Entire vertical graphs in Riemannian product spaces

Abstract

We extend the technique developed by S.T. Yau in [21] in order to investigate the rigidity of entire vertical graphs in a Riemannian product space R × Mⁿ, whose fiber Mⁿ is supposed to have Ricci curvature with strict sign. In this setting, under a suitable restriction on the norm of the gradient of the function u which determines such a graph Σⁿ(u), we are able to prove that Σⁿ(u) must be a slice {t} × Mⁿ.

Keywords: Product spaces, entire vertical graphs, mean curvature, height function

Quaestiones Mathematicae 37(2014), 321–329