Blaschke's problem for timelike surfaces in pseudo-Riemannian space forms

TL;DR

This paper explores solutions to Blaschke's problem for timelike surfaces in pseudo-Riemannian space forms, revealing connections with isothermic, S-Willmore, and minimal surfaces in both spacelike and timelike contexts.

Contribution

It extends classical Blaschke's problem solutions to timelike surfaces in pseudo-Riemannian spaces, linking isothermic and S-Willmore surfaces with minimal surface descriptions.

Findings

Isothermic and S-Willmore surfaces solve Blaschke's problem in pseudo-Riemannian space forms.

Timelike Willmore and isothermic surfaces are characterized by minimal surfaces.

Classical results of Thomsen are generalized to timelike surfaces.

Abstract

We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical results of Thomsen.