Biharmonic surfaces of constant mean curvature

TL;DR

This paper derives a Simons' type formula for biharmonic maps from surfaces, proves rigidity results for biharmonic CMC surfaces, and extends the Hopf theorem to these surfaces in general Riemannian manifolds.

Contribution

It introduces a new formula for stress-energy tensors of biharmonic maps and extends classical theorems to biharmonic CMC surfaces in arbitrary ambient spaces.

Findings

Rigidity results for biharmonic CMC surfaces

Influence of Gaussian curvature on pseudo-umbilicity

Extension of Hopf theorem to biharmonic CMC surfaces

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 pseudo-umbilicity. Finally, the condition of biharmonicity is shown to enable an extension of the classical Hopf theorem to CMC surfaces in any ambient Riemannian manifold.