Conformal type and isoperimetric dimension of Riemannian manifolds
The authors point out that the isoperimetric dimension of a (noncompact) conformally parabolic Riemannian manifold \((M,g)\) [\textit{V. A. Zorich} and \textit{V. M. Kesel'man}, Funct. Anal. Appl. 30, 106-117 (1996; Zbl 0873.53025)] is bounded under conformal changes of the metric, but it is not a conformal invariant. Moreover, the conformal type is determined only by the structure of the manifold at infinity. Hence, they propose to consider isoperimetric relations only with respect to a family \(E=\{D(t)\colon t\in\mathbb{R}_{+}\}\) of domains (e.g., a familiy of geodesic balls) that exhaust the manifold, that is, \(\overline{D(t_1)}\subset D(t_2)\) whenever \(t_1 < t_2\), and \(\bigcup_{t} D(t) =M\). The conformal isoperimetric dimension of the exhaustion \(E\) is then defined as the supremum of the numbers \(p\geq 0\) such that there exists a (complete) metric \(\tilde g\) conformally equivalent to the original one and satisfying \(V^{p-1}(D)\leq c \cdot S^{p}(\partial D)\) with \(D=D(t)\), provided that \(t\) is sufficiently large. The main result is that a Riemannian manifold \((M,g)\) is conformally parabolic (respectively conformally hyperbolic) if and only if there exists a complete metric conformal to \(g\) such that the conformal isoperimetric dimension of the geodesic exhaustion of \(M\) equals \(\dim M\) (resp., it equals \(+\infty\) for any complete metric conformal to \(g\)).
scientific article; zbMATH DE number 1262560
scientific article