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2012.8.3 Oberwolfach

Notes by Willie Wong (wongwwy -at- member.ams.org).

Piotr Bizon : Instability of ADS

ADS: $$-(1 - r^2 / \ell^2) dt^2 + \frac{dr^2}{1+r^2 / \ell^2} + r^2 d\Omega^2_{d-1}$$

or, compactify coordinates $$\frac{\ell^2}{\cos^2 x} (-dt^2 + dx^2 + \sin^2 x d\Omega^2_{d-1})$$ Infinity is $x = \pi/2$, a time-like cylinder.

Is ADS stable:

• Largely unexplored; M Anderson 2006
• Minkowski is stable (Chr Kla)
• Key difference: dissipiation of energy by dispersion for Minkowski through Scri; cannot do so in ADS because it is effectivel bounded (for no flux boundary conditions it acts like a perfect cavity)
• Note that by positive energy theorems both Mink and ADS are the unique "ground states" for their respective ranges of cosmological constant.

Model:

• Make spherical symmetry, add matter field because of Birkhoff's theorem
• Simple model: massless scalar field
• Christodoulou's work; critical phenomenon
• Set $d = 3$ (same results apply for $d > 3$, but not 2.

Parametrise asym-ADS by $$\frac{\ell^2}{\cos^2 x}\left( - A e^{-2\delta} dt^2 + A^{-1} dx^2 + \sin^2x d\Omega^2\right)$$ and solve the initial-boundary value problem for perturbations of ADS with reflecting boundary conditions.

Perturbation expanded as series and solve order by order in a smallness parameter $\epsilon$.

First order: linearised equation $$\ddot{\phi} - \frac{1}{\tan^2 x}(\partial_x \tan^2 x \partial_x \phi)$$ is linearly stable: eigenvalues are squares of odd integers.

Third order captures cubic nonlinearities, which leads to resonance conditions on the frequency supports (expansion by generalised Fourier series).

For single mode initial data, one can avoid resonance at third order due to trigonometric identities. For two-mode initial data, however, resonance manifest at third order, leading to secular modes that drives a shift of energy spectrum to higher and higher frequencies.

Conjectures:

1. ADS is unstable against the formation of a black hole under arbitrarily small generic perturbations
2. Einstein-Scalar-ADS equations admit time-quasiperiodic solutions.

(Diaz, Horowitz, Santos conjectured also the existence of geons for vacuum; 2011)

Minimally coupled scalar field $d=2$ Jalmuzna 2012. Found turbulent instability for small perturbations of ADS-Einstein-Scalar-Field in $d = 2$, but no collapse (black hole mass gap). No indication of for singularity formation. Evidence for equipartition of energy (almost flat spectrum of energy).

"Instability" of Minkowski space enclosed in a cavity: perturb Einstein-scalar inside a ball with non-flux boundary conditions.

Nonlinear waves on bounded domains: no dispersion to infinity and expect weakly turbulent behaviour, with energy moving from low to high frequencies. Toy problem: $$u_{tt} - u_{xx} + m^2 u + u^3 = 0$$ on a bounded interval. When $m > 0$ it is not resonant (spectrum is $\sqrt{j^2 + m^2}$, $j\in \mathbb{N}$), so can build periodic solutions from linear modes by KAM theory. When $m = 0$ resonance prevents the same techniques from working. Conversely, when $m > 0$ the turbulent behaviour takes a long time to set in; when $m = 0$ the turbulent behaviour is exhibited almost immediately.

Qian Wang : Rough solutions to the EVE in CMCSH gauge

(Constant Mean Curvature Spatially Harmonic)

Given initial data set $(\Sigma,g,k)$ with $\Sigma$ compact. Pick a smooth Riemannian metric $\hat{g}$. We shoose $U = 0$ and $\mathrm{tr}k = t$ on $\Sigma _t$, where $U$ is the vector field corresponding to the trace of the difference between Christoffel symbols of $g$ and $\hat{g}$.

Anderson-Moncrief: $s > 5/2$

Theorem (arXiv: 1201.0049) $s > 2$ LWP of the CMCSH Einstein flow.

$$\|\hat\nabla g, k\|_{L^2_tL^\infty_x} + \|\hat\nabla g, k\|_{L^\infty_t H^{s-1}_x} \leq M$$

Highlights: In Klainerman-Rodnianski $2+\epsilon$ the commuting vector field was implemented by paradifferential regularizing $g$ and dropping higher frequency parts by Littlewood-Paley projection. The main drawback is that the Ricci of the projected metric is no longer vanishing. To push this through to gain over the general quasilinear wave (where the approach has a gap), for Einstein's equation one can show that $D Ric_{\leq \lambda} \approx \lambda^{-a}$, that is, the projected metric has "small" Ricci tensor.

In this work the vector field method is put on $g$ directly, without regularisation.