Homeomorphic nature of E_p minimizers in a fixed homotopy class

Determine whether, for closed hyperbolic Riemann surfaces Σ1 and Σ2 of genus g ≥ 2 and for any p ≥ 1, every minimizer of the p-conformal energy E_p among Sobolev mappings of finite distortion in the homotopy class of a homeomorphism f0: Σ1 → Σ2 is itself a homeomorphism. Equivalently, establish that weak minimizers of E_p in a fixed homotopy class are homeomorphic mappings rather than merely weak Sobolev minimizers.

Background

The paper studies the p-conformal energy E_p of Sobolev mappings of finite distortion between closed Riemann surfaces and proves properness properties on Teichmüller space by analyzing extremal mappings between hyperbolic annuli. In the classical Teichmüller setting (p = ∞), the extremal map is quasiconformal with constant distortion.

For finite p ≥ 1, weak minimizers of E_p exist in a homotopy class, but it is not established whether such minimizers are actual homeomorphisms. Resolving this would clarify the structure of E_p-extremals in L^p Teichmüller theory and would align with the conjectural picture suggested in prior work.