On the Canham Problem: Bending Energy Minimizers for any Genus and Isoperimetric Ratio

Abstract

Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in R 3 of genus g and given isoperimetric ratio v, there exists one with minimum bending energy W. We do this by gluing g + 1 small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation ∆(∆ + 2)ϕ = 0 on the (g + 1)-punctured sphere S 2 to construct a comparison surface of genus g with arbitrarily small isoperimetric ratio v ∈ (0, 1) and W < 8π.

The Willmore flow with prescribed isoperimetric ratio

“…has already been studied in [35,17,31]. While the genus zero case was solved in [35], the results in [17,31] combined with recent findings in [34] and [19] show that the infimum in (1.3) is always attained for any g ∈ N 0 and σ ∈ (0, 1); and satisfies β g (σ) < 8π. The energy threshold 8π also plays an important role in the analysis of the Willmore energy, since by the famous Li-Yau inequality [26], any immersion f of a compact surface with W(f ) < 8π has to be embedded.…”

“…However, for tori we have W(f 0 ) ≥ 2π 2 by [27], and hence 2π 2 ≤ W(f 0 ) < 4π σ can only hold for σ < 2 π < 1. On the other hand, for σ ∈ (0, 1 2 ] and arbitrary genus, there exists f 0 with W(f 0 ) < 4π σ since β g (σ) < 8π by [19,Theorem 1.2]. We now use Lemma 4.2 to deduce the time integrability (3.15) for λ in the case of small curvature concentration, which enables us to bound all derivatives of the second fundamental form by Proposition 3.7.…”