Spherical type surfaces via support function

TL;DR

This paper introduces SS-surfaces, a new class of spherical type surfaces defined through support functions, providing a Weierstrass representation, classification of rotational surfaces, and explicit examples, including the sphere.

Contribution

It develops a novel framework for SS-surfaces using support functions, including a Weierstrass representation and classification results.

Findings

Every compact SS-surface is a sphere

Explicit examples of SS-surfaces are provided

A classification of rotational SS-surfaces is achieved

Abstract

In this work we define the surfaces spherical type via support function (in short, SS-surfaces). We present a Weierstrass type representation for SS-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we use this representation to classify the surfaces of rotation. Moreover, we show that every compact and connected SS-surface is the sphere and we give explicit examples of SS-surfaces.