Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces

Abstract

In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form M 2 × R 1 , where M 2 is a connected Riemannian surface and M 2 × R 1 is endowed with the Lorentzian metric , = , M − dt 2 . In particular, when M is a Riemannian surface with non-negative Gaussian curvature K M , we prove that any complete maximal surface in M 2 × R 1 must be totally geodesic. Besides, if M is non-flat we conclude that it must be a slice M × {t 0 }, t 0 ∈ R (here by complete it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product M 2 × R. This allows us to give also a non-parametric version of the Calabi-Bernstein theorem for entire maximal graphs in M 2 × R 1 , under the same assumptions on K M . Moreover, we also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis K M ≥ 0. These examples are constructed via a duality result between minimal and maximal graphs.

Parabolicity of maximal surfaces in Lorentzian product spaces

“…As another consequence of our main result (Theorem 3), we can give a new proof of the following Calabi-Bernstein theorem, first established in [1,Theorem 4.3] (see also [2] for another approach to the parametric version of that result, first established in [1, Theorem 3.3]). Proof By applying Proposition 7 with = M we know that the function φ = r 2 − h 2 is eventually positive and proper on (u).…”

“…That means that is a local diffeomorphism which increases the distance between the Riemannian surfaces and M, and by [8, Chapter VIII, Lemma 8.1] is a covering map (for the details, see Lemma 3.1 in [1]). Therefore, dim −1 (Cut(x 0 )) = dimCut(x 0 ) < 2 and the function r is smooth almost everywhere in .…”

“…In fact, in [1] we considered that definition of parabolicity. However, along this work, we will reserve the term parabolicity for Riemannian surfaces with non-empty boundary, and we will use the term recurrence for Riemannian surfaces without boundary.…”

Mean curvature flow of spacelike graphs

“…6 we obtain long-time existence using elliptic Schauder theory and prove the existence of a convergent sequence of the flow. The use of the Bernstein-type results obtained in [2,15] leads to Theorem 1.1 (2). In Sect.…”

A New Way to Dirichlet Problems for Minimal Surface Systems in Arbitrary Dimensions and Codimensions

“…The hyperbolic angle θ can be defined by (this definition can also be seen in [1,6]) Now, we would like to explain the connection between the spacelike MCF (2.1) and the high-dimensional Dirichlet problem. Let ⊂ R n be a closed and bounded domain, and ψ : → R m be a vector-valued function.…”