Large-scale conformal rigidity in dimension three

Abstract

Let X be a noncompact smooth manifold. Two complete Riemannian metrics on X in a given conformal class can have very different asymptotic geometries. For instance, starting with the Euclidean plane E2 with polar coordinates (r, θ), multiplying the Euclidean metric g0 = dr2 + r2dθ2 by a function equal to 1/r2 outside a compact neighborhood of the origin, one obtains a complete Riemannian metric g which is quasi-isometric to a half-line. The metrics g and g0 are in the same conformal class, but they are not quasi-isometric. In fact, they have different asymptotic dimensions. One can ask whether something similar can happen to two finitely generated groups and 0: can they be ‘coarsely quasi-conformal’ in some sense and yet not quasi-isometric? To give a precise meaning to this question, we will choose Riemannian manifoldsX0,X that are geometric models for 0, and consider conformal mappings between X0 and X. First let us recall some definitions. Two metric spaces (X1,d1) and (X2,d2) are quasi-isometric if there are constants λ ≥ 1 and C ≥ 0 and a map f : X1 → X2 satisfying: λ−1 d1(x, x′) −C ≤ d2(f (x), f (x′)) ≤ λd1(x, x′)+ C ∀ x, x′ ∈ X1 ∀ y ∈ X2, ∃x ∈ X1, d(f (x), y) ≤ C.