Some topics in differential geometry of normed spaces

TL;DR

This paper explores the differential geometry of surfaces in normed spaces, defining curvature and metric analogues, and proves new characterizations of minimal surfaces, global theorems, and properties of surfaces with constant Minkowski width.

Contribution

It introduces and analyzes concepts of curvature and minimal surfaces in normed spaces, extending classical differential geometry results to this setting.

Findings

Characterizations of minimal surfaces in normed spaces

Extensions of Hadamard-type theorems

Results on curvature of surfaces with constant Minkowski width

Abstract

For a surface immersed in a three-dimensional space endowed with a norm instead of an inner product, one can define analogous concepts of curvature and metric. With these concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces, and analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces of constant Minkowski width is also given. Finally, we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.