Helfrich’s energy and constrained minimisation

For every non-negative integer, we construct a smooth surface of the genus given by the integer and embedded into the unit ball such that the embedded manifold has surface area exactly twice as large as the unit sphere and Willmore energy only slightly larger than that of two spheres. From this we deduce that a minimising sequence for Willmore’s energy in the class of surfaces with some prescribed genus and area $8 \pi$ embedded in the unit ball converges to a doubly covered sphere. We obtain the same result for certain Canham–Helfrich energies without genus constraint and show that Canham–Helfrich energies in another parameter regime are not bounded from below in the class of smooth surfaces with prescribed area which are embedded into a fixed bounded domain.

Furthermore, we prove that the class of connected surfaces embedded in a bounded domain with uniformly bounded Willmore energy and area is compact under varifold convergence.

Keywords

Helfrich energy, Willmore energy, constrained minimisation, topological type, varifold

2010 Mathematics Subject Classification

49Q10, 49Q20, 53C80

Full Text (PDF format)

Received 25 February 2017

Accepted 25 July 2017

Published 20 December 2017