On curvature of surfaces immersed in normed spaces

V. Balestro, H. Martini and Ralph Teixeira, On curvature of surfaces immersed in normed spaces, preprint, 2018. (arXiv link)

Comments: In this paper we explore a little deeper the properties of Minkowski Gaussian and mean curvatures of a surface immersed in a three-dimensional space endowed with a norm. Several results analogous to the Euclidean case are obtained. The most interesting, in our opinion, is an extension of Alexandrov’s theorem, which states that any embedded compact surface without boundary whose Minkowski mean curvature is constant must be a Minkowski sphere (that is, a sphere of the ambient norm). Planar versions of some results are also discussed, as well as some problems.