On the geometry of Riemannian manifolds with a Lie structure at infinity

A manifold with a ``Lie structure at infinity'' is a non-compact manifold $M_0$ whose geometry is described by a compactification to a manifold with corners M and a Lie algebra of vector fields on M, subject to constraints only on $M \smallsetminus M_0$. The Lie structure at infinity on $M_0$ determines a metric on $M_0$ up to bi-Lipschitz equivalence. This leads to the natural problem of understanding the Riemannian geometry of these manifolds. We prove, for example, that on a manifold with a Lie structure at infinity the curvature tensor and its covariant derivatives are bounded. We also study a generalization of the geodesic spray and give conditions for these manifolds to have positive injectivity radius. An important motivation for our work is to study the analysis of geometric operators on manifolds with a Lie structure at infinity. For example, a manifold with cylindrical ends is a manifold with a Lie structure at infinity. The relevant analysis in this case is that of totally characteristic operators on a compact manifold with boundary equipped with a ``b-metric.'' The class of conformally compact manifolds, which was recently proved of interest in the study of Einstein's equation, also consists of manifolds with a Lie structure at infinity.Comment: LaTe

Nonlinear morphoelastic plates II: exodus to buckled states

Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed

Nonlinear Morphoelastic Plates I: Genesis of Residual Stress

Volumetric growth of an elastic body may give rise to residual stress. Here a rigorous analysis of the residual strains and stresses generated by growth in the axisymmetric Kirchhoff plate is given. Balance equations are derived via the global constraint principle, growth is incorporated via a multiplicative decomposition of the deformation gradient, and the system is closed by a response function. The particular case of a compressible neo-Hookean material is analyzed and the existence of residually stressed states is established

Asymptotically cylindrical Calabi-Yau manifolds

Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure theorems for $M$; in particular we show that there is no loss of generality in assuming that $M$ is simply-connected and irreducible with Hol$(M)$ $=$ SU$(n)$, where $n$ is the complex dimension of $M$. If $n > 2$ we then show that there exists a projective orbifold $\bar{M}$ and a divisor $\bar{D}$ in $|{-K_{\bar{M}}}|$ with torsion normal bundle such that $M$ is biholomorphic to $\bar{M}\setminus\bar{D}$, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where $\bar{M}$ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair $(\bar{M}, \bar{D})$ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on $\bar{M}\setminus\bar{D}$.Comment: 33 pages, various updates and minor corrections, final versio

Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow

We study constant mean curvature Lorentzian hypersurfaces of $\mathbb{R}^{1,d+1}$ from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric solutions exhibit one of three (future) asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like cylinder isometric to some $\mathbb{R}\times\mathbb{S}^d$ and (iii) infinite expansion to the future converging asymptotically to a time translation of the de Sitter solution. For class (iii) we examine the future stability properties of the solutions under arbitrary (not necessarily spherically symmetric) perturbations. We show that the usual notions of asymptotic stability and modulational stability cannot apply, and connect this to the presence of cosmological horizons in these class (iii) solutions. We can nevertheless show the global existence and future stability for small perturbations of class (iii) solutions under a notion of stability that naturally takes into account the presence of cosmological horizons. The proof is based on the vector field method, but requires additional geometric insight. In particular we introduce two new tools: an inverse-Gauss-map gauge to deal with the problem of cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical errors, with the exception of Remark 1.2 and Section 9.1 which are new and which explain the extrinsic geometry of the embedding in more detail in terms of the stability result. Version 3: updated reference

The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest, and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines, and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitational-field equations that are nevertheless realized in Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American Mathematical Societ