Contact｜Sitemap｜HOME｜Japanese
HOME > Table of Contents and Abstracts > Vol. 57, No. 2
Tohoku Mathematical Journal
2005
June
SECOND SERIES VOL. 57, NO. 2
Tohoku Math. J.
57 (2005), 171200

Title
TOTAL CURVATURE OF COMPLETE SUBMANIFOLDS OF EUCLIDEAN SPACE
Author
Franki Dillen and Wolfgang Kühnel
(Received June 24, 2003, revised July 21, 2004) 
Abstract.
The classical CohnVossen inequality states that for any complete 2dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean
3space 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 conelike ends. This is based on the extrinsic GaussBonnet 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 4dimensional hypersurface in Euclidean 5space where the curvature defect is negative, so that the direct analogue of the CohnVossen 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 conelike ends the total curvature is stationary if and only if each end has vanishing GaussKronecker curvature in the sphere "at infinity". For this case of stationary total curvature we prove a result on the quantization of the total curvature.
2000 Mathematics Subject Classification.
Primary 53C40; Secondary 53A07.


To the top of this page
Back to the Contents