Efimov's theorem in dimension greater than two
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 negative; i.e. \(\lambda_ 1>0\) and \(\lambda_ 2,\lambda_ 3<0\). The Ricci condition then implies in addition that (2) \(\lambda_ 1\geq | \lambda_ k|\) for \(k=2,3\) and thus \(| A| \leq \sqrt{3}\lambda_ 1\). The condition inf \(\Lambda\) \(+=0\) now implies that \(\inf | A| =0\), as desired. (This analysis requires \(n=3\) because the analog of (1) does not hold if \(n\geq 4\), while the analog of (2) is false when \(n=2.)\) \{Reviewer's comment: A result of \textit{S. Chern} [see Abh. Math. Semin. Univ. Hamb. 29, 77-91 (1965; Zbl 0147.209), especially Theorem 4] verifies the conjecture for all n provided M is an entire nonparametric hypersurface in \(R^{n+1}\).\}
scientific article; zbMATH DE number 4046545
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