arXiv research

Existence proof for Einstein equations with small TT-tensor and vanishing Yamabe invariant.

Study stability of Einstein metrics on homogeneous spaces.

We study transverse-tracefree (TT)-tensors on conformally flat 3-manifolds $(M,g)$. The Cotton-York tensor linearized at $g$ maps every symmetric tracefree tensor into one which is TT. The question as to whether this is the general solution to the TT-condition is viewed as a cohomological problem within an elliptic com…

In this short note, we give a construction of solutions to the Einstein constraint equations using the well known conformal method. Our method gives a result similar to the one in [15, 16, 24], namely existence when the so called TT-tensor $σ$ is small and the Yamabe invariant of the manifold is positive. The method we…

In this paper, we prove a far-from-CMC result similar to the ones obtained by Holst, Nagy, Tsogtgerel and Maxwell for the conformal Einstein-scalar field constraint equations on compact Riemannian manifolds with positive (modified) Yamabe invariant.

In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor $σ$ = 0 have a non-trivial solution, and thus answer a que…

Two problems concerning asymptotically hyperbolic manifolds with an inner boundary are studied. First, we study scalar curvature presciption with either Dirichlet or mean curvature prescription interior boundary condition. Then we apply those results to the Lichnerowicz equation with (future or past) apparent horizon i…

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth soluti…

New method for tensor recovery with fewer samples.

Study of critical points for 4D conformally invariant curvature energies.

Tensorized Rademacher projections outperform Gaussian projections in reducing tensor dimensions.

New estimates show all stable Einstein manifolds are linear stable with respect to Perelman's ν-entropy.

Solves the Cauchy problem for linearised Einstein equation on globally hyperbolic spacetimes.

New LSH methods for tensor data improve efficiency and space usage.