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Rigidity of Poincar\'e-Einstein manifolds with flat Euclidean conformal infinity
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In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary defining function. Additionally, we provide examples of Poincar\'e-Einstein manifolds with non-compact conformal infinities. Furthermore, we draw analogies with Ricci-flat manifolds exhibiting Euclidean volume growth, particularly when the compactified metric has non-negative scalar curvature.
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A sharp relative comparison inequality for conformal fillings of Poincar\'e--Einstein manifolds
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