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We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.

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In this note we show how a generalized Pohozaev-Schoen identity due to Gover and Orsted [GO] can be used to obtain some rigidity results for V -static manifolds and generalized solitons. We also obtain an Alexandrov type result for certain hypersurfaces in Einstein manifolds.

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Abstract. Let (M n+1 , g, e −f dµ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded f -minimal hypersurfaces in M with uniform upper bounds on f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n+1 with 2 ≤ n ≤ 6. We also prove some estimates on the f -index of f -minimal hypersurfaces, and give a conformal structure of f -minimal surface with finite f -index in three-dimensional smooth metric measure space.