A relation between cylindrical critical points of Willmore-type energies, weighted areas and vertical potential energies

TL;DR

This paper explores the relationships between different physical energies related to hypersurfaces, focusing on cylindrical critical points, and demonstrates stability and minimality properties of fluid interfaces under these energies.

Contribution

It establishes connections between Willmore-type, weighted area, and vertical potential energies for cylindrical hypersurfaces, revealing conditions under which their generating curves coincide and analyzing stability.

Findings

Generating curves coincide for Willmore-type and weighted area energies.

Generating curves of Willmore-type and vertical potential energies match after parameter adjustments.

Fluid interfaces as graphs are proven to be stable and globally minimizing under vertical potential energies.

Abstract

This paper considers the energies of three different physical scenarios and obtains relations between them in a particular case. The first family of energies consists of the Willmore-type energies involving the integral of powers of the mean curvature which extends the Willmore and Helfrich energies. A second family of energies is the area functionals arising in weighted manifolds, following the theory developed by Gromov, when the density is a power of the height function. The third one is the free energies of a fluid deposited in a horizontal hyperplane when the potentials depend on the height with respect to this hyperplane. In this paper we find relations between each of them when the critical point is a hypersurface of cylindrical type. Cylindrical hypersurfaces are determined by their generating planar curves and for each of the families of energies, these curves satisfy suitable ordinary differential equations. For the Willmore-type energies, the equation is of fourth order, whereas it is of order two in the other two cases. We prove that the generating curves coincide for the Willmore-type energies without area constraint and for weighted areas, and the similar result holds for the generating curves of Willmore-type energies and of the vertical potential energies, after suitable choices of the physical parameters. In all the cases, generating curves are critical points for a family of energies extending the classical bending energy. In the final section of the paper, we analyze the stability of a liquid drop deposited on a horizontal hyperplane with vertical potential energies. It is proven that if the free interface of the fluid is a graph on this hyperplane, then the hypersurface is stable in the sense that it is a local minimizer of the energy. In fact, we prove that the hypersurface is a global minimizer in the class of all graphs with the same boundary.