Celebratio Mathematica — Chern

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article S.-S. Chern: “On the curvatures of a piece of hypersurface in euclidean space,” Abh. Math. Semin. Univ. Hamb. 29 : 1–2 (December 1965), pp. 77–91. Dedicated to Professor Emanuel Sperner on his sixtieth birthday. MR 0188949 Zbl 0147.20901

Let \( z=z(x,y) \) be a \( C^2 \)-surface in euclidean three-space, defined over the disk \( x^2+y^2 < R^2 \) in the \( (x,y) \)-plane. Let \( H \) and \( K \) denote respectively its mean and Gaussian curvatures. In [1955] Heinz proved the folowing theorem:

If \( |H|\geq c > 0 \), then \( R\leq 1/c \). If \( K\geq c > 0 \), then \( R\leq 1/\sqrt{c} \). If \( K\leq -c < 0 \), then \( R\leq e(3/x)^{1/2} \). (\( c= \) const. in all cases).

The purpose of this paper is to extend this theorem to a hypersurface in an euclidean space of dimension \( m+1 \). Having the global problem in mind, we will consider an immersed hypersurface and establish, in so far as possible, the intermediary results in this general setting. In this sense some of our formulations are more general even in the classical case \( m=2 \).

book W. Blaschke: Gesammelte Werke [Collected works], Band 1. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1982. With commentary by K. Strubecker and with introductory material by Walter Benz and Emanuel Sperner. MR 741181 Zbl 0541.01012

Hans Robert Müller	Related
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