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Math Department Seminars MSU M∩Φ MSU AWM

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2012.8.2 Oberwolfach

Notes by Willie Wong (wongwwy -at- member.ams.org). Comments and suggestions welcome.

Piotr Chrusciel : Causality theory

Statement: $(M,g)$ globally hyperbolic smooth, then Cauchy problem for the linear wave equation has unique global solutions. What happens if the metric is not smooth? Colombini, Spagnola, Lerner: there exists $g \in \cap_{\alpha\in (0,1)} C^{0,\alpha} \setminus C^{0,1}$ such that there are no solutions of the linear wave equations in the distributional sense.

If $g$ is Lipschitz, the wave equation always have unique local solutions. For "global" question, a statement depends on what is meant by "globally hyperbolic".

In $C^2$, we introduce normal coordinates. ($C^{1,1}$ you can solve geodesic equation, but the coordinates may not be as good; in $C^2$ you can use implicit function.)

So for rough metrics, the standard causality theory based on normal coordinates and local behaviour need to be modified.

Einstein equation $(\Sigma, g, K)$ where $g\in H^{s+1}$ and $K\in H^{s}$ data. Solve vacuum equations.

1. Solution in local coordinates (harmonic, etc.); if $s > 3/2$ we get LLWP.
2. Patch things together. $s > 5/2$; higher derivatives for uniqueness. Rodnianski-Planchon announced an improvement to $s > 3/2$.
3. Choquet-Bruhat--Geroch: if $s = \infty$ then exists Maximally Globally Hyperbolic extension.

Theorem: MGHE exists if $s = 2$. (Paper[s] on arXiv, 2011; partially joint work with James Grant)

1. Standard PDEs, $s = 2 > 3/2$.
2. Do Causality theory in this case (so you can define Globaly Hyperbolicity).

Focus on part 2. for which we have Theorem: (a) all standard statements of Causality (cf. arxiv:11006706) not concerning geodesics and normal coordinates are true for $C^{0,1}$ metrics (b) several standard facts become non-facts for general $C^{0,\alpha}$ with $\alpha < 1$.

Main Idea: sandwich $g$ between sequences of smooth metrics and see how much one can retain under convergence (in the topology of uniform convergence on compact sets).

Sandwich in the sense of Causal sandwich; consider partial ordering on the space of metrics by containment of the cone of future timelike vectors. So the main question: if $\hat{g}_i \searrow g$ and $\check{g}_i \nearrow g$, is $$\liminf \text{ Causal structure}(\hat{g}_i) \overset{?}{=} \limsup \text{ Causal structure}(\check{g}_i)$$

A point $p$ is said to be time-like relative to $q$ if there exists a differentiable time-like curve connecting the two. (Differentiable can be replaced with Lipschitz with a.e. derivative timelike with no loss under assumption of $C^{0,1}$ metrics. Since when limits are taken of time-like curves we want them to fall inside the class of causal curvest; we don't need the reverse.)

A causal curve is a Lipschitz curve whose derivative (which is known to exists a.e.) is future directed causal where it exists.

We have the obvious inclusions (where all sets defined in some small compact neighborhood of a point) $$\limsup \mathcal{I}(\check{g}) = \cup \mathcal{I}(\check{g}_i) \subseteq \mathcal{I}(g) \subset \overline{\mathcal{I}(g)} \subseteq \mathcal{J}(g) \subseteq \cap\mathcal{J}(\hat{g}_i) = \liminf \mathcal{J}(\hat{g})$$

The question is how much of these can be made to be equality. Claim: for Lipschitz metrics, the two $\subseteq$ can be replaced by $=$.

Example: $$g = - dx^2 + [ (1 - |u|^\lambda) dx + du ]^2 \in C^{\infty}(\mathbb{R}^2 \setminus \{u =0\}) \cap C^{0,\lambda}$$

The surface $u = 0$ is null. The other null is given by $\frac{du}{dx} = |u|^\lambda$. For $\lambda < 1$ you get bifurcation of the ODE solution. You get what is called a "causal bubble", where the difference of $\mathcal{J}$ and $\mathcal{I}$ has, in some sense positive measure.

Jared Speck : On big bang space-times

Joint with Igor Rodnianski.

Recall Penrose-Hawking theorem. (Assume Cosmological space-times, so the topology of spatial slice assume to be $\mathbb{T}^3$.) Strong energy condition + trace of the second fundamental form being uniformly bounded above by $-C$ then we have a singularity.

Fundamental question: is geodesic incompleteness in Penrose-Hawking type theorems a real singularity (e.g. curvature blow-up) or something else (e.g. Cauchy horizon)?

Study Einstein-Stiff-fluid and Einstein -scalar-field. Stiff-fluid has $p = \rho$ in fluid. Scalar field embeds into irrotational stiff-fluid under the assumption that the scalar field has a time-like gradient.

(Generalised) Kasner solutions $$g = - dt^2 + \sum_{j = 1}^3 t^{2q_j} (dx^j)^2$$

with $p = \bar{p}t^{-2}$ and $u^\mu = \delta_0^\mu$. Restrictions from constraints $\sum q_i = 1$ and $\sum q_i^2 = 1 - 2 \bar{p}$. FLRW is isotrpic, so $q = 1/3$.

Study small perturbations of FLRW (some smallness of the deviation of $q$ is assumed; c.f. "rotation of Kasner axes") and see what happens when solving backwards to the big-bang singularity.

Results:

• No symmetry or analyticity assumptions. Perturbation controlled in 8 derivatives Sobolev norm.
• Description of the entire past of the Cauchy slice
• Fully understand geodesic incompleteness
• Verify SCC, verify Weyl Curvature Hypothesis, and demonstrate BKL type behaviour

On the BKL problem, it is true in symmetry classes (esp. Bianchi IX by Ringstrom). There are counterexamples: Taub, spikes [Weaver-Rendall, Lim] stiff fluid/scalar-field.)

Closely related results:

1. Hans Ringstrom: perfect fluid in Bianchi A. We see
2. curvature blow-up at the singularity except for vacuum Taub
3. Bianchi IX with non-stiff: at least 3 limit points, so oscillating. And those limit points are vacuum type II (matter doesn't matter).
4. Bianchi A with stiff fluid: converges to a singular point. No longer this oscillating behaviour. (matter matters)
5. Andersson-Rendall: Quiescent cosmological singularities (roughly 2001). Stiff-fluids / scalar fields
6. Constructed large family of the spatially analytic solutions to the Einstein equations
7. Same number of degrees of freedom as the "general solution"
8. BK-type behaviour near singularity (shedding of spatial parts)
9. Asymptotic to VTD (velocity term dominated) solutions (Einstein without spatial derivatives)
10. The stiff fluid prevents oscillatory mixmaster behaviour

Theorem (Rodnianski-Speck): Consider near-FLRW (in Sobolev sense) data for the Einstein-stiff-fluid on three-torus. We have

• (Gerhardt, Bartnik) MGHD has a CMC slice $\Sigma_t$ near the initial data. By Cauchy stability use that as the initial data
• Use CMC foliation to synchronise the various space-like points
• Big Bang: volme of the CMC slices collapses as $t\searrow 0$ (The CMC slices have the mean curvature $-1/(3t)$, $t\in (0,1]$.)
• Stability: $t^2p, tk_{ij}, t^{-1}\sqrt{|g|}$ have finite, near (Rescaled) FLRW limits as $t\searrow 0$.
• No fluid shocks!!!
• SCC: Riemann invariants (roughly squared of Riemann curvature) blow up like $t^{-4}$
• WCH (optimal): $|W|_g^2 / |Riem|_g^2$ remains small
• Hawking-Penrose: All past-directed timelike geodesics emanating from $\Sigma_t$ are shorter than $Ct^{2/3 - C\epsilon}$, where $\epsilon$ measures the size of the data deviation from FLRW
• Spatial separation: if $q\neq r \in \Sigma_t$ and $t$ sufficiently small, then their pasts are disjoint. ($\implies$ singularity is space-like)
• VTD: most potential terms negligible near $t = 0$.

Main ideas of the proof:

• CMC - transported spatial coordinates
• Renormalised solution variable + equations (using heavily the structure)
• Boa constrictor, no-mercy analysis (no slick tricks. prove all quantitative estimates)
• integration by parts and $L^2$ energy estimates
• Strong $C^M$ estimates via transport: here the special structure of the equations is heavily used.
• Partial decoupling of the system
• Kinetic terms dominante potential terms. (The most important stuff are second fundamental form and pressure)
• Elliptic lapse estimates - synchronise the singularity - two versions of the PDE.
• Amazing cancellation of some renormalised terms
• Decouple and recombine metric and fluid energy identities in a suitable fashion
• New almost-monotonicy energy identity: a rebel term in the fluid equations + lapse equation + Hamiltonian constraint + CMC + Integration by parts$\implies$ rebel term gives unexpected dynamic control of the lapse.

Toy problem:

$$- \partial_t^2 \phi + \sum r^{2q_i} \partial_i^2 \phi = \frac1t \partial_t\phi + Q(\nabla \phi, \nabla \phi)$$ and $$Q(\nabla \phi, \nabla \phi) = \sum t^{-2q_j} (\partial_j\phi)^2$$ Toy FLRW: $\tilde{\phi} = \ln t$ as the background.

Philippe LeFloch : Weakly Regular Spacetimes with Symmetry

Global Cauchy problem under $T^2$ symmetry, Gowdy, or plane. Metrics with weak regularity.

Include: interacting impulsive waes, matter spacetimes with shock waves.

Challenges: define Einstein equations in a weak sense (delta singularities) and existence theory for such weak equations.

(Can't type fast enough...)

L.-Hs. Huang : Spacetime Positive Mass Theorem in dimension $<8$

Joint with Eichmair, Lee, and Schoen

Theorem: $3 \leq n < 8$, $(M^n,g,k)$ AF, DEC, ... then $E \geq |P|$.

Schoen-Yau proved the time-symmetric case in the same dimensions; also showed rigidity: if $E = 0$ we have Euclidean. ('79, '81, '87)

Schoen-Yau '81 proved the space-time case in dimension 3 $E \geq 0$ with rigidity: if $E = 0$ then isometric embeddable in Minkowski space. Eichmair generalised to $3 \leq n < 8$.

Witten '81 $n = 3$ using spinors. AF DEC gives $E \geq |P|$.

Bartnik '87 and others $n \geq 3$ spin.

Equality case by Beig-Chrusciel (96) Chrusciel-Maerten ('98) for spin.

Riemannian PMT:

1. Density argument, change coordinates (harmonic near infinity)
2. $E < 0$ gives hyperplane barriers. Implies existence of minimal surface throat)
3. Height picking: the constructed minimal surface are stable to compact perturbations. Can using height picking to get further stability (rel. translations).
4. Apply stability, get Yamabe class positive: the minimal surface must be smooth in $n < 8$ by geometric measure theory. After a conformal change, can force the minimal surface to have zero scalar curvature and blow-up to AF manifold with $E < 0$.

The proof of the space-time case is similar. Main difficulties are in steps 1 and 3.

For the space-time theorem requires more than just the hamiltonian constraint. Step 1 is more complicated as such. Step 2 follows from previous works of Eichmair and Metzger. In Step 3 we cannot "minimise" since mass is not variational in the non-symmetric context.

$E < |P|$ gives MOTS barriers. (Step 2)

Juan A. Valiente Kroon : A conformal approach to stability result for radiation FRW cosmologies

Joint work with C. Lubbe

Einstein-Euler with cosmological constant. Use $(+---)$ signature. Recall FRW cosmologies. Remark: a perfect fluid model is FRW iff the fluid is shear-, rotation-, and acceleration-free.

A class of equations of states were shown to have future stability for FLRW with de Sitter like cosmological constant by Rod-Speck; personal failure excludes radiation EOS $p = \frac13 \rho$.

Theorem: small perturbations in radiation cosmologies with spatial curvature $+1$ lead to future global existence, future geodescially complete, and remains close to the background FRW.

Remark: spatial curvature $-1,0$ can be treated with bit more technical detail.

Use conformal rescalings: radiation case is conformal, $\mathrm{tr} T = 0$. Denote conformal weight by $\Theta$. $\implies$ Friedrich equations.

arXiv: 1111.4691 [gr-qc]

C. Sormani : Intrisic flat convergence as a gauge invariant means of defining weak convergence of manifold

Joint work with Stefan Wenger, Dan Lee, Sajjad Lakzian.

If $M_i$ are compact oriented Riemannian manifolds, $d_F$ the intrinsic flat distance is zero iff there exists an orientation preserving isometry between them. Based on Federer-Flemming (Annals 1960) + Ambrosio-Kirchhein (spelling? Acta 2000).

Definition: $$d_F(M_1,M_2) = \inf_{Z, \psi} \{ d_{Flat}^Z(\psi_1(M_1), \psi_2(M_2)) : Z \text{ closed metric space}, \psi_i:M_i\to Z \text{ distance preserving}\}$$

$d_{Flat}$ measures a filling volume between the two sets. (See Federer, Geometric Measure Theory)