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Manifolds of dimension 3 with prescribed positive curvature and with nontrivial homology. (English) Zbl 0709.53042

Let D denote the closed unit disc in \({\mathbb{R}}^ 2\) and let \(x_ 0: D\times S^ 1\to S^ 3\) be an embedding whose image is a tubular neighborhood of some great circle in the standard sphere \(S^ 3\subset {\mathbb{R}}^ 4\). Let furthermore f: \({\mathcal S}\to D\) be a 2-sheeted branched covering of a Riemann surface \({\mathcal S}\) of genus g with boundary \(\partial {\mathcal S}\cong \partial D\) onto D. Then the author considers the map \(x=X_ 0\circ (f,id)\) which, in an obvious sense, is a branched immersion of \({\mathcal S}\times S^ 1\) into \({\mathbb{R}}^ 4\) of constant Gauß curvature 1. It is the main result of the paper that the branched immersion x can be embedded in an infinite dimensional family of different branched immersions of \({\mathcal S}\times S^ 1\) into \({\mathbb{R}}^ 4\) of Gauß curvature 1. The proof is quite intricate and uses a “hard” implicit function theorem.
Reviewer: F.Tomi

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35J60 Nonlinear elliptic equations
58C15 Implicit function theorems; global Newton methods on manifolds
35J55 Systems of elliptic equations, boundary value problems (MSC2000)