Efimov's theorem in dimension greater than two
type: Item
instance of: scholarly article
title: Efimov's theorem in dimension greater than two
zbMATH Open document ID: 0642.53007
DOI: 10.1007/BF01389174
author: Brian Smyth
author: Frederico Xavier
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143517
review text: Efimov's celebrated theorem says that if S is a complete surface in \({\mathbb{R}}^ 3 \)whose Gaussian curvature K is everywhere negative then \(\inf | K| =0\); see, for example, the exposition by \textit{T. Klotz Milnor} [Adv. Math. 8, 474-543 (1972; Zbl 0236.53055)]. Several authors have suggested a similar `n-dimensional Efimov conjecture': if M n is a complete isometrically immersed hypersurface in \({\mathbb{R}}^{n+1}\) whose Ricci curvatures are all negative, then these curvatures are not bounded above by a negative constant. When \(n=3\) the authors prove that this conjecture is correct in a very strong sense: the second fundamental form A of such a hypersurface must satisfy \(\inf | A| =0\). In the case \(n>3\), they have a partial result: if a counterexample to the conjecture exists, then its sectional curvatures must take on every real value. To show all this they first prove a result, important in its own right, concerning the values of the nonzero principal curvatures of any complete immersed oriented hypersurface M in \({\mathbb{R}}^{n+1}:\) let \(\Lambda\) \(+\) and \(\Lambda\)- denote the sets of values assumed by the positive and the negative principal curvatures (respectively) on M; then either (a) one of these sets is empty, in which case the other has connected closure, or (b) both are nonempty, in which case inf \(\Lambda\) \(+=\sup \Lambda\) \(- =0\). (The authors also prove an analogous result for submanifolds with arbitrary codimension.) The main tools are theorems of Sacksteder-van Heijenoort and Wu concerning convex hypersurface [see \textit{H. Wu}, J. Differ. Geom. 9, 279-290 (1974; Zbl 0282.53040), and \textit{R. Osserman}'s `convex hull' theorem, ibid. 6, 267-270 (1971; Zbl 0226.53009)]. What distinguishes the case \(n=3\) is that, because the Ricci curvature is negative, then (1) the principal curvatures \(\lambda_ j\) of M can be labelled (after reversing the orientation if needed) so that one of them is positive and the rest are negati…
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4046545
zbMATH Keywords: Hilbert-Efimov theorem
zbMATH Keywords: hypersurface
zbMATH Keywords: Ricci curvatures
zbMATH Keywords: sectional curvatures
zbMATH Keywords: principal curvatures
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4046545