Min-max minimal hypersurfaces in non-compact manifolds (Q309701)

It is well known that there are no immersed closed minimal surfaces in Euclidean \(3\)-space, showing that there are geometric obstructions for a Riemannian manifold to admit such minimal surfaces. The author analyses such obstructions and proves the existence of closed minimal hypersurfaces in certain non-compact Riemannian manifolds of dimension \(3\leq n\leq 7\), that contain a bounded open subset with smooth and strictly mean-concave boundary and that satisfy certain natural conditions on the injectivity radius and the sectional curvature at infinity (Theorem 1.1). Further, it is proved that any region as above, with smooth and strictly mean-concave boundary, of a compact Riemannian manifold of dimension \(3\leq n\leq 7\) intersects a closed minimal hypersurface (Theorem 1.2).NEWLINENEWLINETo prove these results the author develops a modified min-max theory for the area functional to produce minimal hypersurfaces with intersecting properties, following ideas of \textit{F. Almgren} [The theory of varifolds. Princeton (1965)] and \textit{J. T. Pitts} [Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0462.58003)].NEWLINENEWLINEIn the first sections of the paper the author presents the key ideas and main techniques employed to prove the two theorems in a more informal way -- providing the reader with the oversight and red tape to appreciate the later, more technical parts of the paper. This is one of the aspects that make the paper under review not only very interesting, for its results and methods, but also a pleasure to read.