Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index

TL;DR

This paper classifies minimal Lorentz surfaces in indefinite space forms, extending the understanding of minimal surface theory to Lorentzian geometry with arbitrary codimension and index, providing comprehensive classification results.

Contribution

It offers the first complete classification of minimal Lorentz surfaces in pseudo-Euclidean spaces of arbitrary dimension and index, filling a significant gap in Lorentzian minimal surface theory.

Findings

Complete classification of minimal Lorentz surfaces in pseudo-Euclidean spaces

New results on minimal Lorentz surfaces in indefinite space forms

Extension of classical minimal surface theory to Lorentzian geometry

Abstract

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are known. Hence, in this paper we investigate minimal Lorentz surfaces in arbitrary indefinite space forms. As a consequence, we obtain several classification results for minimal Lorentz surfaces in indefinite space forms. In particular, we completely classify all minimal Lorentz surfaces in a pseudo-Euclidean space $\mathbb E^m_s$ with arbitrary dimension $m$ and arbitrary index $s$.