Böhme, Reinhold Manifolds of dimension 3 with prescribed positive curvature and with nontrivial homology. (English) Zbl 0709.53042 Forum Math. 2, No. 5, 407-431 (1990). 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 Cited in 1 ReviewCited in 1 Document 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) Keywords:prescribed Gauss curvature; branched immersion; implicit function theorem × Cite Format Result Cite Review PDF Full Text: DOI EuDML