Bridge principle for constant and positive Gauss curvature surfaces
The original bridge principle states that, given two compact minimal surfaces with boundary, they can be connected by a thin strip in such a way that the resulting surface remains minimal. It was introduced by R. Courant and P. Levy, who gave insights of why it should be true and examples of its uses. Indeed, the bridge principle is a powerful tool for manufacturing interesting examples of minimal surfaces, like a rectifiable curve bounding an infinite number of topologically equivalent minimal surfaces, or a curve bounding a minimal surface of infinite genus. The bridge principle was proved for minimal surfaces by \textit{W. Meeks} and \textit{S. T. Yau} [Math. Z. 179, 151-168 (1992; Zbl 0479.49026)]. Later \textit{N. Smale} [Invent. Math. 90, 505-549 (1987; Zbl 0637.49020)] proved a ``weak version'' -- assuming that the surfaces are connected by a strip which is tangent to both surfaces -- which is also valid for sub-manifolds in arbitrary dimension and codimension, as well as for surfaces of constant, non-zero mean curvature. In the paper under consideration, the author proves a similar weak bridge principle for surfaces of constant Gauss curvature. More precisely, he considers two surfaces \(S_1\) and \(S_2\) with boundary in \(\mathbb{R}^3\), with constant, positive Gauss curvature \(K\). He supposes that both surfaces are analytic up to the boundary, and have no normal Jacobi field vanishing on the boundary. He proves that there exists a path \(\gamma\) going from the boundary of \(S_1\) to the boundary of \(S_2\), such that any neighborhood of \(S_1\cup \gamma\cup S_2\) contains a constant Gauss curvature \(K\) surface diffeomorphic to \(S_1\) and \(S_2\) joined by a strip. The proof uses some of the ideas introduced in the paper by N. Smale quoted above. The author first constructs an ``approximate solution'', and then uses a fixed point argument and a good understanding of the linearized problem to show the existence of a ``real'' solution. Important technical difficulties have to be overcome to complete this program. This paper is the published version of the author's thesis. Perhaps for this reason, it contains a very documented introduction which should be helpful to the non-specialists. This introduction, strangely, is in French, although the rest of the paper is in English. Readers understanding both languages should appreciate this paper.
scientific article; zbMATH DE number 1339897