The Gauss curvature of an H-graph.(English)Zbl 0645.53003

An H-graph is a non-parametric surface of constant mean curvature H. The author considers such a graph, S, of a function u(x,y) defined on the disc of radius R centered at the origin. For such a surface, he provides an a priori estimate of the Gaussian curvature of S at (0, 0, u(0,0)). The estimate depends only on R and H, and does not require that $$\nabla u(0,0)$$ vanish. This result improves on an estimate of J. Spruck [Proc. Am. Math. Soc. 36, 217-223 (1972; Zbl 0256.53007)].
The method of proof is comparison with a majorant surface known as a nodoid; the nodoids being one family of solutions to the ordinary differential equation for a radially symmetric surface of constant mean curvature. The author is also able to provide bounds on the second derivatives of u at (0,0), which depend only on H, R, and $$| \nabla u(0,0)|$$. The relevant quantities in the estimates are expressible explicitly in terms of elliptic integrals.
Reviewer: H.Parks

MSC:

 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q05 Minimal surfaces and optimization

Zbl 0256.53007