Vol. 40, No. 2, 1972

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ISSN: 0030-8730
A three point condition for surfaces of constant mean curvature

Eamon Boyd Barrett

Vol. 40 (1972), No. 2, 269–277
Abstract

Let ϕ(x,y) be a solution to the equation:

(1) (1+ ϕ2y)ϕxx − 2ϕxϕyϕxy + (1 + ϕ2x)ϕyy = 2H (1 + ϕ2x + ϕ2y)3∕2

The quantity H in equation (1) represents the mean curvature of the surface z = ϕ(x,y). In case H = 0, (1) is the minimal surface equation. For minimal surfaces, the wellknown three point condition may be stated as follows:

Theorem 1. Let ϕ(x,y) be a solution to the Dirichlet problem for the minimal surface equation in some bounded region R. Let T be the continuous space curve defined by the values of ϕ(x,y) over ∂R, the boundary of R. Then, if P is a plane tangent to the surface z = ϕ(x,y) for (x,y) in R,P will have at least 4 points in common with T.

The objective of this paper is to establish a natural analogue of the three-point condition for surfaces of positive, constant mean curvature.

Mathematical Subject Classification 2000
Primary: 53A10
Milestones
Received: 23 November 1970
Published: 1 February 1972
Authors
Eamon Boyd Barrett