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Abstract. We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime. In particular, when t = 0, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy-momentum endomorphism Q is the geometric invariant of asymptotically anti-de Sitter spacetimes.

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Abstract. We establish a type of positive energy theorem for asymptotically anti-de Sitter Einstein-Maxwell initial data sets by using Witten's spinoral techniques.

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.