Uniqueness of Clifford torus with prescribed isoperimetric ratio

Abstract

The Marques-Neves theorem asserts that among all the torodial (i.e. genus 1) closed surfaces, the Clifford torus has the minimal Willmore energy ∫ H 2 d A \int H^2 \, dA . Since the Willmore energy is invariant under Möbius transformations, it can be shown that there is a one-parameter family, up to homotheties, of genus 1 Willmore minimizers. It is then a natural conjecture that such a minimizer is unique if one prescribes its isoperimetric ratio. In this article, we show that this conjecture can be reduced to the positivity question of a polynomial recurrence. A proof of the positivity can be found in the companion article by Melczer and Mezzarobba [submitted to J. Comb. Theory (2020)]. This establishes a first uniqueness result for the Canham model of biomembranes.