Ricci curvature, diameter and optimal volume bound

Abstract

\S 1. Motivation and main results.It is well known that a complete Riemannian $n$ -manifold $M$ with the Ricci curvature $Ric(M)\geqq(n-1)k$ and the diameter $d(M)\leqq D$ has the volume bounded above by the volume $v_{h}(D)$ of a $D$ -ball in the simply connected space form $M_{k}^{n}$ with the constant sectional curvature $k$ .In other words, if we rescale and normalize the metric so that $d(M)=\pi$ and consider the class $M_{k}$ of all closed Riemannian $n$ -manifold with $Ric(M)\geqq(n-1)k$ and $d(M)=\pi$ , then the volume defines a function on $M_{k}$ : $vol:M_{k}arrow R^{+}$ with the range in the interval $(0,\tilde{v}_{k}(\pi)]$ .Note that the Myers theorem implies that $k$ must be smaller than or equal to 1 since $d(M)=\pi$ .For $k=1$ , the maximal diameter sphere theorem of Cheng [Ch] implies that $M_{k}$ contains only one element, the $n$ -sphere with its canonical metric can.Hence the range of $vol$ on $M_{1}$ contains the single value $\tilde{v}_{1}(\pi)$ .To see that there is no positive lower bound on the function $vol$ defineded on $M_{k}$ for k$O, one can consider the flat tori: $S^{1}(\epsilon)\cross T^{n-1},$ $\epsilon>0$ where $S^{1}(\epsilon)$ is the circle with radius $\epsilon$ in $R^{2}$ and $T^{n-1}$ is a flat $(n-1)$ -torus.For positive $k<1$ , one can consider the suspension M. of an $(n-1)$ -sphere, $S_{\epsilon}^{n-1}$ , in $R^{n}$ with radius $\epsilon<1$ .Namely, $M_{\epsilon}=$ $S_{\epsilon}^{n-1}x_{\sin}[0, \pi]$ .Note that $M_{\epsilon}$ is the $n$ -sphere $S^{n}$ .Then smooth the two singular points and rescale the metric to obtain a metric $g_{\epsilon}$ on $M_{\epsilon}$ with $d(g_{\epsilon})$ $=\pi$ and $\min Ric(g_{\text{\'{e}}})\geqq 1-\eta_{1}(\epsilon)$ and $vol(g_{\epsilon})\leqq\eta_{2}(\epsilon)$ where the positive functions $\eta_{1}(\epsilon)$ and $\eta_{2}(\epsilon)$ approach zero as $\epsilon$ goes to zero.See also [GP] for a similar construction.This indicates that the lower bound of $vol$ on $M_{k}$ is also zero for positive $k<1$ .For the upper bound of $vol$ on $M_{k}$ , one may ask if the upper bound $\tilde{v}_{k}(\pi)$ is obtainable by some Riemannian $n$ -manifold in $M_{k}$ ?The answer is yes only when $k=1/4$ or 1.They are obtained by $(RP^{n}, 4can)$ and ( $S^{n}$ , can), respectively.Therefore it is natural to ask the following

Locations

• Journal of the Mathematical Society of Japan - View - PDF