Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy

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.

arXiv comment29 pages. Final version published in Archive for Rational Mechanics and Analysis

References in corpus (4)

• Willmore minimizers with prescribed isoperimetric ratio
• Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint
• Immersed Spheres of Finite Total Curvature into Manifolds
• Rigidity and stability of spheres in the Helfrich model

Cited by in corpus (5)

• Li-Yau inequalities for the Helfrich functional and applications
• On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves
• Embedded Delaunay tori and their Willmore energy
• A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint
• Lower-semicontinuity for the Helfrich problem