Rigidity theorems for minimal submanifolds in a hyperbolic space

Abstract

Abstract. Let M be a complete immersed minimal hypersurface in a hyperbolic space. In this paper we establish conditions on the first eigenvalue of the stability and super stability operators and the L d norm of the length of the second fundamental form of M to imply that M is totally geodesic. Similar results for minimal submanifolds in a hyperbolic space are also proven.

Rigidity and stability estimates for minimal submanifolds in the hyperbolic space

“…Simon's work has been of great interest to differential geometers, and in the last decade several interesting gap theorems for submanifolds have been successfully obtained. We refer the reader to [1], [2], [3], [4], [5], [7], [9], [10], [11], [13], [14], and the references therein.…”

“…In this direction, the first author and Wang studied in [3] the stability operator for complete minimal submanifolds M n in H n+m . They showed that if the condition (1.2) is satisfied and the first eigenvalue of the super stability operator is greater than a certain constant, then a Simon's type theorem holds if 2 ≤ n ≤ 5, and for all n = 3 if m = 1.…”

“…In our next result we present a version of [3,Theorem 1.2] for minimal submanifolds M n of the hyperbolic space H n+m , with a condition involving the norm of the second fundamental form of M n and the first eigenvalue of the super stability operator for dimensions n ≥ 6. We will denote the first eigenvalue λ 1 (L |A| 2 −n , M ) of the super stability operator of M by λ 1 .…”