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In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number $\gamma>0$, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as $t\nearrow\infty$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate $(2t+1)^{(\gamma-1)/2}$. (2) In a neighbourhood of the tip, the Type-IIb blow-up of the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface has a precise growth rate depending on $\gamma$., Comment: Comments are welcome. arXiv admin note: text overlap with arXiv:1911.07282, arXiv:1603.01664

We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a Type-II rate no slower than $(T-t)^{-1}$. Secondly, there is a "far" class of initial data which lead to solutions developing a local Type-I nondegenerate neckpinch under mean curvature flow. These numerical findings further suggest the existence of a "critical" class of initial data which conjecturally lead to mean curvature flow of noncompact hypersurfaces forming local Type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than $(T-t)^{-1}$., Comment: 23 pages, 11 figures. Comments are welcome!

We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized [Formula: see text]-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant [Formula: see text]. This stability result generalizes the results proven in Ames E

We prove the nonlinear stability of the asymptotic behavior of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarised $T^2$-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant $\Lambda$. This stability result generalizes the results proven in [3], which focus on the $\Lambda=0$ case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for $\Lambda=0$, the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarised $T^2$-symmetric vacuum solutions than those considered in [3] and [26]. Our results establish that the areal time coordinate takes all values in $(0, T_0]$ for some $T_0 > 0$, for certain families of polarised $T^2$-symmetric solutions with cosmological constant., Comment: 20 pages

We prove stability of the family of Kasner solutions within the class of polarized $T^2$-symmetric solutions of the vacuum Einstein equations in the contracting time direction with respect to an areal time foliation. All Kasner solutions for which the asymptotic velocity parameter $K$ satisfies $|K-1|>2$ are non-linearly stable, and all sufficiently small perturbations exhibit asymptotically velocity term dominated (AVTD) behavior and blow-up of the Kretschmann scalar., 28 pages. Fixed minor typos and made several clarifications. Agrees with published version

The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state $g_0$ exists for all time and converges to a stable fixed point, then the flows of solutions that start near $g_0$ also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries., 18 pages

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.

We prove a nonpolarised analogue of the asymptotic characterization of $T^2$-symmetric Einstein Flow solutions completed recently by LeFloch and Smulevici. In this work, we impose a condition weaker than polarisation and so our result applies to a larger class. We obtain similar rates of decay for the normalized energy and associated quantities for this class. We describe numerical simulations which indicate that there is a locally attractive set for $T^2$-symmetric solutions not covered by our main theorem. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarised asymptotics are unstable., 19 pages, 5 figures

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.

We present a procedure for asymptotic gluing of hyperboloidal initial data sets that preserves the shear-free condition. Our construction is modeled on a previous gluing construction by the last three named authors, but with significant modifications that incorporate the shear-free condition. We rely on the special H\"older spaces, and the corresponding theory for elliptic operators on weakly asymptotically hyperbolic manifolds, introduced by the authors and applied to the Einstein constraint equations in two previous papers., Comment: 50 pages, 1 figure

We prove an asymptotic stability result for a linear coupled hyperbolic-elliptic system on a large class of singular background spacetimes in CMC gauge on the n-torus. At each spatial point these background spacetimes are perturbations of Kasner-like solutions of the Einstein-scalar field equations which are not required to be close to the homogeneous and isotropic case. We establish the existence of a homeomorphism between Cauchy data for this system and a set of functions naturally associated with the asymptotics in the contracting direction, which we refer to as asymptotic data. This yields a complete characterization of the degrees of freedom of all solutions of this system in terms of their asymptotics. Spatial derivative terms can in general not be fully neglected which yields a clarification of the notion of asymptotic velocity term dominance (AVTD)., 58 pages. Fixed several further typos. Further clarifications of the results. Results unchanged. Agrees with published version

We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber $\{s\}\times\mathrm{SU}(2)$ a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch singularities, and that they asymptotically approach round product metrics in space-time neighborhoods of their singular sets, in precise senses. These are the first examples of Ricci flow solutions without rotational symmetry that become asymptotically rotationally symmetric locally as they develop local finite-time singularities., Comment: We correct a miscalculation of some terms in equation (22) and its subsequent uses. The main results of the paper are unchanged, because the methods employed in the proofs are robust enough to give the needed estimates, with only insignificant changes of constants

We prove local existence for the second order Renormalization Group flow initial value problem on closed Riemannian manifolds (M, g) in general dimensions, for initial metrics whose sectional curvatures KP satisfy the condition 1 + αKP > 0, at all points p ∈ M and planes P ⊂ TpM . This extends results previously proven for two and three dimensions. The second order approximation of the Renormalization Group flow for the nonlinear sigma model of quantum field theory, which we label the RG-2 flow, is specified by the PDE system (1) ∂ ∂t g = −2 Rc− 2 Rm . Here g is a Riemannian metric, Rc is its Ricci curvature, Rmij = g ggRiklmRjpqn, and α is a positive parameter. We note that for our purposes here, α can assume any real value. For α = 0, this system (1) reduces to the Ricci flow. One can see that the sign of the right hand side, which is roughly 1+α×Curvature, should have an impact on the behavior of the flow, and this has been confirmed in various settings: in particular, the size of the term influences the parabolicity of the flow. Oliynyk has shown in [10] that on a two-dimensional manifold, if the Gaussian curvature K satisfies the condition 1 + αK > 0, then the flow is (weakly) parabolic; while if 1 + αK 0 is satisfied for all sectional curvatures KP . In this note we extend this curvature criterion for short-time existence for RG-2 flow to all dimensions, as first announced in [7]. Our main result is the following: Theorem 1. Let (M, g0) be a closed n-dimensional Riemannian manifold. If 1 + αKP > 0 for all sectional curvatures KP (g0), at all points p ∈M and planes P ⊂ TpM , then there exists a unique solution g(t) of the initial value problem ∂ ∂tg = −2 Rc− α 2 Rm , g(0) = g0, on some time interval [0, T ). Remark 2. In [10], Oliynyk finds open subspaces of the space of smooth metrics that are invariant under the two-dimensional RG-2 flow, and for which the flow remains parabolic (resp. backward parabolic). We are currently investigating this for general dimensions. Proof. To prove the theorem, we calculate the principal symbol of the DeTurck-modified version of RG-2 flow, which is generated by the PDE system (compare with (1) above) (2) ∂ ∂t gij = −2Rij + LWu,ggij − α 2 Rmij . Here Wu,g = −giju−1 jk gg(∇puql− 1 2∇lupq) is the standard vector field usually chosen to modify the Ricci flow into the related (parabolic) DeTurck version of Ricci flow, with u a fixed metric. Letting φt be the one-parameter family of diffeomorphisms generated by the vector field −Wu,g, then φt g is a solution of the RG-2 flow (see also [8]). As in the analogous Ricci flow case, if one can show (for a class of choices of the initial metric) that the PDE system (2) is parabolic, then short-time existence holds for the RG-2 flow (1) as well as for the DeTurck-modified flow (2). Date: January 7, 2014. KG is partially supported by the NSF under grant DGE-1144155. CG is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 283083. JI is partially supported by the NSF under grant PHY-1306441 at the University of Oregon. He also wishes to thank the Mathematical Sciences Research Institute in Berkeley, California, for support under grant 0932078 000. Some of this work was carried out while JI was in residence at MSRI during the fall of 2013. 1 2 KARSTEN GIMRE, CHRISTINE GUENTHER, AND JAMES ISENBERG To calculate the symbol of the system (2), we first linearize the flow. For the first two terms of the right hand side of (2), this linearization effectively produces the Laplacian (see [4], or Theorem 2.1 in [5]). For the remaining term, Rm, it is useful to recall the formula for the variation of the Riemann curvature tensor with respect to the metric (see pg. 74 in [2]): [DRmg(h)] l ijk = [ ∂ ∂e Rm(g + eh) ∣∣∣∣ e=0 ]l

We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all \textit{closed} curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such `generalized Taub-NUT' spacetimes and show that, generically, they admitted \textit{only} the single (horizon-generating) Killing field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as `ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the \textit{conjectures} that every such spacetime with an \textit{ergodic} horizon is trivially constructable from the flat Kasner solution by making certain `irrational' toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case., Comment: 55 pages. arXiv admin note: substantial text overlap with arXiv:0805.1451

The Ricci flow has been of fundamental importance in mathematics, most famously through its use as a tool for proving the Poincare conjecture and Thurston’s geometrization conjecture. It has a parallel life in physics, arising as the first-order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory. There recently has been interest in the second-order approximation of this flow, called the RG-2 flow, which mathematically appears as a natural nonlinear deformation of the Ricci flow. A curvature flow arising from quantum field theory seems to us to capture the spirit of Yvonne Choquet-Bruhat’s extensive work in mathematical physics, and so in this commemorative article we give a geometric introduction to the RG-2 flow. A number of new results are presented as part of this narrative: short-time existence and uniqueness results in all dimensions if the sectional curvatures K ij satisfy certain inequalities; the calculation of fixed points for n = 3 dimensions; a reformulation of constant curvature solutions in terms of the Lambert W function; a classification of the solutions that evolve only by homothety; an analogue for RG flow of the 2-dimensional Ricci flow solution known to mathematicians as the cigar soliton, and discussed in the physics literature as Witten’s black hole. We conclude with a list of open problems whose resolutions would substantially increase our understanding of the RG-2 flow both physically and mathematically.

We generalize Ringstr\"om's global future causal stability results (Ringstr\"om 2009) for certain expanding cosmological solutions of the Einstein-scalar field equations to solutions of the Einstein-Maxwell-scalar field system. In particular, after noting that the power law inflationary spacetimes $(M^{n+1}, \hat{g}, \hat{\phi})$ considered by Ringstr\"om in Ringstr\"om (2009) are solutions of the Einstein-Maxwell-scalar field system (with exponential potential) as well as of the Einstein-scalar field system (with the same exponential potential), we consider (nonlinear) perturbations of initial data sets of these spacetimes which include electromagnetic perturbations as well as gravitational and scalar perturbations. We show that if (as in Ringstr\"om, 2009) we focus on pairs of relatively scaled open sets $U_{R_0} \subset U_{4R_0}$ on an initial slice of $(M^{n+1}, \hat{g})$, and if we choose a set of perturbed data which on $ U_{4R_0}$ is sufficiently close to that of $(M^{n+1}, \hat{g},\hat{\phi},\hat{A}=0)$, then in the maximal globally hyperbolic spacetime development $(M^{n+1},g,\phi,A)$ of this data via the Einstein-Maxwell-scalar field equations, all causal geodesics emanating from $U_{R_0}$ are future complete (just as in $(M^{n+1}, \hat{g})$). We also verify the controlled future asymptotic behavior of the fields in the spacetime developments of the perturbed data sets., Comment: 56 pages

Einstein’s theory of general relativity models the physical universe using spacetimes which satisfy Einstein’s gravitational field equations. To date, Einstein’s theory has been enormously successful in modeling observed gravitational phenomena, both at the astrophysical and the cosmological levels. The collection of spacetime solutions of Einstein’s equations which have been effectively used for modeling the physical universe is a very small subset of the full set of solutions. Among this larger set, there are many spacetimes in which strange phenomena related to time are present: There are solutions containing regions in which determinism and the predictability of experimental outcomes breaks down (the Taub-NUT spacetimes), and there others in which the breakdown of determinism occurs everywhere (the Godel universe). Should the existence of these strange solutions lead us to question the usefulness of Einstein’s theory in modeling physical phenomena? Should it instead lead us to seriously search for strange time phenomena in physics? Or should we simply treat these solutions as anomalous (if embarrassing) distractions which we can ignore? In this essay, after introducing some basic ideas of special and general relativity and discussing what it means for a spacetime to be a solution of Einstein’s equations, we explore the use of spacetime solutions for modeling astrophysical events and cosmology. We then examine some of the spacetime solutions in which determinism and causal relationships break down, we relate such phenomena to Penrose’s “Strong Cosmic Censorship Conjecture”, and finally we discuss the questions noted above.

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes., 27 pages

Gu and Zhu (2008 Commun. Anal. Geom. 16 467–94) have shown that type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on . In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.

Given a collection of N solutions of the (3 + 1) Einstein constraint equations which are asymptotically Euclidean and vacuum near infinity, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier work which handled the time-symmetric case, thus providing a construction of large classes of initial data for the many body problem in general relativity.

Given a collection of N asymptotically Euclidean ends with zero scalar curvature, we construct a Riemannian manifold with zero scalar curvature and one asymptotically Euclidean end, whose boundary has a neighborhood isometric to the disjoint union of a specified collection of sub-regions of the given ends. An application is the construction of time-symmetric solutions of the constraint equations which model N-body initial data.

We prove that, starting at an initial metric \(g(0)=e^{2u_{0}}(dx^{2}+dy^{2})\) on ℝ2 with bounded scalar curvature and bounded u0, the Ricci flow ∂tg(t)=−Rg(t)g(t) converges to a flat metric on ℝ2.

We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave gauges. These results show that the AVTD property, which is known to hold for solutions in areal coordinates, is stable to perturbations with respect to the gauge source functions. Our proof is based on an analysis of the singular initial value problem for the Einstein vacuum equations in the generalized wave gauge formalism, and provides a framework which we anticipate to be useful for more general spacetimes., 39 pages

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $\gamma>1/2$, there is a solution with the highest curvature blowing up at the rate $(T-t)^{-(\gamma +1/2)}$. (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter $\gamma$., Comment: Change of title. Final version to appear in Journal f\"ur die reine und angewandte Mathematik

We consider the Einstein–Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conformal boundary at future null infinity; see (Andersson and Chruściel 1994 Commun. Math. Phys. 161 533–68). We work with initial data sets in a variety of regularity classes, primarily considering those data sets whose geometries are weakly asymptotically hyperbolic, as defined in (Allen et al 2015 arXiv:1506.03399). These metrics are C 1,1 conformally compact, but not necessarily C 2 conformally compact. In order to ensure that the data sets we construct are indeed shear-free, we make use of the conformally covariant traceless Hessian introduced in (Allen et al 2015 arXiv:1506.03399). We furthermore construct a class of initial data sets with weakly asymptotically hyerbolic metrics that may be only C 0,1 conformally compact; these data sets are insufficiently regular to make sense of the shear-free condition.