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Front. Phys., 26 October 2020 | https://doi.org/10.3389/fphy.2020.571250

Rigidity of Complete Minimal Submanifolds in Spheres

  • 1School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
  • 2School of Mathematics and Statistics, Fuyang Normal University, Fuyang, China

Let M be an n-dimensional complete minimal submanifold in an (n + p)-dimensional sphere 𝕊n+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 (x,u(x)) be a minimal graph in ℝ2 × ℝ, which means that u(x) solves the equation

div(u1+|u|2)=0.

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

limRBx0(R)|h|2dvR2q+2=0,     q<2n,

where Bx0(R) denotes the geodesic ball of radius R centered at x0M. Many interesting generalizations of the do Carmo-Peng theorem have been obtained (see, e.g., [7, 15, 16, 18]). By definition, the hyperbolic space ℍn+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 spacen+p, and let h be the second fundamental form of M. Assume that

limRsupBx0(R)|h|2dvR2=0.    (1.1)

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

M|h|ndv<C,

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 spacen+p such that n and p satisfy (n2-6n+1)+8p>0. Assume that

limRsupBx0(R)|h|ddvR2=0,

where d is a constant with the following properties:

(1) if p = 1 and n ≥ 4, then

d(n-1n,(n-2)(n-1)n);

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

d(n1)22n(114(n1)2(n2p), 1                                 +14(n1)2(n2p)).

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

M|h|ndv<C.

The unit sphere 𝕊n+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 𝕊n+p and |h|2np2p-1, then either M is totally geodesic or |h|2=np2p-1. 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 |h|223n everywhere on M. This result improves the pinching constant in Simons' formula. Deshmukh [5] studied n-dimensional compact minimal submanifolds in 𝕊n+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 𝕊n+p. We further assume that (1.1) holds. If

M|h|ndv<Cn(n,p)

with C(n,p)=(c(n))-1(n2+7)-12(2b(p))-1+(npb(p))-1, where c(n)=2n(1+n)1+1n(n-1)-1ωn-1n, ωn is the volume of the unit ball inn, b(1) = 1, and b(p)=32 if 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 𝕊n+p. If

M|h|ndv<C~n(n,p)

with C~(n,p)=(nc(n))-12(n-1)(n2+7)b(p)+4n(n2+7)pb(p), where c(n)=2n(1+n)1+1n(n-1)-1ωn-1n, ωn is the volume of the unit ball inn, b(1) = 1, and b(p)=32 if 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 C~(n,p) 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 fC0(M),

M(|f|2-(|h|2+Ric¯(ν,ν))f2)dv0,    (1.2)

where Ric¯ 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

limRsupBx0(R)|h|2δdvR2=0,     1-2n<δ<1+2n,    (1.3)

where Bx0(R) denotes the geodesic ball of radius R centered at x0M, then M is totally geodesic.

2. Preliminaries

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

1A,B,C,n+p,     1i,j,k,n,                             n+1α,β,γ,n+p.

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

ωiα=jhijαωj,     h=i,j,αhijαωiωjeα,     ξ=1ni,αhiiαeα,

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

H=|ξ|,     |h|2=i,j,α(hijα)2,     |h|2=i,j,k,α(hijkα)2,

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

12Δ|h|2=|h|2+n|h|2-i,j,k,l,α,βhijαhijβhklαhklβ                   -i,j,α,β(k(hikαhkjβ-hjkαhkiβ))2.    (2.1)

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

-i,j,k,l,α,βhijαhijβhklαhklβ-i,j,α,β(k(hikαhkjβ-hjkαhkiβ))2-b(p)|h|4,    (2.2)

with b(1) = 1 and b(p)=32 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 Mn+p(k). Then

|h|2||h||22np||h||2.

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 ψC01(M). Then

(Mψnn-1dv)n-1nc(n)M(|ψ|+n|H|ψ)dv,

where c(n)=2n(1+n)1+1n(n-1)-1ωn-1n and ωn is the volume of the unit ball inn.

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 𝕊n+p and let ψC01(M). Then

(Mψ2nn-2dv)n-2n2(n2+7)c2(n)M(|ψ|2+|ψ|2)dv.

3. Proofs of the Main Theorems

PROOF OF THEOREM 1.3: Noting that

12Δ|h|2=||h||2+|h|Δ|h|,

it follows from (2.1) and (2.2) that

||h||2+|h|Δ|h||h|2+n|h|2b(p)|h|4.

From Lemma 2.1, we have

|h|Δ|h|2np||h||2+n|h|2b(p)|h|4.    (3.1)

Given ηC0(M), multiplying (3.1) by η2 and integrating over M gives

Mη2|h|Δ|h|dv+Mb(p)η2|h|4dv2npM||h||2η2dv                                       +nM|h|2η2dv,    (3.2)

which implies

-M2η|h||h|,ηdv+Mb(p)η2|h|4dv(1+2np)M||h||2η2dv+nM|h|2η2dv.    (3.3)

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

Mη2|h|4dv(M|h|ndv)2n(M(η|h|)2nn2dv)n2n                               2(n2+7)c2(n)(M|h|ndv)2nM(|(η|h|)|2                                +|h|2η2)dv.    (3.4)

Setting

l=2b(p)(n2+7)c2(n)(M|h|ndv)2n,

from (3.3) and (3.5) we may estimate

lM|η|2|h|2dv+(l-1)M2η|h||h|,ηdv      (1+2np-l)M||h||2η2dv+(n-l)M|h|2η2dv.    (3.5)

By assumption,

(M|h|ndv)1n<c-1(n)(n2+7)-12(2b(p))-1+(npb(p))-1,

and it is easy to see that

1+2np-l>0.

Therefore, we can find a θ > 0 such that

1+2np-lθ.

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

(l-1)M2η|h||h|,ηdv|l-1|εMη2||h||2dv                                                               +|l-1|ε-1M|η|2|h|2dv.    (3.6)

Thus, when |l-1|εθ2, we obtain

(l+|l-1|ε-1)M|η|2|h|2dvθ2M||h||2η2dv                                                                       +(n-l)M|h|2η2dv.    (3.7)

Fix a point x0M and choose ηC0(M) as

η={1on Bx0(R),0on M\Bx0(2R),|η|1Ron Bx0(2R)\Bx0(R),    (3.8)

with 0 ≤ η ≤ 1, where Bx0(R) denotes the geodesic ball of radius R centered at x0M. Substituting the above η into (3.7) and letting R → ∞, we deduce that

M||h||2dv0,     M|h|2dv0.

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

Proof of Theorem 1.4: Direct computation yields

Δ|h|δ=δ(δ-1)|h|δ-2||h||2+δ|h|δ-1Δ|h|.    (3.9)

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

|h|δΔ|h|δ=δ1δ||h|δ|2+δ|h|2δ2|h|Δ|h|                         δ1δ||h|δ|2+2δnp||h||2|h|2δ2+nδ|h|2δ                         δb(p)|h|2δ+2                         =(δ1δ+2npδ)||h|δ|2+(nδb(p)δ|h|2)|h|2δ.    (3.10)

Let ηC0(M). Multiplying (3.10) by η2 and integrating over M yields

Mη2|h|δΔ|h|δdv(δ-1δ+2npδ)Mη2||h|δ|2dv                                              +nδMη2|h|2δdv                                                   -b(p)δMη2|h|2δ+2dv.    (3.11)

It follows from the divergence theorem and (3.11) that

b(p)δMη2|h|2δ+2dv-M2η|h|δη,|h|δdv     (2+2-npnpδ)Mη2||h|δ|2dv+nδMη2|h|2δdv.    (3.12)

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

Mη2|h|2δ+2dv(M|h|ndv)2n(M(η|h|δ)2nn2dv)n2n                                      2n2c2(n)(M|h|ndv)2nM(|(η|h|δ)|2                                       +|h|2δη2)dv.    (3.13)

Substituting (3.13) into (3.12) yields

δlM|h|2δ|η|2dv+(δl-1)M2η|h|δη,|h|δdv         (2+2-npnpδ-lδ)Mη2||h|δ|2dv         +(nδ-lδ)Mη2|h|2δdv,    (3.14)

where l=2b(p)(n2+7)c2(n)(M|h|n)2n. Further, using the Cauchy-Schwarz inequality, for each ε > 0 we obtain

(δl-1)M2η|h|δη,|h|δdv|δl-1|εMη2||h|δ|2dv+|δl-1|ε-1M|h|2δ|η|2dv.    (3.15)

Therefore

(δl+|δl-1|ε-1)M|h|2δ|η|2dv     (2+2-npnpδ-lδ-|δl-1|ε)Mη2||h|δ|2dv     +(nδ-lδ)Mη2|h|2δdv.    (3.16)

By the assumption in the theorem that

(nc(n))-12(n-1)(n2+7)b(p)+4n(n2+7)pb(p)>(M|h|ndv)1n,

we have

2+2-npnpδ-lδ>0.

Choosing ε sufficiently small, we can get

2+2-npnpδ-lδ-|δl-1|ε>0.

Defining the cut-off function as in (3.8) and taking δ=n2 in (3.16), we obtain

(n2l+|n2l-1|ε-1)M|h|ndvR2(n2l+|n2l-1|ε-1)Bx0(2R)|h|ndvR2(2+4-2npn2p-n2l-|n2l-1|ε)Bx0(R)||h|n2|2dv+(12n2-n2l)Bx0(R)|h|ndv.    (3.17)

Since

(M|h|ndv)1n<C~(n,p),

upon taking R → ∞ we have

M|h|ndvR20.

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 ηC0(M). Replacing f by η|h|δ in (1.2) and taking Ric¯(ν,ν)=n give

M|(η|h|δ)|2dvMη2|h|2δ+2dv+nMη2|h|2δdv,

that is,

M|η|2|h|2δdv+Mη2||h|δ|2dv            +2Mη|h|δη,|h|δdv-nMη2|h|2δdv                                           Mη2|h|2δ+2dv.    (3.18)

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

δM|η|2|h|2δdv+2(δ-1)Mη|h|δη,|h|δdv     (2+2-nnδ-δ)Mη2||h|δ|2dv+2nδMη2|h|2δdv.    (3.19)

Using 1-2n<δ<1+2n, we see that

2+2-nnδ-δ>0.

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

ε|δ-1|Mη2||h|δ|2dv+ε-1|δ-1|M|η|2|h|2δdv       2(δ-1)Mη|h|δη,|h|δdv.    (3.20)

Combining (3.20) and (3.19) gives

(δ+ε-1|δ-1|)M|η|2|h|2δdv     (2+2-nnδ-δ-ε|δ-1|)Mη2||h|δ|2dv     +2nδMη2|h|2δdv.    (3.21)

Choosing ε sufficiently small, we can obtain

2+2-nnδ-δ-ε|δ-1|>0.

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 𝕊n+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.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Funding

This work was supported by the Natural Science Foundation of Anhui Province Education Department (grant nos. KJ2017A341 and KJ2018A0330), the Talent Project of Fuyang Normal University (grant no. RCXM201714), and the Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (grant no. SX201805).

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The author expresses sincere gratitude to the reviewers and the editors for their careful reading of the manuscript and constructive recommendations.

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Keywords: rigidity, minimal submanifold, totally geodesic submanifold, stable hypersurface, Sobolev inequality

Citation: Zhou J (2020) Rigidity of Complete Minimal Submanifolds in Spheres. Front. Phys. 8:571250. doi: 10.3389/fphy.2020.571250

Received: 10 June 2020; Accepted: 17 August 2020;
Published: 26 October 2020.

Edited by:

Manuel Asorey, University of Zaragoza, Spain

Reviewed by:

Sania Qureshi, Mehran University of Engineering and Technology, Pakistan
Kazuharu Bamba, Fukushima University, Japan

Copyright © 2020 Zhou. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jundong Zhou, zhou109@mail.ustc.edu.cn