Periodic Reporting for period 1 - MCFBeyondAndApp (Mean curvature flow: singularity formation beyond 2 convexity and applications)
Reporting period: 2023-09-01 to 2026-02-28
Geometric flows as a mean to attack problems in topology, geometry and physics, had been demonstrated to be an extremely powerful tool. The most successful such flow to date is the Ricci flow (RF), which was used in the proof of the geometrization conjecture. The mean curvature flow (MCF) - the most natural geometric flow for sub-manifolds in an ambient space, had also been successfully applied to address such problems. Nevertheless, the most striking potential applications of MCF are still out of reach. The goal of the proposed research is to advance the understanding of the formation of singularities in MCF, and to study a particular application of MCF for general relativity. More concretely, we continue the systematic study of the formation of bubble-sheet singularities in 4-space, initiated by Choi, Haslhofer and the PI, with the goal of obtaining a mean convex neighbourhood theorem in this setting. We also study the formation of singularities more generally, and in particular, the structure of the singular set. The second objective of the proposed research is to employ MCF in Lorentzian spacetime satisfying the Einstein equation with positive cosmological constant to obtain versions of the cosmic no hair conjecture, namely, geometric convergence results to de Sitter space.
The research conducted by me and my team-mates have produced one publication and two preprints. Within the context of the study of bubble sheet singularities, Kyeongsu Choi, Robert Haslhofer and the PI have derived a more refined estimate for the structure of ovals and the HIWM translators Fundamentally, this provides better understanding of these (implicitly defined) solution, which makes their analysis more tractable. Indeed, in the subsequent work we have used this more refined understanding (and more) to study the moduli space of translators: Previously, we knew that all translators belong to the HIMW family, which was a topological interval, but it remained open whether this family is in fact smooth, and parametrized by its tip curvature, This was the main aim of the above mentioned paper was to show that this was indeed a smooth family parametrized by its tip curvature. This required developing a theory for the linearized translator equation in well crafted spaces (to deal with the degeneracy of the ellipticity at infinity) and substituting geometric estimates (foliations and avoidance principles) with PDE estimates.We have also further explored these PDE substitutes, deriving a gradient estimate that replaces Hamilton Harnack inequality. Showing that non compact non collapsed mean convex flows asymptotic to a bubble sheet, which are not the bubble sheet are translators was achieved by Choi and Haslhofer, without the PI involvement.
With respect to the mean curvature flow approach to the the cosmic no-hair conjecture, rather than working with symmetry, we took the avenue of going from rigidity to stability to convergence. The first step of this was achieved in the preprint “mean curvature flow in de Sitter space, co-authored with Leonardo Senatore, where we show that in de Sitter space itself, every initial graphical, mean convex surface converges to the flat slicing of de Sitter space. This had produced a version of “extending pseudolcality” where the contrast of the non linearity of the equation and the expanding ambient space interact nicely. This should be thought of as a test case for the approach to the conjecture - the approach is to show the convergence to de Sitter space is by showing that the flow has to converge to the flat slicing of it. If it doesn’t work without assuming initial de Sitter, there is no hope. On the more analytical side, an MSc thesis of my student Nimrod Gabison (supported by the grant) dealt with the construction of mean curvature flow starting from graphical mean convex data on the entire, non compact, de Sitter space. I had hoped that this would lead to a pseudo-locality result which holds without an upper bound on H (a bound which is natural in this context) but this part was not yet achieved
Finally, conforming the the main topic of this grant of better understanding the singularity formation, in anticipation of some future application, together with Joshua Daniel Holgate with embarked on the study of backwards uniqueness of mean curvature flow past singularities. This is interesting both from the PDE standpoint, and is relevant so my remaining major conjecture for surfaces. compact This has resulted two achivements: backwards uniqueness, past singularities, for mean curvature flow with asymptotically conical singularities, and a maximal rate estimate for how far can two flows be close to a compact singularity without being the same. The former result is very significant (and quite surprising) as it’s the first instance of backwards uniqueness for any singular geometric flow that does not assume global self similarity. In particular, no other result for any geometric flow applied to compact initial data
With respect to the mean curvature flow approach to the the cosmic no-hair conjecture, rather than working with symmetry, we took the avenue of going from rigidity to stability to convergence. The first step of this was achieved in the preprint “mean curvature flow in de Sitter space, co-authored with Leonardo Senatore, where we show that in de Sitter space itself, every initial graphical, mean convex surface converges to the flat slicing of de Sitter space. This had produced a version of “extending pseudolcality” where the contrast of the non linearity of the equation and the expanding ambient space interact nicely. This should be thought of as a test case for the approach to the conjecture - the approach is to show the convergence to de Sitter space is by showing that the flow has to converge to the flat slicing of it. If it doesn’t work without assuming initial de Sitter, there is no hope. On the more analytical side, an MSc thesis of my student Nimrod Gabison (supported by the grant) dealt with the construction of mean curvature flow starting from graphical mean convex data on the entire, non compact, de Sitter space. I had hoped that this would lead to a pseudo-locality result which holds without an upper bound on H (a bound which is natural in this context) but this part was not yet achieved
Finally, conforming the the main topic of this grant of better understanding the singularity formation, in anticipation of some future application, together with Joshua Daniel Holgate with embarked on the study of backwards uniqueness of mean curvature flow past singularities. This is interesting both from the PDE standpoint, and is relevant so my remaining major conjecture for surfaces. compact This has resulted two achivements: backwards uniqueness, past singularities, for mean curvature flow with asymptotically conical singularities, and a maximal rate estimate for how far can two flows be close to a compact singularity without being the same. The former result is very significant (and quite surprising) as it’s the first instance of backwards uniqueness for any singular geometric flow that does not assume global self similarity. In particular, no other result for any geometric flow applied to compact initial data
The result regarding backwards uniqueness for AC singularities is potentially a breakthrough result. While there had been numerous results about backwards uniqueness for geometric flows, none could deal with flows with singularities - the main theme of research in geometric flows in the last 30 years. If one further assumes global self similarity (a case which was previously studied by Wang and Kotschwar) this backwards uniqueness has led to some dramatic applications. Technically our current result is already a breakthrough, Hopefully this will allow for some additional applications which have (in terms of statement) nothing to do with backwards uniqueness