Generalized polar transforms of spacelike isothermic surfaces

TL;DR

This paper extends the concept of polar transforms of spacelike isothermic surfaces to higher-dimensional pseudo-Riemannian spaces, introduces new classes of such surfaces, and explores their properties, including invariance of Willmore functional and transformation permutability.

Contribution

It generalizes polar transforms to n-dimensional pseudo-Riemannian spaces and characterizes the resulting surfaces as generalized H-surfaces with null minimal sections.

Findings

Existence of c-polar spacelike isothermic surfaces in various space forms.

c-polar surfaces preserve the Willmore functional under certain conditions.

Derived permutability theorems for c-polar, Darboux, and spectral transforms.

Abstract

In this paper, we generalize the polar transforms of spacelike isothermic surfaces in $Q^4_1$ to n-dimensional pseudo-Riemannian space forms $Q^n_r$. We show that there exist $c-$polar spacelike isothermic surfaces derived from a spacelike isothermic surface in $Q^n_r$, which are into $S^{n+1}_r(c)$, $H^{n+1}_{r-1}(c)$ or $Q^n_r$ depending on $c>0,<0,$ or $=0$. The $c-$polar isothermic surfaces can be characterized as generalized $H-$surfaces with null minimal sections. We also prove that if both the original surface and its $c-$polar surface are closed immersion, then they have the same Willmore functional. As examples, we discuss some product surfaces and compute the $c-$polar transforms of them. In the end, we derive the permutability theorems for $c-$polar transforms and Darboux transform and spectral transform of isothermic surfaces.