Metrics of negative Ricci curvature. (English) Zbl 0824.53033

Lohkamp, Joachim

Ann. Math. (2) 140, No. 3, 655-683 (1994).

Using some deformation techniques the author is able to construct Riemannian metrics \(g\) of negative Ricci curvature \(r(g)\) and to prove in this way the following remarkable results: (i) For any \(n \geq 3\) there exist constants \(a(n) > b(n) > 0\) such that any manifold \(M\) with \(\dim M \geq 3\) admits a complete Riemannian metric \(g\) for which \(-a(n) < r(g) < -b(n)\). (ii) Any manifold \(M\) with \(\dim M \geq 3\) admits a complete Riemannian metric \(g\) such that \(r(g) < -1\) and \(\text{Vol} (M,g) < \infty\). (iii) If \(M\) is closed, \(\dim M \geq 3\) and \(G \subset \text{Diff} (M)\) is a finite group, then \(G\) coincides with the group of isometries of a Riemannian metric \(g\) on \(M\) with \(r(g) < 0\).
Roughly speaking, the construction starts with a Riemannian metric \(g^ -_ 3\) on \(\mathbb{R}^ 3\) such that (1) \(r(g^ -_ 3) < 0\) on the unit ball \(B_ 1 (0)\), and (2) \(g\) coincides with the standard Euclidean metric outside the ball.
Then, the analogous metrics \(g^ -_ n\) on \(\mathbb{R}^ n\) are obtained by induction for \(n = 4, 5, \dots\) Finally, Ricci negatively curved balls are distributed over the whole \(M\) to get Riemannian metrics with the desired properties.

Reviewer: P.Walczak (Łódź)

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

negative Ricci curvature; group of isometries