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@article{2386463, author = {Blitz, Samuel Harris and Gover, A Rod and Waldron, Andrew}, article_number = {6}, doi = {http://dx.doi.org/10.1512/iumj.2023.72.9518}, keywords = {Extrinsic conformal geometry; hypersurface embeddings; Poincare-Einstein metrics; Yamabe problem}, language = {eng}, issn = {0022-2518}, journal = {Indiana University Mathematics Journal}, title = {Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition}, url = {https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml}, volume = {72}, year = {2023} }
TY - JOUR ID - 2386463 AU - Blitz, Samuel Harris - Gover, A Rod - Waldron, Andrew PY - 2023 TI - Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition JF - Indiana University Mathematics Journal VL - 72 IS - 6 SP - 2215-2284 EP - 2215-2284 PB - INDIANA UNIV MATH JOURNAL SN - 00222518 KW - Extrinsic conformal geometry KW - hypersurface embeddings KW - Poincare-Einstein metrics KW - Yamabe problem UR - https://www.iumj.indiana.edu/oai/2023/72/9518/9518.xml N2 - An important problem is to determine under which circumstances a metric on a conformally compact manifold is conformal to a Poincare-Einstein metric. Such conformal rescalings are in general obstructed by conformal invariants of the boundary hypersurface embedding, the first of which is the trace-free second fundamental form and then, at the next order, the trace-free Fialkow tensor. We show that these tensors are the lowest-order examples in a sequence of conformally invariant higher fundamental forms determined by the data of a conformal hypersurface embedding. We give a construction of these canonical extrinsic curvatures. Our main result is that the vanishing of these fundamental forms is a necessary and sufficient condition for a conformally compact metric to be conformally related to an asymptotically Poincare-Einstein metric. More generally, these higher fundamental forms are basic to the study of conformal hypersurface invariants. Because Einstein metrics necessarily have constant scalar curvature, our method employs asymptotic solutions of the singular Yamabe problem to select an asymptotically distinguished conformally compact metric. Our approach relies on conformal tractor calculus as this is key for an extension of the general theory of conformal hypersurface embeddings that we further develop here. In particular, we give in full detail tractor analogs of the classical Gauss Formula and Gauss Theorem for Riemannian hypersurface embeddings. ER -
BLITZ, Samuel Harris, A Rod GOVER and Andrew WALDRON. Conformal Fundamental Forms and the Asymptotically Poincare-Einstein Condition. \textit{Indiana University Mathematics Journal}. INDIANA UNIV MATH JOURNAL, 2023, vol.~72, No~6, p.~2215-2284. ISSN~0022-2518. Available from: https://dx.doi.org/10.1512/iumj.2023.72.9518.
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