Department of MathematicsWe discuss the Willmore Problem of minimizing the bending energy W := ∫H^2 among embedded surfaces in R^3 of genus g, and the related Canham Problem, where the isoperimetric ratio v := 36πVol^2/Area^3 ∈ (0, 1) is prescribed as well.
I. Existence for the Canham problem involves constructing a comparison surface of each genus g with W less than 8π and arbitrarily small isoperimetric ratio v by gluing g+1 small catenoidal bridges to the bigraph of a singular solution to the linearized Willmore equation ∆(∆+2)φ = 0 on the (g+1)-punctured sphere S^2 (joint with Peter McGrath).
II. More evidence for the conjecture that the Lawson surfaces with W less than 8π solve the Willmore problem: they are all W-stable, and they also minimize W among all surfaces with the same symmetries and genus (joint with Peng Wang and Ying Lü).
I. Existence for the Canham problem involves constructing a comparison surface of each genus g with W less than 8π and arbitrarily small isoperimetric ratio v by gluing g+1 small catenoidal bridges to the bigraph of a singular solution to the linearized Willmore equation ∆(∆+2)φ = 0 on the (g+1)-punctured sphere S^2 (joint with Peter McGrath).
II. More evidence for the conjecture that the Lawson surfaces with W less than 8π solve the Willmore problem: they are all W-stable, and they also minimize W among all surfaces with the same symmetries and genus (joint with Peng Wang and Ying Lü).