A characterization of spheres, circles and cardioids (Q689775)

We define the notions of supported mean curvature, \(pH\), and normalized supported mean curvature, \(\overline pH\), for immersed submanifolds of Euclidean spaces. They are generalizations of the classical product of the support function, \(p\), and the mean curvature, \(H\), for surfaces in \(\mathbb{R}^ 3\). We recall that constant \(pH\) characterizes compact spherical submanifolds, and we conjecture that constant \(\overline pH\) characterizes the spheres in the class of compact hypersurfaces in \(\mathbb{R}^ n\). We support this conjecture by showing that constant \(\overline pH\) actually does characterize circles and cardioids among simple closed curves in \(\mathbb{R}^ 2\), and that it is impossible to deform the \(m\)-sphere through a smooth family of immersed manifolds with the same constant \(\overline pH\).