Remarks on the boundary curve of a constant mean curvature topological disc

Abstract

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.

Keywords:

• Constant mean curvature
• holomorphic quadratic differentials
• capillary surface

AMS Subject Classifications:

• Primary: 53A10
• 49Q10
• Secondary: 35J60
• 76D45

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second-named author has been partially supported by the MINECO/FEDER   [grant number MTM2014-52368-P].