Application of a unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian three-space

TL;DR

This paper unifies Kenmotsu-type formulas for describing surfaces in Euclidean and Lorentzian three-space, enabling new insights into constant mean curvature surfaces through a single comprehensive equation.

Contribution

It presents a unified Kenmotsu-type formula applicable to both Euclidean and Lorentzian 3-spaces, simplifying the analysis of constant mean curvature surfaces.

Findings

Unified formula written in a single equation

Application to rotational and helicoidal surfaces

Analysis of surfaces with constant mean curvature

Abstract

Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rotational or helicoidal surfaces with constant mean curvature. Before that, we show that the three formulas above can be written in a unified single equation.