Characterizations of umbilic hypersurfaces in warped product manifolds

Abstract

We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$

“…This gives (1). Using Lemma 2.6 we obtain (2). From the Gauss equation, see Proposition 2.4, we have…”

“…In [1], Simon Brendle shows that surfaces having constant mean curvature, immersed into certain warped product spaces are actually, totally umbilical surfaces (this result can be apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds). More recently, Shanze Gao and Hui Ma characterized immersed totally umbilical surfaces into certain warped product spaces with some additional condition on the mean curvature, [2]. For the simply-connected homogeneous 3-dimensional manifolds having 4-dimensional isometry group, usually labelled by E(κ, τ) for real numbers κ and τ satisfying κ = 4τ 2 , such classification was made in two steps.…”

“…In order o simplify the computations, we prove the item (1), the item (2) is similar. Consider Σ R 2 f × R a totally umbilical, rotationally-invariant surface, immersed into the warped product R 2 f × R. We consider, as in section (3.2), the vertical plane for some constant c 0 ∈ R. Notice that, the problem of finding rotationally-invariant surfaces immersed into the warped product R 2 f × R consists in determine the admissible expressions of the profile curve α(s), we mean, we wish to find, for each c 0 ∈ R, the possible integral curve of the ODE's system (4.1). Therefore, in order to find some example of such surface, we consider without loss of generality c 0 = 1.…”