Biharmonic surfaces of constant mean curvature

Eric Loubeau and Cezar Oniciuc

Vol. 271 (2014), No. 1, 213–230

DOI: 10.2140/pjm.2014.271.213

Abstract

We compute a Simons-type formula for the stress-energy tensor of biharmonic maps from surfaces. Specializing to Riemannian immersions, we prove several rigidity results for biharmonic CMC surfaces, putting in evidence the influence of the Gaussian curvature on pseudoumbilicity. Finally the condition of biharmonicity is shown to enable an extension of the classical Hopf theorem to CMC surfaces in any ambient Riemannian manifold.

Keywords

biharmonic maps, constant mean curvature, stress-energy tensor

Mathematical Subject Classification 2010

Primary: 53C42, 53C43, 58E20

Milestones

Received: 11 June 2013

Accepted: 21 December 2013

Published: 10 September 2014