Nonlinear stability for a class of doubly warped spacetimes is proved. The background spacetimes have negative Einstein factors. It shown that for dimension D ≥ 11 there is a full parameter family of solutions to the vacuum Einstein equations which has Kasner-like singularity and Friedmann like asympotics in the future. In particular, these spacetimes have crushing singularity and are globally ...

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