New applications of mapping degrees to minimal surface theory (Q1099453)

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 \(\neq 0\).