Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 018, 11 pages      arXiv:2403.06782      https://doi.org/10.3842/SIGMA.2025.018

Mass from an Extrinsic Point of View

Alexandre de Sousa a and Frederico Girão b
a) Escola de Ensino Fundamental e Médio Santa Luzia, Fortaleza, 60110-300, Brazil
b) Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, 60455-760, Brazil

Received October 22, 2024, in final form March 03, 2025; Published online March 18, 2025

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.

Key words: Gauss-Bonnet-Chern mass; asymptotically Euclidean submanifolds; positive mass theorem; Hsiung-Minkowski identities.

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