White, Brian New applications of mapping degrees to minimal surface theory. (English) Zbl 0638.58005 J. Differ. Geom. 29, No. 1, 143-162 (1989). This paper uses global analysis to show: (1) For most of the known examples of area minimizing hypercones C with isolated singularities, there exists a complete singular minimal hypersurface that is asymptotic to C at infinity but that is not a cone, and a complete area minimizing hypersurface that is asymptotic to C at infinity but that is neither a cone nor any leaf of the Hardt-Simon minimal foliation associated to a cone. (2) If N is a compact Riemannian 3-manifold with boundary, \(\partial N\) is mean-convex and diffeomorphic to the 2-sphere, and if N is not diffeomorphic to the 3-ball or if N contains a compact minimal surface without boundary, then there exists a sequence of smooth embedded minimal disks \(D_ i\subset N\) such that \(\partial D_ i\subset \partial N\) converges to a smooth embedded curve and such that \(Area(D_ i)\to \infty.\) (3) If \(\Phi\) is a smooth even parametric elliptic functional on \(R^ 3\) and if \(\gamma\) is a smooth embedded curve on the boundary of a convex set in \(R^ 3\), then \(\gamma\) bounds at least one smooth embedded \(\Phi\)-stable disk. Almost every such \(\gamma\) bounds an odd number of \(\Phi\)-stationary embedded disks and an even number of embedded \(\Phi\)- stationary surfaces of each genus \(\neq 0\). Reviewer: B.White Cited in 11 Documents MSC: 58E12 Variational problems concerning minimal surfaces (problems in two independent variables) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q05 Minimal surfaces and optimization Keywords:minimal hypersurface; Hardt-Simon minimal foliation; minimal surfaces; parametric elliptic functionals PDFBibTeX XMLCite \textit{B. White}, J. Differ. Geom. 29, No. 1, 143--162 (1989; Zbl 0638.58005) Full Text: DOI