arXiv subjects

James Isenberg

Publications and source records attributed to James Isenberg.

At least 19 recordsLinked to original sources

arXiv2026

Yvonne Choquet-Bruhat 1923-2025

This is a memorial article for Yvonne Choquet-Bruhat, who was one of the great pioneers of mathematical general relativity and of partial differential equations. Starting with her 1952 result on local existence of solutions of the vacuum Einstein field equations, she obtained many results on the Einstein evolution equations, the Einstein constraint equations, and the equations of supergravity. Her methods have also been important for numerical relativity. She also wrote several textbooks and a memoir. An abridged version of this article has been submitted to AMS Notices.

gr-qc↗

arXiv2026

Big bang stability and isotropisation for the Einstein-scalar field equations in the ekpyrotic regime

It has been shown that, in spacetime dimensions $n\geq 3$, that the Kasner-scalar field solutions to the Einstein-scalar fields equations with potential $V_0 e^{-s \phi}$, where $s s_c$ and $V_0<0$. Such scalar field potentials are known in the literature as \textit{ekpyrotic}. In particular, we prove that the FLRW solutions to the Einstein-scalar field equations are nonlinearly stable to the past and terminate at a quiescent, crushing AVTD big bang singularity. A distinguishing property of these perturbed spacetimes is that they isotropise towards the big bang.

gr-qc↗

arXiv2025

Asymptotically Euclidean Solutions of the Constraint Equations with Prescribed Asymptotics

We demonstrate that in constructing asymptotically flat vacuum initial data sets in General Relativity via the conformal method, certain asymptotic structures may be prescribed a priori through the specified seed data, including the ADM momentum components, the leading- and next-to-leading-order decay rates, and the anisotropy in the metric's mass term, yielding a recipe to construct initial data sets with desired asymptotics. We numerically construct a simple explicit example of an initial data set, with stronger asymptotics than have been obtained in previous work, such that the evolution of this initial data set does not exhibit the conjectured antipodal symmetry between future and past null infinity.

gr-qc↗

arXiv2025

Linear and nonlinear stability for the Bach flow, I

In this paper we prove the linear stability of a gauge-modified version of the Bach flow on any complete manifold (M, h) of constant curvature. This involves some intricate calculations to obtain spectral bounds, and in particular introduces a higher order generalization of the well-known Koiso identity. We also prove nonlinear stability for the Bach flow if (M, h) is hyperbolic space, and more generally any Poincar\'e-Einstein space sufficiently close to h. In the forthcoming Part II of this project, we study the nonlinear stability question if M is either compact or else noncompact and flat, since those cases require different considerations involving a center manifold.

math.DG↗

arXiv2025

Local singularities of compact multiply warped Ricci flow solutions

We demonstrate that any four-dimensional shrinking Ricci soliton $(\mathcal B \times {\mathbb S^2}, g)$, where $\mathcal B$ is any two-dimensional complete noncompact surface and $g$ is a warped product metric over the base $\mathcal B$, has to be isometric to the generalized cylinder $\mathbb R^2\times\mathbb S^2$ equipped with the standard cylindrical metric. After completing this classification, we study Ricci flow solutions that are multiply warped products -- but not products -- and provide rigorous examples of the formation of generalized cylinder singularity models $\mathbb R^k\times\mathbb S^\ell$.

math.DG↗

arXiv2024

Initial Data for First-order Causal Viscous Conformal Fluids in General Relativity

We solve the Einstein constraint equations for a first-order causal viscous relativistic hydrodynamic theory in the case of a conformal fluid. For such a theory, a direct application of the conformal method does not lead to a decoupling of the equations, even for constant-mean curvature initial data. We combine the conformal method applied to a background perfect fluid theory with a perturbative argument in order to obtain the result.

gr-qc↗

arXiv2024

Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

Kr\"oncke has shown that the Fubini-Study metric is an unstable generalized stationary solution of Ricci flow [Kr\"o20]. In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini-Study metric develop local singularities modeled by the blowdown soliton discovered in [FIK03].

math.DG↗

arXiv2023

A Numerical Study of the Expanding Direction of $T^2$-Symmetric Spacetimes

The asymptotic behavior of expanding, generic, $T^2$-Symmetric, vacuum spacetimes is examined via numerical simulations. After validation of the numerical methods, the properties of these generic spacetimes are explored and compared to non-generic subfamilies where proven results exist. The non-generic subfamilies within this class, including the Kasner, the Gowdy, the pseudo-homogeneous, and the $B=0$ spacetimes, all have known asymptotic behaviors in the expanding direction which have been determined either from the explicit solutions or using analytic methods. For the $B\ne 0$ spacetimes, the generic case within the $T^2$-Symmetric vacuum solutions, the asymptotic behavior has not been determined analytically. In this work, we use numerical simulations to explore the asymptotic behavior of the $B\ne 0$ spacetimes. Our results indicate that, for these generic spacetimes, the asymptotic behavior in the expanding direction differs from that seen in the non-generic subfamilies. In addition to differences in asymptotic power laws, an apparent quasi-periodic exchange of energy from one gravitational mode to the other for the generic non-polarized solutions is observed.

gr-qc↗

arXiv2023

Convergence stability for Ricci flow on manifolds with bounded geometry

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on our recent work where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in H\"older norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric.

math.DG↗

arXiv2022

Wellposedness of nonlinear flows on manifolds of bounded geometry

We present simple conditions which ensure that a strongly elliptic operator $L$ generates an analytic semigroup on H\"older spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that $L$ is "sectorial", a condition that specifies the decay of the resolvent $(\lambda I - L)^{-1}$ as $\lambda$ diverges from the H\"older spectrum of $L$. As one step, we prove existence of this resolvent if $\lambda$ is sufficiently large, and on this general class of manifolds, use a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of $L$ and $e^{-tL}$ we obtain can then be used to prove wellposedness of a wide class of nonlinear flows. We illustrate this by proving wellposedness on H\"older spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.

math.AP↗

arXiv2022

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that for small rotationally-symmetric perturbations of the initial embeddings that are sufficiently far away from the tip, MCF develops Type-I "neckpinch" singularities. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally-symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence -- becoming round -- and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this work is an adaptation of the "overlap" method introduced in [GIKW21] and permits angular dependence.

math.DG↗

arXiv2021

Sectoriality of the Laplacian on Asymptotically Hyperbolic Spaces

We prove that both the Laplacian on functions, and the Lichnerowicz Laplacian on symmetric 2-tensors with respect to asymptotically hyperbolic metrics, are sectorial maps in weighted H\"older spaces. As an application, the machinery of analytic semigroups then applies to yield well-posedness results for parabolic evolution equations in these spaces.

math.AP↗

arXiv2021

Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant

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.

gr-qc↗

arXiv2021

A Phase Space Approach to the Conformal Construction of Non-Vacuum Initial Data Sets in General Relativity

We present a uniform (and unambiguous) procedure for scaling the matter fields in implementing the conformal method to parameterize and construct solutions of Einstein constraint equations with coupled matter sources. The approach is based on a phase space representation of the space-time matter fields after a careful $n+1$ decomposition into spatial fields $B$ and conjugate momenta $\Pi_B$, which are specified directly and are conformally invariant quantities. We show that if the Einstein-matter field theory is specified by a Lagrangian which is diffeomorphism invariant and involves no dependence on derivatives of the space-time metric in the matter portion of the Lagrangian, then fixing $B$ and $\Pi_B$ results in conformal constraint equations that, for constant-mean curvature initial data, semi-decouple just as they do for the vacuum Einstein conformal constraint equations. We prove this result by establishing a structural property of the Einstein momentum constraint that is independent of the conformal method: For an Einstein-matter field theory which satisfies the conditions just stated, if $B$ and $\Pi_B$ satisfy the matter Euler-Lagrange equations, then (in suitable form) the right-hand side of the momentum constraint on each spatial slice depends only on $B$ and $\Pi_B$ and is independent of the space-time metric. We discuss the details of our construction in the special cases of the following models: Einstein-Maxwell-charged scalar field, Einstein-Proca, Einstein-perfect fluid, and Einstein-Maxwell-charged dust. In these examples we find that our technique gives a theoretical basis for scaling rules, such as those for electromagnetism, that have worked pragmatically in the past, but also generates new equations with advantageous features for perfect fluids that allow direct specification of total rest mass and total charge in any spatial region.

gr-qc↗

arXiv2021

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

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 $(T-t)^{-1}$.

math.DG↗

arXiv2021

Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes

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.

gr-qc↗

arXiv2020

Singularity formation of complete Ricci flow solutions

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.

math.DG↗

arXiv2020

On the precise asymptotics of Type-IIb solutions to mean curvature flow

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 $\gamma$.

math.DG↗