Critical Points of Modified Geroch Functional in Higher Dimension

Abstract

In this paper, we study several aspects of the modified Geroch functional of n-dimensional Einstein submanifold Σn embedded in an (n + 1)-dimensional Einstein manifold Mn+1,Σn ⊂ Mn+1. We parameterized the embedding by a parameter t and using the first variation of MGF, we show that the submanifold Σn with non-negative mean curvature H ≥ 0 is a critical point of MGF if and only if is minimal or umbilic.