Embedding and surrounding with positive mean curvature (Q760684)

This article divides into two parts: a) Let \(M\) be a compact connected manifold with non-empty boundary \(\partial M\). According to a theorem of Gromov, \(M\) carries a Riemannian metric of strictly positive sectional curvature \(K\). The authors investigate under which topological assumptions one can achieve, in addition, that \(\partial M\) has positive mean curvature \(H\). Theorem: \(M (\dim M\neq 4)\) carries a Riemannian metric with \(K>0\) and \(H>0\) iff \(M\) is 1-thin. b) Let \(N\) be a closed manifold without boundary and \(f: N\to \bar M\) a (normalized) hypersurface embedding into an arbitrary Riemannian manifold \(\bar M\). Under which conditions is the embedding of positive mean curvature? Theorem: If \(dim \bar M\neq 4\) and \(f\) bounds a compact domain \(D_ f\) in \(\bar M,\) then \(f\) is strongly isotopic to an embedding of positive mean curvature if \(D_ f\) is 1-thin. The basic tool is a handle-attaching theorem stating a method for constructing new hypersurfaces with \(H>0\) out of a method for constructing new hypersurfaces with \(H>0\) out of given ones by attaching ambient handles of codimension \(\geq 2\).