Mass from an Extrinsic Point of View

Abstract

We express the 𝑞-th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total (2𝑞)-th mean curvature and the integral, over the entire manifold, of the inner product between the (2𝑞 + 1)-th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each 𝑞, a geometric inequality that holds whenever the positive mass theorem (for the 𝑞-th Gauss-Bonnet-Chern mass) holds.