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 × M n , whose fiber M n 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 Σ n (u), we are able to prove that Σ n (u) must be a slice {t} × M n . (2010): Primary 53C42; Secondary 53B30, 53C50. Mathematics Subject Classification

Hypersurfaces with prescribed angle function

“…In [Aquino andLima 2011] and[Lima andParente 2012], we applied the wellknown generalized maximum principle of Omori [1967] and Yau [1975], and an extension of it due to Akutagawa [1987], in order to obtain rigidity theorems concerning complete vertical graphs with constant mean curvature in ‫ޒ‬ × ‫ލ‬ n . In [Lima 2014], the first author extended the technique developed in [Yau 1976] in order to investigate the rigidity of entire vertical graphs in a Riemannian product space ‫ޒ‬ × M n , whose base M n is assumed to have Ricci curvature with strict sign. Under a suitable restriction on the norm of the gradient of the function u which determines such a graph n (u), he proved that n (u) must be a slice {t} × M n .…”