  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.
