The isoperimetric inequality on manifolds of conformally hyperbolic type (Q5951267)

Let \((M,g)\) be a noncompact conformally hyperbolic manifold. The author obtain the existence of a metric \(\widetilde g\) of \(M\) conformally equivalent to \(g\) such that the inequality (1) \(V(D)\leq S(\partial D)\) holds for any regular domain \(D\) of \(M\), where \(V(D)\) is the volume of \(D\) and \(S(\partial D)\) is the area of \(\partial D\) with respect to \(\widetilde g\). An \(n\)-dimensional manifold \(M\) is said to be conformally hyperbolic if \(\text{cap}(C,M)=\inf\int_M|\nabla f|^ndv>0\), where \(C\) is a continuum in \(M\). Here the infimum is taken over all smooth compactly supported functions \(f\), \(0\leq f\leq 1\) such that \(f=1\) on \(C\). The Euler-Lagrange equation of this variational problem is \(\text{div} (|\nabla u|^{n-2} \nabla u)=0\). By using the solution of this equation (with the boundary condition \(u|_C=1)\), and the inequality \(\int_M|\nabla u|^n dv>0\), it is shown \(\widetilde g=((n-1) {|\nabla u|\over u})^2g\) is the desired metric (Section 2). It is also shown that this inequality (1) is asymptotically sharp (Supplement 1) and that the volume of the geodesic ball of radius \(r\) with respect to the metric \(\widetilde g\) is of order \(e^r\) as \(r\to\infty\) (Supplement 2). For a general \(n\)-dimensional Riemannian manifold \(M\), if the inequality \(P(V(D))\leq S(\partial D)\) holds, \(P\) is said to be an isoperimetric function of \(M\). The authors conjectured \(P(x)= x^{(n-1)/n}\) if \(M\) is parabolic \((\text{cap}(C,M)=0)\), and \(P(x)=x\) if \(M\) is hyperbolic in an earlier paper [\textit{V. A. Zorich}, Geom. Funct. Anal. 9, 393-411 (1999; Zbl 0981.53018)]. This paper solves half of this conjecture and shows the equivalence of the following three properties on \(M\): 1) the conformal isoperimetric dimension (supremum of \(m\) such that \(x^{(m-1)/m}\) can be taken as an isoperimentric function) is greater than \(n\); 2) this dimension is infinite; 3) \(M\) is of conformally hyperbolic type (Section 3). The inequality (1) holds for an arbitrary (not necessarily precompact) finite volume domain \(D\). This is proved in Section 5. It is also remarked that regularity of \(\widetilde g\) is not known owing to the lack of knowledge on the regularity of \(u\). But the authors announce that they solved this problem and that the proof will appear soon (Section 6).