Rigidity of minimal submanifolds in hyperbolic space

Rigidity of Minimal Submanifolds in Space Forms

(1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal imme...

Minimal Lagrangian submanifolds in the complex hyperbolic space

In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1-parameter families with cohomogeneity one. We characterize them as the only minimal Lagrangian submanifolds in CHn foliated by umbilical hypersurfaces of Lagrangian subspaces RHn of CHn. Several suitable generalizations of the above constructio...

Rigidity of Compact Minimal Submanifolds in a Sphere

In this paper we study n-dimensional compact minimal submanifolds in S with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. T...

Minimal Lagrangian Submanifolds in Complex Euclidean Space

Geometric, Topological and Differentiable Rigidity of Submanifolds in Space Forms

Let M be an n-dimensional submanifold in the simply connected space form F(c) with c + H > 0, where H is the mean curvature of M . We verify that if M(n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies RicM ≥ (n−2)(c+H2), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n+1)-sphere with n = even, or CP ( 4 3 (c+H )) ...