Rigidity in the Class of Orientable Compact Surfaces of Minimal Total Absolute Curvature

TL;DR

This paper proves that orientable compact surfaces in 3D with minimal total absolute curvature are rigid under certain curvature sign change and nondegeneracy conditions, meaning any isometric surface is congruent by a Euclidean motion.

Contribution

It establishes a rigidity theorem for such surfaces, extending classical results to those with sign-changing Gaussian curvature under specific conditions.

Findings

Surfaces with minimal total absolute curvature are rigid under given conditions.

Any isometric surface differs by at most a Euclidean motion.

The result applies to surfaces with Gaussian curvature changing sign to finite order.

Abstract

Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion.