Geometric properties of surfaces with the same mean curvature in R³ and L³

2016/04/14 by Alma L. Albujer, Albujer, Alma L., Magdalena Caballero +1
Mathematics · Physics and Astronomy · #35J93 #53C42 #53C50 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1604.04187

openalex publication_date 2016/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Spacelike surfaces in the Lorentz-Minkowski space L³ can be endowed with two different Riemannian metrics, the metric inherited from L³ and the one induced by the Euclidean metric of R³. It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. One of our main results states that those surfaces have non-positive Gaussian curvature in R³. As an application of this result, jointly with a general argument on the existence of elliptic points, we present several geometric consequences for the surfaces we are considering. Finally, as any spacelike surface in L³ is locally a graph over a domain of the plane x₃=0, our surfaces are locally determined by the solutions to the H_R=H_L surface equation. Some uniqueness results for the Dirichlet problem associated to this equation are given.

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