Concentration of Total Curvature of Minimal Surfaces in H × R Ricardo Sa Earp and Eric Toubiana

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space H × R, where H is the hyperbolic plane. Under some geometric conditions on the asymptotic boundary of an oriented stable minimal surface immersed in H × R, it has infinite total curvature. In particular, we infer that a minimal graph M in H × R whose asymptotic boundary is a graph over an arc of ∂∞H 2 × {0}, different from the asymptotic boundary of ∂M , has infinite total curvature. Consequently, if M is a stable minimal surface immersed into H × R with compact boundary, such that its asymptotic boundary is a graph over the whole ∂∞H ×{0}, then it has infinite total curvature. We exhibit an example of a minimal graph such that in a domain whose asymptotic boundary is a vertical segment the total curvature is finite, but the total curvature of the graph is infinite, by the theorem cited before. We also present some simple and peculiar examples of infinite total curvature minimal surfaces in H ×R and their asymptotic boundaries.