VOLUME ESTIMATE VIA TOTAL CURVATURE IN HYPERBOLIC SPACES
Published online by Cambridge University Press: 20 March 2003
Abstract
Let $D\subset H^n(-k^2)$ be a convex compact subset of the hyperbolic space $H^n(-k^2)$ with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of $\partial D$. More precisely, $(n-1)k^n{\rm Vol}(D)+ {\rm Vol}(S^{n-1})\leq \int_{\partial D}K$, where K denotes the Gauss–Kronecker curvature of $\partial D$ and Vol$(S^{n-1})$?> denotes the Euclidean volume of the sphere.
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