Bridge principle for constant and positive Gauss curvature surfaces
type: Item
instance of: scholarly article
title: Bridge principle for constant and positive Gauss curvature surfaces
zbMATH Open document ID: 0993.53004
author: L. Hauswirth
published in: Communications in Analysis and Geometry
publication date: 1999-11-23T00:00:00Z
review text: 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 techni…
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1339897
zbMATH Keywords: bridge principle
zbMATH Keywords: constant Gauss curvature
zbMATH Keywords: convex surface
reviewed by: Jean-Marc Schlenker
MaRDI profile type: Publication
DOI: 10.4310/CAG.1999.V7.N3.A2
description: scientific article; zbMATH DE number 1339897