Weil-Petersson convexity of the energy functional on classical and universal Teichmüller spaces.
Let \(N\) be a closed surface of genus \(\geq 2\) and let \(\mathcal T\) be its Teichmüller space, equipped with the Weil-Petersson metric. There is a natural energy functional defined on \(\mathcal T\), \({\mathcal E}: {\mathcal T}\to {\mathbb R}\), and the main result of the author is that this function is a strictly convex function (that is, it is strictly convex along any Weil-Petersson geodesic). Now let \(M\) be a (not necessarily connected) \(n-\)dimensional manifold equipped with a Riemannian metric \(g\). For any hyperbolic metric \(G\) on \(N\), let \(u:(M,g)\to (N,G)\) be a harmonic map and suppose that the induced map \[ u_*:\pi(M)\to\pi(N) \] is surjective. Then, there is natural associated energy functional defined on \(\mathcal T\). As an application of the above result, the author proves that this function is strictly convex and proper, and hence it has a unique minimum in \(\mathcal T\). The last statement is reminiscent in some sense of a famous result of S. Kerckhoff in which the domain \(M\) is a disjoint union of circles which are mapped to a collection of filling geodesics in the surface \(N\) [see \textit{S. Kerckhoff}, Ann. Math. (2) 117, 235-265 (1983; Zbl 0528.57008)]. Kerckhoff's result states that the sum of the length of these geodesics is a proper and convex functional with respect to the Teichmüller metric. \textit{S. Wolpert} proved an analogous result for the Weil-Petersson metric [J. Differ. Geom. 25, 275-296 (1987; Zbl 0616.53039)].
scientific article; zbMATH DE number 1465059