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Alexandre de Sousa

Publications and source records attributed to Alexandre de Sousa.

4 recordsLinked to original sources
arXiv2026

On Umbilical Real Hypersufaces of Products of Complex Space Forms

Tashiro and Tachibana proved that there exist no totally umbilical hypersurfaces in complex space forms with nonzero constant holomorphic sectional curvature, and it is also known that the shape operator of such hypersurfaces cannot be parallel. Motivated by these results, we study real hypersurfaces in products of complex space forms. We establish rigidity and nonexistence results for totally umbilical real hypersurfaces in this setting. In particular, we show that if a real hypersurface in a product of complex space forms does not admit a local product structure, then its shape operator cannot be parallel. Moreover, we provide a classification of totally umbilical real hypersurfaces, showing that those admitting a local almost product structure are necessarily totally geodesic or extrinsic hyperspheres.

math.DG↗
arXiv2024

Mass from an Extrinsic Point of View

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.

math.DG↗
arXiv2019

Min-max minimal surfaces, horizons and electrostatic systems

We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality relating the area and the charge of a minimal surface of index one in a Cauchy data satisfying the Dominant Energy Condition for non-electromagnetic matter fields. Moreover, we explore a global version of this inequality, and the rigidity in the case of the equality, using a result proved by Marques and Neves.

math.DG↗