Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

TL;DR

This paper explores spacelike Willmore surfaces in 4D Lorentzian space forms, introducing new transforms, duality theorems, and classifying certain surfaces, advancing the understanding of Lorentzian conformal geometry.

Contribution

It introduces two types of transforms for spacelike Willmore surfaces, establishes duality theorems, classifies spacelike Willmore 2-spheres, and constructs homogeneous Willmore tori.

Findings

Duality theorem for polar and adjoint surfaces

Classification of spacelike Willmore 2-spheres

Construction of homogeneous spacelike Willmore tori

Abstract

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.