Einstein manifolds with boundary

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We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal emebeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a non-degenerate boundary condition is used to construct metrics with nonpositive Einstein constant.

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