Two classification results for stationary surfaces of the least moment of inertia

TL;DR

This paper classifies stationary surfaces in Euclidean space based on a specific energy functional, revealing that ruled surfaces are planes or elongated helicoids, and circle-foliated surfaces are rotational, with special cases for certain alpha values.

Contribution

It provides a comprehensive classification of alpha-stationary surfaces, identifying all ruled surfaces and those foliated by circles, including special cases for specific alpha values.

Findings

Ruled alpha-stationary surfaces are planes or elongated helicoids.

Surfaces foliated by circles are rotational unless alpha is -4 or -2.

Special non-spherical cyclic (-2)-stationary surfaces are constructed.

Abstract

A surface in Euclidean space $\r^3$ is said to be an $\alpha$-stationary surface if it is a critical point of the energy $\int_\Sigma|p|^\alpha$, where $\alpha\in\r$. We prove that all ruled $\alpha$-stationary surfaces are vector planes (for all $\alpha$) and a type of elongated helicoids (for $\alpha=1$). The second result of classification asserts that if $\alpha\not=-2,-4$, any $\alpha$-stationary surface foliated by circles must be a rotational surface. If $\alpha=-4$, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If $\alpha=-2$, we find many non-spherical cyclic $(-2)$-stationary surfaces.