A new renormalized volume-type invariant (Q6656376)

Let \(M\) be a complete noncompact hyperbolic surface of finite topological type that can be conformally compactified to a bounded domain in \(\mathbb{C}\) and let \(\Omega \subset \mathbb{C}\) be a fixed conformal compactification of \(M\). Then there exists a smooth function \(u\) on \(\Omega\) such that \((M,e^{-2u}g_{M})\) is isometric to \((\Omega, g_{E})\). Let \(v=e^{-u}\), then the asymptotic behaviour of \(v\) near a \(C^{3,\alpha}\) boundary component of \(\Omega\) is given by\N\[\Nv(z)=d(z)-\frac{1}{2}\kappa(y)d(z)^{2}+c_{3}(y)d(z)^{3}+O(d(z)^{3+\alpha})\N\]\Nwhere \(d(z)\) is the distance from \(z\) to the boundary of \(\Omega\) and \(y\) is a point in \(\partial \Omega\) such that \(d(z)=|z-y|\), and \(\kappa(y)\) is the curvature at \(y\). Assuming that the outermost boundary component \(C\) of \(\Omega\) is \(C^{3,\alpha}\), one can define\N\[\N\lambda(\Omega, v) = - \int_{C} dl \int_{C}c_{3}(y) dl(y)\N\]\Nwhere the line integral is taken with respect to the Euclidean metric. By varying \(\Omega\) among all conformal compoactifications of \((M,g_{M})\) with outermost booundary component of class \(C^{3,\alpha}\), the author defines the quantity\N\[\N\Lambda(M,g_{M})=\inf_{\Omega}\lambda(\Omega,v).\N\]\NThe main result of the paper is the following:\N\NTheorem. Let \((M, g_M)\) be a complete noncompact hyperbolic doubly connected surface with conformal compactification \((\Omega_{0},g_{E})\), where \(\Omega_{0}\subset \mathbb{C}\) is bounded. Then\N\[\N\Lambda(M,g_{M})=\frac{2\pi^{2}}{3}\left[\left(\frac{\pi}{\log\beta}\right)^{2}+1\right]\N\]\Nwhere \(0<\beta< 1\) is the exponent of the modulus of continuity of \(\Omega_{0}\), so that \(\Omega_{0}\) is biholomorphic to \(B_{1} \setminus \overline{B}_{\beta}\). Moreover, for any bounded \(\Omega\) biholomorphic to \(\Omega_{0}\) with \(C^{3,\alpha}\) outermost boundary component, the inequality\N\[\N\lambda(\Omega,v)=\frac{2\pi^{2}}{3}\left[\left(\frac{\pi}{\log\beta}\right)^{2}+1\right]\N\]\Nand equality holds if and only if \(\Omega\) is the image of \(B_{1} \setminus \overline{B}_{\beta}\) under a composition of translation and homotheties.