The existence of surfaces of constant mean curvature with free boundaries (Q1103877)

For a given surface S in \({\mathbb{R}}^ 3\) diffeomorphic to \(S^ 2\) the author considers the problem of finding disc-type surfaces of prescribed constant mean curvature which intersect S orthogonally along their boundary. He uses the parametric approach and looks at the corresponding evolution problem. If I denotes the radius of the smallest ball containing S he is able to show the existence of such a (nonconstant) surface having mean curvature H provided \(H\in {\mathfrak H}\subset [- 1/L,1/L]\). Here, \(0\in {\mathfrak H}\) and \({\mathfrak H}\) is shown to be dense in \([-1/L,1/L].\) Furthermore, the solution surface stays in that ball of radius L containing S. This theorem includes as a special case \((H=0)\) an earlier result by the author [Invent. Math. 75, 547-560 (1984; Zbl 0537.35037)] but uses the direct method developed in [Comment. Math. Helv. 60, 558-581 (1985; Zbl 0595.58013)]. It remains an open problem to prove the existence of such surfaces for any \({\mathfrak H}\subset {\mathbb{R}}\). [A printing error on p. 62 has been corrected in a later volume of this journal.]