Geometric properties of surfaces with the same mean curvature in R^3 and L^3

TL;DR

This paper investigates spacelike surfaces in Lorentz-Minkowski space that have the same mean curvature with respect to both the Lorentzian and Euclidean metrics, revealing they have non-positive Gaussian curvature and exploring related geometric properties.

Contribution

It characterizes spacelike surfaces with equal mean curvature in both metrics, showing they possess non-positive Gaussian curvature and providing uniqueness results for associated boundary value problems.

Findings

Surfaces with equal mean curvature in both metrics have non-positive Gaussian curvature.

Several geometric consequences and properties of these surfaces are established.

Uniqueness results for the Dirichlet problem of the surface equation are provided.

Abstract

Spacelike surfaces in the Lorentz-Minkowski space L^3 can be endowed with two different Riemannian metrics, the metric inherited from L^3 and the one induced by the Euclidean metric of R^3. 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 R^3. 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 L^3 is locally a graph over a domain of the plane x_3=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.