Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C^1,α boundary to ℝ³ which belongs to the critical Sobolev space W^2,2 is C¹ up to the boundary. More generally C¹ regularity up to the boundary holds for all scalar functions V ∈ W^2,2(S) which satisfy det ∇²V=0. If S has only Lipschitz boundary we show such V can be approximated in W^2,2 by functions V_k ∈ W^1,∞∩W^2,2 with det ∇²V_k=0.     

Codimensionm isometric immersions between pseudo-spheres

We describe all isometric immersionsf:S _p ⁿ S _p ^+1/n+m ,n –p 2m, whose first normal space is -parallel in the complement of totally geodesic points while the set of totally geodesic points does not disconnectS _p ⁿ .     

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

Rigidity of isometric immersions of higher codimension

Given an isometric immersionf:M ⁿ → ℝ ^N into Euclidean space, we provide sufficient conditions onf so that any 1-regular isometric immersion ofM ⁿ into ℝ ^N+1 is necessarily obtained as a composition off with a local isometric immersion ℝ ^N ⊃U → ℝ ^N+1 . This result has several applications.     

On the first Hodge eigenvalue of isometric immersions

We give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on -forms on a compact manifold without boundary isometrically immersed in or . The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature.

     

Existence of isometric immersions into nilpotent Lie groups

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups.     

Uniqueness of isometric immersions with the same mean curvature

Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also prove similar results if the mean curvature condition is replaced by an ${\sigma }_2$-curvature condition.