Negative Ricci curvature and isometry group
Given positive constants \(\lambda\), \(\Lambda\), \(i_ 0\), \(V\) and a positive integer \(n\) let \({\mathcal M} = {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\) denote the class of Riemannian \(n\)-manifolds \(M\) satisfying \(- \Lambda \leq \text{Ric} \leq -\lambda < 0\), \(\text{inj} \geq i_ 0 > 0\) and \(\text{vol} \leq V\), where Ric, inj, and vol denote the Ricci curvature, injectivity radius and volume of \(M\) respectively. M. Anderson has shown that \(\mathcal M\) is precompact in the \(C^{1,\alpha}\) topology and has obtained uniform \(L^ p\) bounds for the curvature tensor of manifolds in \(\mathcal M\). Let \(I(M)\) and \(| I(M)|\) denote the isometry group of \(M\) and its cardinality, respectively. In this article the authors prove the following Theorem A. Let \(\{M_ i\}\) be a sequence of manifolds in \({\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\) that converges to a \(C^{1,\alpha}\) Riemannian manifold \(M\). Then (a) \(| I(M)|\) is finite, and (b) \(\lim_{i \to \infty} \sup | I(M_ i)| \leq | I(M)|\). From this result one obtains the following Corollary. There exists a positive integer \(N = N(n, \lambda, \Lambda, i_ 0, V)\) such that \(| I(M)| \leq N\) for every \(M \in {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\). Katsuda has previously proved the corollary under the additional hypothesis that the sectional curvatures of manifolds in \({\mathcal M}(n,\lambda, \Lambda, i_ 0, V)\) are uniformly bounded below. By a result of T. Yamaguchi one can drop the lower bound on the injectivity radius if one assumes that the sectional curvature satisfies \(-\Lambda \leq K < 0\). The theorem above is a consequence of the next result on the displacement function \(d_ \varphi : x \to d(x,\varphi x)\) of an isometry \(\varphi\). Theorem B. Let \(M \in {\mathcal M}(n, \lambda, \Lambda, i_ 0, V)\). Then there is a constant \(\varepsilon_ 0 = \varepsilon_ 0 (n, \lambda, \Lambda, i_ 0, V)\) such that if \(\varphi\) is an isometry of \(M\) satisfying \(d_ \varphi(x) \leq \varepsilon_ 0\) for all \(x\in M\), then \(\varphi\) is the identity map. The proof of Theorem B follows from the Sobolev and Hölder inequalities and the following estimate of the Laplacian of a small displacement function \(d\varphi\). Lemma. Given \(M\in {\mathcal M}\) and \(\delta > 0\) there exists \(\varepsilon = \varepsilon(n, \lambda, \Lambda, i_ 0, V)\) such that if \(d_ \varphi(x) \leq \varepsilon\) for all \(x \in M\), then in the sense of distributions one has \(\Delta d_ \varphi \geq \lambda d_ \varphi - \delta G d_ \varphi\), where \(G\) is a function on \(M\) with a uniform \(L^ p\) bound \(\| G \|_ p \leq c_ p(n, \lambda, \Lambda,i_ 0,V)\) for all \(1 \leq p < \infty\). The proof of the lemma involves some uniform estimates of Jacobi vector field growth that use a sharp lower bound estimate of the Laplacian of the distance function from a fixed point due to \textit{R. Brocks} [Diplomarbeit, University of Münster, Germany (1993)].
scientific article; zbMATH DE number 727819
scientific article