Total curvature of complete submanifolds of Euclidean space

Abstract

The classical Cohn-Vossen inequality states that for any complete 2-dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean 3-space this curvature defect can be interpreted in terms of the length of the curve "at infinity''. The goal of this paper is to investigate higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends. This is based on the extrinsic Gauss-Bonnet formula for compact submanifolds with boundary and its extension "to infinity''. It turns out that the curvature defect can be positive, zero, or negative, depending on the shape of the ends "at infinity''. We give an explicit example of a 4-dimensional hypersurface in Euclidean 5-space where the curvature defect is negative, so that the direct analogue of the Cohn-Vossen inequality does not hold. Furthermore we study the variational problem for the total curvature of hypersurfaces where the ends are not fixed. It turns out that for open hypersurfaces with cone-like ends the total curvature is stationary if and only if each end has vanishing Gauss-Kronecker curvature in the sphere "at infinity''. For this case of stationary total curvature we prove a result on the quantization of the total curvature.