Communications in Mathematical Sciences
Volume 15 (2017)
Helfrich’s energy and constrained minimisation
Pages: 2373 – 2386
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.
Helfrich energy, Willmore energy, constrained minimisation, topological type, varifold
2010 Mathematics Subject Classification
49Q10, 49Q20, 53C80
Paper received on 25 February 2017.
Paper accepted on 25 July 2017.