Modified Hawking Mass and Geroch Monotonicity on 2-Surfaces

In this paper, we study several aspects of the modified Hawking mass on 2-surface defined as(1) where R, H, and λ are the scalar curvature of the surface ∑, the mean curvature of the surface ∑, and a constant, respectively. Finally, using Geroch approach we study the monotonicity properties of the Hawking mass and show that m H(∑ t ) is a nondecreasing function of t