Convexity of energy functions of harmonic maps homotopic to covering maps of surfaces
Journal article, 2023

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil-Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases.

Teichmuller space

energy function

Harmonic map



Inkang Kim

Korea Institute for Advanced Study

Xueyuan Wan

Chongqing University of Technology

Chalmers, Mathematical Sciences, Algebra and geometry

University of Gothenburg

Genkai Zhang

University of Gothenburg

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Communications in Contemporary Mathematics

0219-1997 (ISSN) 17936683 (eISSN)

Vol. In press 2350054

Representations of Lie groups. Harmonic and complex analysis on symmetric and locally symmetric spaces

Swedish Research Council (VR), 2019-01-01 -- 2022-12-31.

Swedish Research Council (VR) (2022-02861), 2023-01-01 -- 2026-12-31.

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Mathematical Analysis



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