Helfrich's Energy and Constrained Minimisation

TL;DR

This paper constructs specific surfaces with minimal Willmore energy, analyzes minimising sequences, and explores boundedness and compactness properties of surfaces under energy constraints, advancing understanding of Helfrich's energy in geometric analysis.

Contribution

It provides explicit constructions of low-energy surfaces, proves convergence of minimising sequences to doubly covered spheres, and establishes compactness results for surfaces with bounded Willmore energy.

Findings

Minimising sequences converge to doubly covered spheres for all genus g.

Surfaces with bounded Willmore energy and area form a compact class under varifold convergence.

Canham-Helfrich energies with non-positive k are bounded below; positive k energies are not.

Abstract

For every $g\in\mathbb{N}_0$ and $\epsilon>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8\pi$ and Willmore energy smaller than $8\pi + \epsilon$. From this we deduce that a minimising sequence for Willmore's energy in the class of genus $g$ surfaces embedded in the unit ball with area $8\pi$ converges to a doubly covered sphere for all $g\in\mathbb{N}_0$. We obtain the same result for certain Canham-Helfrich energies with $\chi_K\leq 0$ without genus constraint and show that Canham-Helfrich energies with $\chi_K>0$ are not bounded from below in the class of smooth surfaces with area $S$ embedded into a domain $\Omega\Subset \mathbb{R}^3$. Furthermore, we prove that the class of connected surfaces embedded in a domain $\Omega\Subset\mathbb{R}^3$ with uniformly bounded Willmore energy and area is compact under varifold convergence.