New geometrical perspectives in general relativity

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Two driving forces have fueled general relativity research in the past few decades. One is the onslaught of data from ``big science'', such as the cosmic microwave background (CMB) observations and LIGO gravitational wave observations. The other is the explosive growth of geometric analysis, one of the most successful fields in mathematics in the last half-century. This proposal aims to leverage geometric analytical tools and to apply them to problems in general relativity. One such problem is that of the possible topology of black hole horizons. This in turn is part of the black hole uniqueness problem. While these problems are mostly well understood in 4 spacetime dimensions, the theory will not be complete until we understand them in all dimensions and for nonzero cosmological constant. Two tools to exploit here are the equations of static vacuum Einstein metrics and the near horizon geometry equations. These are both types of Bakry-Émery Ricci "quasi-Einstein" equations, as are the Ricci soliton equations which are important in the theory of Ricci flow. I propose to study these equations from a unified viewpoint, which will enable me to borrow techniques from one kind of Bakry-Émery Ricci equation and exploit it to study another. This strategy has already yielded early results, including a surprising application to interpretation of CMB data: Galloway, Khuri, and I have been able to show that the allowed topologies for the Universe can be constrained whenever the Universe is closed, even if the Universe does not achieve closure density (which is a sufficient but not necessary condition for a closed Universe). A second, related part of the proposal is the study of asymptotically hyperbolic spacetimes, especially static vacuum ones. A vexing problem is an old conjecture of Horowitz and Myers that there exists a kind of positive mass theorem (allowing some negative mass, but with lower bound) for isolated systems whose conformal infinity is that of a torus rather than a sphere, if the cosmological constant is negative. Recent progress has shown that the conjecture holds in some quite special circumstances, and I propose to extend those results. But the main test of the conjecture will come when special symmetry assumptions are removed. In particular, can the conjecture hold when conformal infinity is a torus but the spacetime contains an array of topologically spherical black holes whose interaction potential energy may contribute negative mass-energy? Finally, I propose to study soliton solutions of the curve shortening flow (CSF) and related extrinsic geometric flows. A curve evolves by curve shortening if it deforms in the normal direction with speed given by its curvature. Solitons are curves that evolve self-similarly under the flow. But little is known of CSF solitons in curved manifolds. This will be a rich source of research projects for undergraduates. I expect to apply the results to cosmic string evolution in cosmology.