The isometry groups of compact manifolds with almost negative Ricci curvature
Let \(M\) be a compact Riemannian manifold of dimension \(n\geq3\). Bochner showed that if the Ricci tensor of \(M\) is negative, then there are no non-trivial Killing vector fields and thus the isometry group \(I(M)\) is finite. The authors generalize this result to assume that \(M\) has negative Ricci curvature except in small portions. To make this precise, consider the Sobolev inequality \[ \|f\|^2_{\frac{2n}{n-2}}\leq A\|\nabla f\|^2_2+B\|f\|^2_2\,. \] Let \(\varrho(P):=\max_{X\in T_PM,\|X\|=1}\mathrm{Ric}(X,X)\). Let \((\varrho+\omega)_+=\max(0,\varrho+\omega)\) where \(\omega\) is a positive constant. Theorem. Suppose there exists \(\omega>0\) so that \(\|(\varrho+\omega)_+\|_{\frac n2}<\min\left(\frac1A,\frac\omega B\right)\). Then \(M\) has no non-trivial Killing vector fields and consequently \(I(M)\) is finite. If one assumes that \(\|(\rho+\omega)_+\|<\min\left(\frac1{2A},\frac\omega{2B}\right)\), the authors discuss bounds for the isometry group in terms of various geometric quantities.
scientific article; zbMATH DE number 6996534