Otis Chodosh

Time-Periodic Einstein–Klein–Gordon Bifurcations of Kerr

We construct one-parameter families of solutions to the Einstein–Klein–Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein–Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein–Klein–Gordon equations.

On the topology and index of minimal surfaces

We show that for an immersed two-sided minimal surface in R 3 , there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R 3 of index 2, as conjectured by Choe (Cho90). Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.

EXPANDING RICCI SOLITONS ASYMPTOTIC TO CONES

We show that an expanding gradient Ricci soliton which is asymptotic to a cone at infinity in a certain sense must be rotationally symmetric.

Large Isoperimetric Regions in Asymptotically Hyperbolic Manifolds

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres are uniquely isoperimetric in manifolds that are Schwarzschild–anti-deSitter at infinity. These results have important repercussions for our understanding of spacelike hypersurfaces in Lorentzian space-times which are asymptotic to null infinity. In fact, as an application of our results, we verify the asymptotically hyperbolic Penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes.

A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least −6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter–Schwarzschild metrics.

Minimal hypersurfaces with bounded index

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold

On discontinuity of planar optimal transport maps

(M^{n},g)

Isoperimetric structure of asymptotically conical manifolds

(Mn,g),