Mean curvature and compactification of surfaces in a negatively curved Cartan–Hadamard manifold

where χ(S) is the Euler characteristic of the surface, B r denotes the geodesic r-ball in Hn(b) and Vol(S 2∩Bb,n r ) Vol(B r ) is the volume growth of the domains S2 ∩B r . A natural question arises in this context: can we prove the finiteness of the topology of a not necessarily minimal surface in a Cartan–Hadamard manifold and, moreover, establish a Chern–Osserman-type inequality for its Euler characteristic? (At this point we are referring to the work [19], where the finiteness of the topology and a Chern–Osserman inequality are proven for not necessarily minimal surfaces in the Euclidean spaces Rn.) In this paper, we provide a partial answer to this question. We consider a complete and connected surface S properly immersed in a Cartan–Hadamard manifold N with sectional curvatures KN bounded from above by b < 0. As in [6], we assume that ∫ S ‖AS‖2dσ < ∞ and that the sectional curvatures of the ambient manifold N satisfy ∫ S(b−KN )dσ < ∞. On the other hand,