WILLMORE FLOW

∂X/∂t = -(ΔH + 2H(H²−K)) — surfaces minimizing bending energy
Willmore energy:
W = ∫H²dA ≥ 4π

Minimum at 4π: round sphere.
For tori: W(Clifford) = 2π²

Willmore conjecture (1965):
min W(torus) = 2π²
Proved by Marques-Neves 2012!

Current W:
W/4π: