arXiv subjects

James Isenberg

Publications and source records attributed to James Isenberg.

At least 37 records · Page 2Linked to original sources

arXiv2019

Asymptotic gluing of shear-free hyperboloidal initial data sets

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 Hölder 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.

math.DG↗

arXiv2019

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate $(T-t)^{-1}$. (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface approaches a collapsing cylinder at an exponential rate.

math.DG↗

arXiv2019

Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations

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).

gr-qc↗

arXiv2019

Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

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.

math.DG↗

arXiv2018

Stability Within $T^2$-Symmetric Expanding Spacetimes

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.

math.AP↗

arXiv2018

Symmetries of Cosmological Cauchy Horizons with Non-Closed Orbits

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.

gr-qc↗

arXiv2018

Convergence Stability for Ricci Flow

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.

math.DG↗

arXiv2017

Asymptotic Behavior in Polarized {\bf T}$^2$-symmetric Vacuum Spacetimes

We use Fuchsian Reduction to study the behavior near the singularity of a class of solutions of Einstein's vacuum equations. These solutions admit two commuting spacelike Killing fields like the Gowdy spacetimes, but their twist does not vanish. The spacetimes are also polarized in the sense that one of the `gravitational degrees of freedom' is turned off. Examining an analytic family of solutions with the maximum number of arbitrary functions, we find that they are all asymptotically velocity-term dominated as one approaches the singularity.

gr-qc↗

arXiv2017

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

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 $γ>1/2$, there is a solution with the highest curvature blowing up at the rate $(T-t)^{-(γ+1/2)}$. (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 $γ$.

math.DG↗

arXiv2016

A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges

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.

gr-qc↗

arXiv2016

Weakly asymptotically hyperbolic manifolds

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.

math.DG↗

arXiv2016

General Relativity, Time and Determinism

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 Gödel 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.

gr-qc↗

arXiv2016

Existence and Blowup Results for Asymptotically Euclidean Initial Data Sets Generated by the Conformal Method

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.

gr-qc↗

arXiv2015

The shear-free condition and constant-mean-curvature hyperboloidal initial data

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. 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 [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 [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.

math.DG↗

arXiv2015

On Strong Cosmic Censorship

For almost half of the one hundred year history of Einstein's theory of general relativity, Strong Cosmic Censorship has been one of its most intriguing conjectures. The SCC conjecture addresses the issue of the nature of the singularities found in most solutions of Einstein's gravitational field equations: Are such singularities generically characterized by unbounded curvature? Is the existence of a Cauchy horizon (and the accompanying extensions into spacetime regions in which determinism fails) an unstable feature of solutions of Einstein's equations? In this short review article, after briefly commenting on the history of the SCC conjecture, we survey some of the progress made in research directed either toward supporting SCC or toward uncovering some of its weaknesses. We focus in particular on model versions of SCC which have been proven for restricted families of spacetimes (e.g., the Gowdy spacetimes), and the role played by the generic presence of Asymptotically Velocity Term Dominated behavior in these solutions. We also note recent work on spacetimes containing weak null singularities, and their relevance for the SCC conjecture.

gr-qc↗

arXiv2014

Ricci flow neckpinches without rotational symmetry

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.

math.DG↗

arXiv2014

General Relativity and Gravitation: A Centennial Perspective

To commemorate the 100th anniversary of general relativity, the International Society on General Relativity and Gravitation (ISGRG) commissioned a Centennial Volume, edited by the authors of this article. We jointly wrote introductions to the four Parts of the Volume which are collected here. Our goal is to provide a bird's eye view of the advances that have been made especially during the last 35 years, i.e., since the publication of volumes commemorating Einstein's 100th birthday. The article also serves as a brief preview of the 12 invited chapters that contain in-depth reviews of these advances. The volume will be published by Cambridge University Press and released in June 2015 at a Centennial conference sponsored by ISGRG and the Topical Group of Gravitation of the American Physical Society.

gr-qc↗

arXiv2014

Short-time existence for the second order renormalization group flow in general dimensions

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 $K_P$ satisfy the condition $1+αK_P > 0$, at all points $p \in M$ and planes $P \subset T_p M$. This extends results previously proven for two and three dimensions.

math.DG↗