Circles-foliated stationary surfaces of the Dirichlet energy

TL;DR

This paper investigates surfaces with constant anisotropic mean curvature related to the Dirichlet energy, extending classical minimal surface examples to anisotropic settings and classifying such surfaces foliated by circles.

Contribution

It constructs and classifies surfaces with zero anisotropic mean curvature foliated by circles, including non-rotational examples, and extends classical minimal surface theory to anisotropic contexts.

Findings

Existence of non-rotational circle-foliated surfaces with zero anisotropic mean curvature.

Classification of all zero anisotropic mean curvature surfaces foliated by circles.

Extension of Riemann minimal examples to anisotropic Dirichlet energy setting.

Abstract

In Euclidean space we study surfaces with constant anisotropic mean curvature $\Lambda$ of the Dirichlet energy $\int_\Omega( |Du|^2+\Lambda u)$. We prove the existence of non-rotational surfaces with $\Lambda=0$ and foliated by a one-parameter family of circles contained in horizontal planes obtaining a geometric description of them. These surfaces extend the known Riemann examples of the theory of minimal surfaces to the anisotropic context of the Dirichlet energy. More general, we classify all surfaces with zero anisotropic mean curvature foliated by circles proving that either the surface is axially symmetric about the $z$-axis or the surface belongs to one of the above examples. We also study the case that the anisotropic mean curvature is a non-zero constant.