# Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy

## TL;DR

This paper proves the existence and regularity of minimizers for the Canham-Helfrich energy in the class of weak immersions of the 2-sphere, solving a long-standing problem in modeling lipid bilayer membranes.

## Contribution

It establishes the existence and regularity of minimizers for the Canham-Helfrich energy in the spherical case, including weak and branched immersions, and proves lower semicontinuity of the energy.

## Key findings

- Existence of minimizers for the Canham-Helfrich energy on the 2-sphere.
- Regularity results for these minimizers.
- Lower semicontinuity of the energy under weak convergence.

## Abstract

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.