Stationary surfaces of height-dependent weighted area functionals in $\mathbb{R}^3$ and $\mathbb{L}^3$

TL;DR

This paper establishes a correspondence between weighted minimal surfaces in Euclidean space and weighted maximal surfaces with singularities in Lorentzian space, providing representation formulas and analyzing their asymptotic behaviors and singularities.

Contribution

It introduces a general framework linking weighted minimal and maximal surfaces, including a Weierstrass representation and criteria for classifying singularities.

Findings

Established a correspondence between weighted minimal and maximal surfaces.

Provided a Weierstrass representation for these surfaces.

Analyzed asymptotic behavior and classified singularities.

Abstract

We describe a general correspondence between weighted minimal surfaces in $\mathbb{R}^3$ and weighted maximal surfaces with some admissible singularities in $\mathbb{L}^3$, for a class of functions $\varphi$ which provides the corresponding weight. For these families of surfaces, we provide a Weierstrass representation when $\dot{\varphi}\neq 0$ and analyze in detail the asymptotic behavior of both such a weighted maximal surface around its singular set and its corresponding weighted minimal immersion around the nodal set of its angle function, establishing criteria that allow us to easily determine the type of singularity and classify the associated moduli spaces.