Rigidity of Complete Minimal Submanifolds in Spheres

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.


INTRODUCTION
Let x, u(x) be a minimal graph in R 2 × R, which means that u(x) solves the equation div ∇u The celebrated Bernstein theorem states that the complete minimal graphs in R 3 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 R n+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 R 3 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 R n+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 R n+1 is a hyperplane if where B x 0 (R) denotes the geodesic ball of radius R centered at x 0 ∈ M. Many interesting generalizations of the do Carmo-Peng theorem have been obtained (see, e.g., [7,15,16,18] where d is a constant with the following properties: (1) if p = 1 and n ≥ 4, then d ∈ n − 1 n , (n − 2)(n − 1) n ; (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 M |h| n dv < C.
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 |h| 2 ≤ np 2p−1 , then either M is totally geodesic or |h| 2 = np 2p−1 . In the latter case, Chern et al. [3] further proved that M is either a Clifford hypersurface or a Veronese surface in S 4 . Li and Li [14] and Chen and Xu [4] proved independently that M is either a totally geodesic submanifold or a Veronese surface in S 4 if |h| 2 ≤ 2 3 n 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 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 M |h| n dv < C n (n, p) with C(n, p) = (nc(n)) −1 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 f ∈ C ∞ 0 (M), 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 S n+1 . If where B x 0 (R) denotes the geodesic ball of radius R centered at x 0 ∈ M, then M is totally geodesic.

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 {e 1 , e 2 , . . . , e n+p } in S n+p such that, restricted to M, {e 1 , e 2 , . . . , 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 h α ijk is the component of the covariant derivative of h α ij . When M is minimal, we obtain the Simons' formula [3,17] The last terms in (2.1) can be estimated as [14] − i,j,k,l,α,β 2) with b(1) = 1 and b(p) = 3 2 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.  Given η ∈ C ∞ 0 (M), multiplying (3.1) by η 2 and integrating over M gives

PROOFS OF THE MAIN THEOREMS
(3.5) 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 |l − 1|ε ≤ θ 2 , we obtain Fix a point x 0 ∈ M and choose η ∈ C ∞ 0 (M) as with 0 ≤ η ≤ 1, where B x 0 (R) denotes the geodesic ball of radius R centered at x 0 ∈ M. Substituting the above η into (3.7) and letting R → ∞, we deduce that Hence |h| 2 = 0, that is, M n is totally geodesic.

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.

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.