On total absolute curvature of nonclosed submanifolds
In this paper certain results about the total absolute curvature for compact submanifolds of \(E^ n\) are carried over to the case of noncompact complete submanifolds. The general assumption is made that this total absolute curvature is finite. For curves in \(E^ 3\) it is shown that \(\int \kappa ds\) is greater or equal to the angle between the limit tangents at infinity. Furthermore if \(\int \kappa ds<2\pi\) then the curve is unknotted in \(S^ 3\) and a lower bound for this integral involves the minimal number of generators of the knot group. For immersions \(f: X^ n\to E^ N\) of higher dimensional manifolds the additional assumption is made that there are only finitely many so-called limit directions \(\lim_{x\to \infty}f(x)/\| f(x)\|\). In this case the Gauss-Bonnet formula \(\int K dV=c\cdot \chi (X^ n)\) is proved. Consequently the total absolute curvature is greater or equal to \(| \chi |\). For surfaces of negative Euler characteristic this follows also from Cohn-Vossen's inequality. Then for immersed surfaces \(X^ 2\) a better inequality for the total absolute curvature is shown involving a quantity from the limit points. The conjecture is made that for surfaces in general (i.e. without the assumption of finitely many limit points) there holds \(\int K dV+H^ 1(d_{\infty})=2\pi \cdot \chi\) where \(H^ 1(d_{\infty})\) is the 1- dimensional length of the set of limit points. Then a notion of tightness for such surfaces is introduced and an analogue of Chern-Lashof's inequality is proved. Finally an inequality for knotted surfaces is given involving a limit term and the minimal number of generators of the knot group. Here a Morse theory for manifolds with boundary is used. The counterpart in the compact case has been shown by the author [Beitr. Algebra Geom. 10, 87-96 (1980; Zbl 0472.53057)]; see also the recent paper by \textit{N. H. Kuiper} and \textit{W. Meeks III} reviewed below.
scientific article; zbMATH DE number 3881435
scientific article