Some Applications of Surface Curvatures in Theoretical Physics

TL;DR

This survey explores how surface curvatures are applied in theoretical physics, specifically in biophysics for cell shape modeling and in astrophysics for cosmic string analysis, highlighting new energy functionals and mathematical challenges.

Contribution

It introduces a scale-invariant anisotropic bending energy extending the Canham energy and analyzes its minimizers, along with topological bounds, in biophysics, and discusses the geometric framework for matter accretion in astrophysics.

Findings

Existence obstruction for Helfrich energy minimizers on ring tori.

Unique toroidal energy minimizer for the proposed anisotropic energy.

Topological bounds depending on genus for the new energy functional.

Abstract

In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.