The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres

It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

We consider the Dirichlet problem u_t &= \Delta u + f(x, u, \nabla u)+ h(x, t),& \qquad &(x, t) \in \Omega \times (0, \infty), u &= 0, & \qquad &(x, t) \in \partial\Omega \times (0, \infty), on a bounded domain $\Omega \subset \mathbb{R}^N$. The domain and the nonlinearity $f$ are assumed to be invariant under the reflection about the $x_1$-axis, and the function $h$ accounts for a nonsymmetric decaying perturbation: $h(\cdot, t)\to 0$ as $t\to\infty$. In one of our main theorems, we prove the asymptotic symmetry of each bounded positive solution $u$. The novelty of this result is that the asymptotic symmetry is established even for solutions that are not assumed uniformly positive. In particular, some equilibria of the limit time-autonomous problem (the problem with $h\equiv 0$) with a nontrivial nodal set may occur in the $\omega$-limit set of $u$ and this prevents one from applying common techniques based on the method of moving hyperplanes. The goal of our second main theorem is to classify the positive entire solutions of the time-autonomous problem. We prove that if $U$ is a positive entire solution, then one of the following applies: (i) for each $t\in \mathbb{R}$, $U(\cdot,t)$ is even in $x_1$ and decreasing in $x_1>0$, (ii) $U$ is an equilibrium, (iii) $U$ is a connecting orbit from an equilibrium with a nontrivial nodal set to a set consisting of functions which are even in $x_1$ and decreasing in $x_1>0$, (iv) is a heteroclinic connecting orbit between two equilibria with a nontrivial nodal set

The Constraint Equations

We review the properties of the constraint equations, from their geometric origin in hypersurface geometry through to their roles in the Cauchy problem and the Hamiltonian formulation of the Einstein equations. We then review properties of the space of solutions and construction techniques, including the conformal and conformal thin sandwich methods, the thin sandwich method, quasi-spherical and generalized QS methods, gluing techniques and the Corvino-Schoen projection.Comment: 34 pages, LaTeX, to appear in the proceedings of the 2002 Cargese meeting "50 Years of the Cauchy Problem, in honour of Y. Choquet-Bruhat", editors P.T.Chru\'sciel and H. Friedric

Numerical validation of blow-up solutions with quasi-homogeneous compactifications

We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing a new compactification, which shall be called a quasi-parabolic compactification. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up times as well as rigorous profile enclosures of blow-up solutions.Comment: 42 pages. Validation codes are available at http://www.risk.tsukuba.ac.jp/~takitoshi/codes/NVbQC.zi

On long-time existence for the flow of static metrics with rotational symmetry

B List has proposed a geometric flow whose fixed points correspond to solutions of the static Einstein equations of general relativity. This flow is now known to be a certain Hamilton-DeTurck flow (the pullback of a Ricci flow by an evolving diffeomorphism) on RxM^n. We study the SO(n) rotationally symmetric case of List's flow under conditions of asymptotic flatness. We are led to this problem from considerations related to Bartnik's quasi-local mass definition and, as well, as a special case of the coupled Ricci-harmonic map flow. The problem also occurs as a Ricci flow with broken SO(n+1) symmetry, and has arisen in a numerical study of Ricci flow for black hole thermodynamics. When the initial data admits no minimal hypersphere, we find the flow is immortal when a single regularity condition holds for the scalar field of List's flow at the origin. This regularity condition can be shown to hold at least for n=2. Otherwise, near a singularity, the flow will admit rescalings which converge to an SO(n)-symmetric ancient Ricci flow on R^n.Comment: 30 pages; typos fixed; accepted version for Commun Anal Geo

Black Hole Initial Data with a Horizon of Prescribed Geometry

The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic geometry. To do this, we use the parabolic partial differential equation for prescribing scalar curvature. In this equation the horizon geometry is contained within the freely specifiable part of the metric. This contrasts with the conformal method in which the geometry of the horizon can only be specified up to a conformal factor

Black hole initial data with a horizon of prescribed intrinsic and extrinsic geometry

The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic and extrinsic geometry. To do this, we use the parabolic partial differential equation for prescribing scalar curvature. In this equation the horizon geometry is contained within the freely specifiable part of the metric. This contrasts with the conformal method in which the geometry of the horizon can only be specified up to a conformal factor.Comment: to appear in Contemporary Mathematic

Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initial metrics which belong to this class and are rotationally symmetric, we prove that if the sectional curvature in planes tangent to the orbits of symmetry is initially nonpositive, the flow starting from such an initial metric exists for all time. Moreover, if the sectional curvature in planes tangent to these orbits is initially negative, the flow converges at an exponential rate to standard hyperbolic space. This restriction on sectional curvature automatically rules out initial data admitting a minimal hypersphere.Comment: 28 pages, replaced one-word error in two locations on page

Constraints as evolutionary systems

The constraint equations for smooth $[n+1]$-dimensional (with $n\geq 3$) Riemannian or Lorentzian spaces satisfying the Einstein field equations are considered. It is shown, regardless of the signature of the primary space, that the constraints can be put into the form of an evolutionary system comprised either by a first order symmetric hyperbolic system and a parabolic equation or, alternatively, by a symmetrizable hyperbolic system and a subsidiary algebraic relation. In both cases the (local) existence and uniqueness of solutions are also discussed.Comment: 18 pages; exposition improved concerning the algebraic hyperbolic system; references added; to appear in CQ

Einstein metrics with anisotropic boundary behaviour

We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity