Sharp estimates for the size of balls in the complement of a hypersurface (Q2574975)

The authors obtain some results on upper estimates for the radii of balls in the complements to hypersurfaces in Euclidean spaces assuming that these hypersurfaces enjoy the Jordan-Brouwer property which means that the complement to a hypersurface is the union of two disjoint nonempty open sets for which the hypersurface is the common boundary. In particular, they prove that if an \(n\)-dimensional hypersurface \(M\) meets at least one of the following two conditions for some positive constant \(\lambda\): 1) \(| \sigma| ^2 \geq \frac{n}{\lambda^2}\) and \(| h| > \frac{n-2}{n\lambda}\), where \(| \sigma| ^2 \) is the squared length of the second fundamental form and \(h\) is the mean curvature; 2) for some \(k=1,\dots,n\) the \(k\)-mean curvature \(H_k\) satisfies the inequality \(| H_k| \geq {1}/{\lambda^k}\) and, moreover, if \(k \geq 2\) then there is a point of the hypersurface at which all principal curvatures do not change sign, then there is a component of the complement to \(M\) such that any ball contained in it has radius less than \(\lambda\) unless the hypersurface is a sphere of radius \(\lambda\). These theorems generalize the previous statements known before for compact hypersurfaces. The same technique allows also to prove that \(\inf | h| =0\) for the graph of a function \(f: {\mathbb R}^n \to {\mathbb R}\) and, assuming that the mean curvature does not change sign, \(\inf | \sigma| \) also vanishes.