Comparison surfaces for the Willmore problem

Robert Kusner

Vol. 138 (1989), No. 2, 317–345

DOI: 10.2140/pjm.1989.138.317

Abstract

The infimum of the conformally invariant functional W = ∫ H² is estimated for each regular homotopy class of immersed surfaces in R³. Consequently, we obtain rather sharp bounds on the maximum multiplicity and branching order of a W-minimizing surface. In the case of RP² we provide an example of a symmetric W-minimizing Boy’s surface (W = 12π)—as well as symmetric static surfaces of higher index—thereby solving part of the Willmore problem.

Mathematical Subject Classification 2000

Primary: 53A10

Secondary: 53A30, 53C42, 58E12

Milestones

Received: 4 May 1987

Revised: 16 May 1988

Published: 1 June 1989