Moebius rigidity for compact deformations of negatively curved manifolds

2018/12/12 by Biswas, Kingshook
#53C24 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.04888

Abstract

Let (X, g₀) be a complete, simply connected Riemannian manifold with sectional curvatures K_g0 satisfying -b² ≤ K_g0 ≤ -1 for some b ≥ 1. Let g₁ be a Riemannian metric on X such that g₁ = g₀ outside a compact in X, and with sectional curvatures K_g1 satisfying K_g1 ≤ -1. The identity map id : (X, g₀) → (X, g₁) is bi-Lipschitz, and hence induces a homeomorphism between the boundaries at infinity of (X, g₀) and (X, g₁), which we denote by id_{g0, g1} : ∂_g0 X → ∂_g1 X. We show that if the boundary map id_{g0, g1} is Moebius (i.e. preserves cross-ratios), then it extends to an isometry F : (X, g₀) → (X, g₁).

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