CMC hypersurfaces with polynomial volume growth in warped products and the nonexistence of entire solutions to the minimal hypersurface equation

Abstract

We investigate constant mean curvature (CMC) complete two-sided hypersurfaces with polynomial volume growth in a class of warped products satisfying a suitable curvature constraint. In this setting, we establish the nonexistence of such a CMC hypersurface under mild hypotheses involving the mean curvature and the warping function. Applications to Einstein warped product, pseudo-hyperbolic, Schwarzschild and Reissner–Nordström spaces are also given. Furthermore, we present a nonparametric version of our main result which, in particular, guarantees the nonexistence of entire solutions with finite C² norm of the the minimal hypersurface equation on a complete Riemannian manifold with polynomial volume growth.