Concentration of total curvature of minimal surfaces in H²xR

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space H^2xR; where H^2 is the hyperbolic plane. Under some geometric conditions on the asymptotic boundary of an oriented stable minimal surface immersed in H^2xR, it has infinite total curvature. In particular, we infer that a minimal graph M in H^2xR whose asymptotic boundary is a graph over an arc of the asymptotic boundary of H^2, different from the asymptotic boundary of the boundary of M, has infinite total curvature. Consequently, if M is a stable minimal surface immersed into H^2xR with compact boundary, such that its asymptotic boundary is a graph over the whole asymptotic boundary of H^2; 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^2xR and their asymptotic boundaries.