Some topics in differential geometry of normed spaces

V. Balestro, H. Martini and R. Teixeira, Some topics in differential geometry of normed spaces, to appear in Adv. Geom., 2020. (arXiv link)

Comments: When one extend the “core” concepts of Euclidean differential geometry to surfaces immersed in normed spaces a lot of topics are worth investigating. This paper is devoted to shed some light on some of this topics. We study minimal immersions, proving a lot of characterizations which are very similar to the Euclidean subcase. We also prove global theorems such as Hadamard-type theorems and an extension of the classical Bonnet theorem. Surfaces with constant Minkowski width are also investigated. Further on, we briefly study the induced ambient metric, obtaining an upper bound for the perimeter of a compact, connected surface (in the sense of Schäffer).