Harmonic maps of conic surfaces with cone angles less than \(2\pi\) (Q748344)

In the paper under review, the author proves the existence and uniqueness of harmonic maps in degree-one homotopy classes of closed, orientable surfaces of positive genus, where the target has non-positive Gaussian curvature and conic points. Since [\textit{M. Gromov} and \textit{R. Schoen}, Publ. Math., Inst. Hautes Étud. Sci. 76, 165--246 (1992; Zbl 0896.58024)], many authors study harmonic maps into singular spaces. Particulary detailed results are available in the case of maps of surfaces. For instance, in [Math. Z. 221, No. 3, 421--436 (1996; Zbl 0871.58028)], \textit{E. Kuwert} studied degree-one harmonic maps of closed Riemann surfaces into flat, conic surfaces with cone angles bigger than \(2\pi\). Kuwert showed that the minimizing maps can be obtained as limits of diffeomorphisms, and that the inverse image under a degree-one harmonic map of each point in the singular set is the union of a finite number of vertical arcs of the Hopf differential. In [ibid. 242, No. 4, 633--661 (2002; Zbl 1044.58021)], \textit{C. Mese} proved the same result when the target is a metric space with curvature bounded above, in particular, when it is a conic surface with cone angle bigger than \(2\pi\). In this paper, the author studies cone angles less than \(2\pi\). Let \(\Sigma\) be a closed surface of positive genus, equipped with a conformal structure and a Riemannian metric, with a conic point \(p\) of cone angle less than \(2\pi\) and non-positive Gaussian curvature away from \(p\). The author proves that if \(\phi: \Sigma \to \Sigma\) is a homeomorphism, then there is a unique map \(u: \Sigma \to \Sigma\) which minimizes energy in the homotopy class of \(\phi\), and this map \(u\) satisfies (i) \(u^{-1}(p)\) is a single point, and (ii) \(u: \Sigma -u^{-1}(p) \to \Sigma -p\) is a diffeomorphism. When the cone angle is less than \(\pi\), the author obtains a more refined result. Namely, if the cone angle is less than or equal to \(\pi\), then for each \(q \in\Sigma\) and each homeomorphism \(\phi\) of \(\Sigma\) with \(\phi(q) = p\), there is a unique map \(u : \Sigma \to \Sigma\) with \(u(q) = p\) which minimizes energy in the relative \(q\)-homotopy class of \(\phi\). The author also discusses the regularity of these maps near the inverse images of the cone points in detail. Finally, when the genus is zero, he proves the same relative minimization provided there are at least three cone points of cone angle less than or equal to \(\pi\).