Spheres

Abstract

Let M be an n-dimensional complete minimal submanifold in an (n + p)-dimensional sphere 𝕊^n+p, and let h be the second fundamental form of M. In this paper, it is shown that M is totally geodesic if the L² norm of |h| on any geodesic ball of M is of less than quadratic growth and the Lⁿ norm of |h| on M is less than a fixed constant. Further, under only the latter condition, we prove that M is totally geodesic. Moreover, we provide a sufficient condition for a complete stable minimal hypersurface to be totally geodesic.

1. Introduction

Let be a minimal graph in ℝ² × ℝ, which means that u(x) solves the equation

The celebrated Bernstein theorem states that the complete minimal graphs in ℝ³ are planes. The works of Fleming [9], Almgren [1], and Neto and Wang [16] tell us that the Bernstein theorem is valid for complete minimal graphs in ℝⁿ⁺¹ provided that n ≤ 7. Counterexamples to the theorem for n ≥ 8 have been found by Bombieri et al.[2] and, later, by Lawson [13]. On the other hand, do Carmo and Peng [6] and Fischer-Colbrie and Schoen [10] proved independently that a completely stable minimal surface in ℝ³ must be a plane, a result that generalizes the Bernstein theorem. For the high-dimensional case, it is an open question whether the completely oriented stable minimal hypersurfaces in ℝⁿ⁺¹ (for 3 ≤ n ≤ 7) are hyperplanes. However, it has been proved by do Carmo and Peng [6] that a complete stable minimal hypersurface M in ℝⁿ⁺¹ is a hyperplane if

where B_x0(R) denotes the geodesic ball of radius R centered at x₀ ∈ M. Many interesting generalizations of the do Carmo-Peng theorem have been obtained (see, e.g., [7, 15, 16, 18]). By definition, the hyperbolic space ℍ^n+p is a Riemannian manifold with sectional curvature −1 which is simply connected, complete, and (n + p)-dimensional. In hyperbolic space, some results similar to the do Carmo-Peng theorem have been derived. Xia and Wang [20] studied complete minimal submanifolds in a hyperbolic space and obtained the following result.

Theorem 1.1. [20] For n ≥ 5, let M be an n-dimensional complete immersed minimal submanifold in a hyperbolic space ℍ^n+p, and let h be the second fundamental form of M. Assume that

If there exists a positive constant C depending only on n and p such that

then M is totally geodesic.

Recently, de Oliveira and Xia [8] improved Theorem 1.1 as follows.

Theorem 1.2. [8] For n ≥ 4, let M be an n-dimensional complete immersed minimal submanifold in a hyperbolic space ℍ^n+psuch that n and p satisfy. Assume that

where d is a constant with the following properties:

(1) if p = 1 and n ≥ 4, then

(2) if p > 1 and n > 5, then

Then there exists a positive constant C depending only on n, p, and d such that M is totally geodesic if

The unit sphere 𝕊^n+p is a Riemannian manifold with sectional curvature 1 which is simply connected, complete, and (n + p)-dimensional. Many results are available on the classification of compact minimal submanifolds in the unit sphere. Simons [17] calculated the Laplacian of |h|² of minimal submanifolds in a space form. As a consequence of Simons' formula, if M is a compact minimal submanifold in 𝕊^n+p and , then either M is totally geodesic or . In the latter case, Chern et al.[3] further proved that M is either a Clifford hypersurface or a Veronese surface in 𝕊⁴. Li and Li [14] and Chen and Xu [4] proved independently that M is either a totally geodesic submanifold or a Veronese surface in 𝕊⁴ if everywhere on M. This result improves the pinching constant in Simons' formula. Deshmukh [5] studied n-dimensional compact minimal submanifolds in 𝕊^n+p with scalar curvature S satisfying the pinching condition S > n(n − 2) and proved that for p ≤ 2 these submanifolds are totally geodesic.

The above results are rigidity theorems valid in the unit sphere, which characterize the behavior of minimal submanifolds. In this paper, we use the methods of minimal submanifolds in Euclidean space and hyperbolic space to investigate the rigidity of complete minimal submanifolds in spherical space. The main theorems are as follows.

Theorem 1.3. For n ≥ 3, let M be an n-dimensional complete minimal submanifold in the unit sphere 𝕊^n+p. We further assume that (1.1) holds. If

with, where, ω_nis the volume of the unit ball in ℝⁿ, b(1) = 1, andif p > 1, then M is totally geodesic.

In [20], Xia and Wang believed that the condition (1.1) is not necessary. It is therefore interesting to see whether we can remove condition (1.1) from Theorem 1.3. In this case, we get a positive answer.

Theorem 1.4. For n ≥ 3, let M be an n-dimensional complete minimal submanifold in the unit sphere 𝕊^n+p. If

with, where, ω_nis the volume of the unit ball in ℝⁿ, b(1) = 1, andif p > 1, then M is totally geodesic.

Remark 1.5. By using Simons' formula and the technique developed in do Carmo and Peng's paper, we obtain Theorem 1.4. The constant in Theorem 1.4 is smaller than C(n, p) in Theorem 1.3.

We also investigate stable minimal hypersurfaces in the unit sphere and obtain a result similar to do Carmo and Peng's theorem. A minimal hypersurface M in a Riemannian manifold N is said to be stable if for each ,

where is the Ricci curvature of N and ν is the unit normal vector of M.

Theorem 1.6. For n ≥ 2, let M be an n-dimensional complete stable minimal hypersurface in the unit sphere 𝕊ⁿ⁺¹. If

where B_x0(R) denotes the geodesic ball of radius R centered at x₀ ∈ M, then M is totally geodesic.

2. Preliminaries

Let M be an n-dimensional complete submanifold in the (n + p)-dimensional unit sphere 𝕊^n+p. We will use the following convention on the range of indices unless specified otherwise:

We choose a local field of orthonormal frame {e₁, e₂, …, e_n+p} in 𝕊^n+p such that, restricted to M, {e₁, e₂, …, e_n} is tangent to M and {e_n+1, …, e_n+p} normal to M. Let {ω_A} be the field of dual frame and {ω_AB} the connection 1-form of 𝕊^n+p. Restricting these forms to M, we have

where h and ξ are the second fundamental form and the mean curvature vector of M, respectively. We define

where is the component of the covariant derivative of . When M is minimal, we obtain the Simons' formula [3, 17]

The last terms in (2.1) can be estimated as [14]

with b(1) = 1 and if p > 1. We need the following estimate:

Lemma 2.1. [19] Let M be an n-dimensional immersed submanifold with parallel mean curvature in the space form M^n+p(k). Then

We also need the following Hoffman-Spruck Sobolev inequality.

Lemma 2.2. [12] Let M be an n-dimensional complete submanifold in a Hadamard manifold and let. Then

whereand ω_nis the volume of the unit ball in ℝⁿ.

From Lemma 2.2, we have the following estimate.

Lemma 2.3. [11] For n ≥ 3, let M be an n-dimensional complete minimal submanifold in 𝕊^n+pand let. Then

3. Proofs of the Main Theorems

Proof of Theorem 1.3: Noting that

it follows from (2.1) and (2.2) that

From Lemma 2.1, we have

Given , multiplying (3.1) by η² and integrating over M gives

which implies

Further, applying Hölder's inequality and taking ψ = |h|η in Lemma 2.3, one verifies that

Setting

from (3.3) and (3.5) we may estimate

By assumption,

and it is easy to see that

Therefore, we can find a θ > 0 such that

On the other hand, for any ε > 0 we have

Thus, when , we obtain

Fix a point x₀ ∈ M and choose as

with 0 ≤ η ≤ 1, where B_x0(R) denotes the geodesic ball of radius R centered at x₀ ∈ M. Substituting the above η into (3.7) and letting R → ∞, we deduce that

Hence |h|² = 0, that is, Mⁿ is totally geodesic.     □

Proof of Theorem 1.4: Direct computation yields

Multiplying (3.9) by |h|^δ and using (3.1), we infer that

Let . Multiplying (3.10) by η² and integrating over M yields

It follows from the divergence theorem and (3.11) that

Applying Hölder's inequality and taking ψ = |h|^δη in Lemma 2.3, we have

Substituting (3.13) into (3.12) yields

where . Further, using the Cauchy-Schwarz inequality, for each ε > 0 we obtain

Therefore

By the assumption in the theorem that

we have

Choosing ε sufficiently small, we can get

Defining the cut-off function as in (3.8) and taking in (3.16), we obtain

Since

upon taking R → ∞ we have

This and (3.17) imply ∇|h| = 0 and |h| = 0, that is, Mⁿ is totally geodesic.     □

Proof of Theorem 1.6: Since M is a stable minimal hypersurface in the unit sphere 𝕊ⁿ⁺¹, (1.2) holds on M. Let . Replacing f by η|h|^δ in 1.2) and taking give

that is,

Substituting (3.18) into (3.12) and noting that b(1) = 1, we obtain

Using , we see that

Further, for any ε > 0, it follows from the Cauchy-Schwarz inequality that

Combining (3.20) and (3.19) gives

Choosing ε sufficiently small, we can obtain

Furthermore, defining the cut-off function as in (3.8) and using the assumption (1.3) yield ∇|h| = 0 and |h| = 0, that is, Mⁿ is totally geodesic.     □

4. Conclusion

In this paper, by using Simons' formula, a Sobolev-type inequality as in Chen and Xu [4], and the technique of do Carmo and Peng, we obtain rigidity theorems for minimal submanifolds in 𝕊^n+p. Compared with Theorem 1.1, Theorem 1.4 removes the condition on the growth of the norm of the second fundamental form. Moreover, our results require only n ≥ 3, whereas Theorems 1.1 and 1.2 require n ≥ 5 and n ≥ 4, respectively. Whether the pinching constant for the total curvature in Theorem 1.4 is optimal remains an open question and is a topic of future research.

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