Acta Mathematica

Under these assumptions, we show that if $u$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y), 0)$ at time $t=0$, then the velocity field $u$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first non-linear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved.

Our main result is a calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton’s formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [42]. The formula on the Picard stack is obtained from [42] for target varieties $\mathbb{CP}^n$ in the limit $n\to \infty$. The result may be viewed as a universal calculation in Abel–Jacobi theory.

As a consequence of the calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton’s formula (as conjectured in [28, Appendix] and [72]). The comparison result of fundamental classes proven in [40] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated with Pixton’s formula.

The proof essentially boils down to the fact that the equation does not have purely non-radiative multi-soliton solutions. The proof overcomes the fundamental obstruction for the extension of the 3D case (treated in [21]) by reducing the study of a multisoliton solution to a finite-dimensional system of ordinary differential equations on the modulation parameters. The key ingredient of the proof is to show that this system of equations creates some radiation, contradicting the existence of pure multi-solitons.

The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In a previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces.

Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger–Kleiner–Naor (2009) and Lafforgue–Naor (2012), they have several noteworthy implications. We deduce that the $L_1$ distortion of a word-ball of radius $n\geqslant 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\operatorname{log}n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work (2017) showed that the distortion of a word-ball of radius $n \geqslant 2$ is of order $\sqrt{\operatorname{log} n}$. We also show that, for any $p \gt 2$, there is a metric space that embeds into both $\ell_1$ and $\ell_p$, yet not into a Hilbert space. This answers the classical question of whether there is a metric analogue of the Kadec–Pełczyński theorem (1962), which implies that a normed space that embeds into both $L_p$ and $L_q$ for $p \lt 2 \lt q$ is isomorphic to a Hilbert space. Another consequence is that for any $p \gt 2$ there is a Lipschitz function $f : \ell_p \to \ell_1$ that cannot be factored through a subset of a Hilbert space using Lipschitz functions, i.e., there are no Lipschitz functions $g : \ell_p \to \ell_2$ and $h : g (\ell_p) \to \ell_1$ such that $f = h \circ g \,$; this answers the question, first broached by Johnson–Lindenstrauss (1983), whether there is an analogue of Maurey’s theorem (1974) that such a factorization exists if $f$ is linear. Finally, we obtain conceptually new examples that demonstrate the failure of the Johnson–Lindenstrauss dimension reduction lemma (1983) for subsets of $\ell_1$; these are markedly different from the previously available examples (Brinkman–Charikar, 2003) which do not embed into any uniformly convex normed space, while for any $p \gt 2$ we obtain subsets of $\ell_1$ for which the Johnson–Lindenstrauss lemma fails, yet they embed into $\ell_p$.

To the contrary, the “spectral implies tiling” direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ under the a priori assumption that $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ is a polytope) and could not be treated using the previously developed techniques.

In this paper we fully settle Fuglede’s conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $\Omega \subset \mathbb{R}^d$ is a spectral set, then $\Omega$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric “weak tiling” condition necessary for a set $\Omega \subset \mathbb{R}^d$ to be spectral.