Luis José Alias Linares
Departamento de Matemáticas, Universidad de Murcia
On the global behaviour of the curvature of spacelike zero mean curvature surfaces in Lorentzian spaces
Spacelike zero mean curvature surfaces in Lorentzian spaces arise naturally as critical points
of the area functional. From a physical point of view, they also have interest because of their
applications in general relativity and conformal geometry.
As is well known, the Gaussian curvature of a minimal surface (zero mean
curvature surface) immersed into a Riemannian space of constant curvature is bounded from above
by the constant curvature of the ambient space, with equality precisely at the totally geodesic
points of the surface. In contrast to this, when the
dimension of the ambient Lorentzian space is greater than 3 (codimension greater than 1),
one cannot deduce any \textit{a priori} regularity on the behaviour of the curvature of
spacelike zero mean curvature surfaces, due to the fact that the normal bundle is
indefinite. This motivated our interest on the subject.
For instance, maximal surfaces (spacelike zero mean curvature surfaces) in the flat Minkowski
3-space $\mathbb{L}^3$ all have
non-negative Gaussian curvature. However, in the Minkowski 4-space, complete minimal surfaces in
the Euclidean 3-space $\mathbb{E}^3\subset\mathbb{L}^4$ furnish many examples of complete
spacelike zero mean curvature surfaces in $\mathbb{L}^4$ with non-positive
Gaussian curvature. It is therefore natural to inquire about the existence
of complete spacelike zero mean curvature surfaces in $\mathbb{L}^4$ with
non-negative Gaussian curvature. This fact is also related to one of the most important global
results about maximal surfaces in $\mathbb{L}^3$, the so called Calabi-Bernstein
theorem, which states that the only complete maximal surfaces in $\mathbb{L}^3$ are the
spacelike planes.
In recent years, we have studied in depth the global behaviour of the curvature
of spacelike zero mean curvature surfaces in 3-dimensional and 4-dimensional Lorentzian spaces.
In this talk we will report on our recent advances on this topic.