On the asymptotics for minimizers of Donaldson functional in Teichmüller theory
Gabriella Tarantello
Dipartimento di Matematica, Universita’ di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
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This book chapter is published open access.
Abstract
We discuss the asymptotic behavior of minimizers for a Donaldson functional of interest in Teichmüller theory. For example, such minimizers allow one to parametrize the moduli space of constant mean curvature immersions of a closed surface of genus into a -manifold with sectional curvature , by elements of the tangent bundle of the Teichmüller space of . The minimizers are governed by a system of PDEs which include a Gauss equation of Liouville type and a holomorphic -differential.
In our asymptotic analysis, we face the difficulty to describe the possible blow-up behavior of minimizers, especially when it occurs at a point where different zeroes of the holomorphic -differential coalesce. Therefore, we need to pursue accurate estimates of the blow-up profile of solutions for Liouville type equations, in the “collapsing” case.