Stationary rotating surfaces in Euclidean space

TL;DR

This paper investigates the geometric properties of stationary rotating surfaces in Euclidean space, which model rotating fluid drops without gravity, focusing on boundary relationships and special cases like axisymmetric and embedded surfaces.

Contribution

It provides a detailed analysis of the geometry of these surfaces, especially their boundary behavior and specific subclasses such as axisymmetric and embedded configurations.

Findings

Characterization of boundary relationships for stationary rotating surfaces

Analysis of axisymmetric solutions and their properties

Insights into embedded surfaces with planar boundaries

Abstract

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature $H$ at each point $x$ satisfies $2H(x)=a r^2+b$, where $r$ is the distance from $x$ to a fixed straight-line $L$, and $a$ and $b$ are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about $L$ with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.