In this talk, we study isometric immersions of surfaces into simply connected three-dimensional unimodular Lie groups endowed with left-invariant metrics, both in the Riemannian and Lorentzian settings (assuming in the latter case that Milnor’s operator is diagonalizable). We present global coordinate models for these metric Lie groups, depending analytically on their structure constants, and discuss several fundamental results that characterize such immersions. In particular, we investigate to what extent an immersion can be reconstructed from natural geometric data, including (a) the tangent projections of a left-invariant frame, (b) the left-invariant Gauss map, and (c) the shape operator. As a main result, we show that an isometric immersion is essentially determined by its left-invariant Gauss map, up to certain well-controlled angular companions, which are characterized in terms of a PDE.

This is a joint work with I. Castro and J. S. Santiago available at https://arxiv.org/abs/2507.16728.

I will explain that to every small and decaying solution of the linearized constraint equations about Minkowski initial data, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity, the correction is given by Kerr black hole initial data, up to a term that decays faster than the linearized solution, and that has Schwartz decay if the linearized solution has Schwartz decay. The main tool is a right inverse (up to necessary integrability conditions) for the linearized constraint operator about Minkowski initial data, that has optimal mapping properties relative to weighted b-Sobolev spaces, where the weights measure decay towards infinity. Using a recent result, one obtains that the solutions of the Einstein equations with these initial data admit a regular conformal compactification along null and timelike infinity.

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