Minimal diffeomorphism between hyperbolic surfaces with cone singularities (Q2286176)

Let \(\Sigma\) be a closed oriented surface and let \(\mathfrak p =\{p_1, p_2, \dots, p_n\}\subset \Sigma\) be a finite set. Denote by \(\Sigma_{\mathfrak p}:= \Sigma\setminus \mathfrak p\) and let \(\alpha:=(\alpha_1, \alpha_2, \dots, \alpha_n)\in (0, \frac{1}{2})^n\) be such that \(\chi(\Sigma_{\mathfrak p}) - \sum_{i=1}^n (\alpha_i-1)<0\). By a theorem of \textit{R. C. McOwen} [Proc. Am. Math. Soc. 103, No. 1, 222--224 (1988; Zbl 0657.30033)] and \textit{M. Troyanov} [Enseign. Math. (2) 32, 79--94 (1986; Zbl 0611.53035)], this condition implies the existence of hyperbolic metrics with cone singularities of angle \(\alpha\). Recall that a metric \(g\) on \(\Sigma_{\mathfrak p}\) is a metric with cone singularities of angle \(\alpha\) if \(g\) is \(C^2\) on each compact subset \(K \subset \Sigma_{\mathfrak p}\) and for each puncture \(p_i \in \mathfrak p\), there exists a holomorphic chart \(z\) from a neighborhood of \(p_i\) to a disk of the form \(g = ce^{2\mu} |z|^{2(\alpha_i-1)}|dz|^2\), where \(\mu\) satisfies some Hölder regularity. Denote by \({\mathcal M}_\alpha(\Sigma_{\mathfrak p})\) the space of such metrics and define \({\mathcal M}^{-1}_\alpha(\Sigma_{\mathfrak p})\) as the space of hyperbolic metrics with cone singularities of \(\alpha\). The moduli space \({\mathcal F}_\alpha(\Sigma_{\mathfrak p})\), called the Fricke space, which is considered in this paper is the quotient of the infinite-dimensional space \({\mathcal M}^{-1}_\alpha(\Sigma_{\mathfrak p})\) of hyperbolic cone metrics on \(\Sigma_{\mathfrak p}\) by the action of the group of diffeomorphims isotopic to the identity. The author proves the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces. More precisely, for given \(\alpha, \alpha' \in (0, \frac{1}{2})^n, \, g_1 \in {\mathcal F}_\alpha(\Sigma_{\mathfrak p})\) and \(g_2 \in {\mathcal F}_\alpha'(\Sigma_{\mathfrak p})\), there exists a minimal diffeomorphism \(\Psi : (\Sigma_{\mathfrak p}, g_1) \to (\Sigma_{\mathfrak p}, g_2)\) isotopic to the identity when the cone angles of \(g_1\) and \(g_2\) are different and smaller than \(\pi\). Recall that a diffeomorphism \(f: (M, g) \to (N, h)\) between two Riemannian manifolds is called minimal if its graph is a minimal submanifold of \((M\times N, g\oplus h)\). The author also shows that if, moreover, for all \(i \in \{1,2,\cdots, n\}\), \(\alpha_i < \alpha'_i\), then \(\Psi\) is unique.