A Weierstrass type representation for translating solitons and singular minimal surfaces

TL;DR

This paper introduces a Weierstrass representation formula for translating solitons and singular minimal surfaces in three-dimensional space, enabling new insights into their geometric properties and solving related Cauchy problems.

Contribution

It provides a novel Weierstrass representation for translating solitons and singular minimal surfaces, expanding the tools for analyzing these geometric objects.

Findings

Characterization of when the Euclidean Gauss map has a harmonic argument

Solution to a general Cauchy problem for these surfaces

New representation formula for translating solitons and singular minimal surfaces

Abstract

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in ${\mathbb{R}^3}$. As application we study when the euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.