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Free keywords:
General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG
Abstract:
We analyze (the harmonic map representation of) static solutions of the
Einstein Equations in dimension three from the point of view of comparison
geometry. We find simple monotonic quantities capturing sharply the influence
of the Lapse function on the focussing of geodesics. This allows, in
particular, a sharp estimation of the Laplacian of the distance function to a
given (hyper)-surface. We apply the technique to asymptotically flat solutions
with regular and connected horizons and, after a detailed analysis of the
distance function to the horizon, we recover the Penrose inequality and the
uniqueness of the Schwarzschild solution. The proof of this last result does
not require proving conformal flatness at any intermediate step.
