The Variational Geometry of Surfaces in the Conformally Flat Space

TL;DR

This paper investigates the geometric properties of surfaces in conformally flat spaces, showing conditions for minimality, characterizing critical points of mean curvature, and deriving related Euler-Lagrange equations.

Contribution

It provides new insights into the variational geometry of surfaces in conformally flat spaces, including conditions for minimal surfaces and characterizations of critical points.

Findings

Surfaces tangent to conformal vector fields with constant mean curvature are minimal.

Critical points of the mean curvature functional are topologically spheres.

Euler-Lagrange equations for mean curvature and Willmore functionals are derived.

Abstract

In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change then surface is minimal. Also, it is determined that the critical point of mean curvature functional is homeomorphic to the sphere. Furthermore, the Euler-Lagrange equations associated to the mean curvature and Willmore functionals are determined.