2018/12/31 by Biswas, Kingshook
#53C24 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1812.11724
Let X, Y be complete, simply connected Riemannian surfaces with pinched negative curvature -b2 ≤ K ≤ -1. We show that if f : ∂ X → ∂ Y is a Moebius homeomorphism between the boundaries at infinity of X, Y, then f extends to an isometry F : X → Y. This can be viewed as a generalization of Otal's marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal's theorem asserts that if X, Y admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map f is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of X, Y may be trivial.