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We study the geometry of geodesics on $\mathsf{SL}(n)$, equipped with the Hilbert-Schmidt metric which makes it a Riemannian manifold. These geodesics are known to be related to affine motions of incompressible ideal fluids. The $n = 2$ case is special and completely integrable, and the geodesic motion was completely described by Roberts, Shkoller, and Sideris; when $n = 3$, Sideris demonstrated some interesting features of the dynamics and analyzed several classes of explicit solutions. Our analysis shows that the geodesics in higher dimensions exhibit much more complex dynamics. We generalize the Virial-identity-based criterion of unboundedness of geodesic given by Sideris, and use it to give an alternative proof of the classification of geodesics in 2D obtained by Roberts--Shkoller--Sideris. We study several explicit families of solutions in general dimensions that generalize those found by Sideris in 3D. We additionally classify all "exponential type" geodesics, and use it to demonstrate the existence of purely rotational solutions whenever $n$ is even. Finally, we study "block diagonal" solutions using a new formulation of the geodesic equation in first order form, that isolates the conserved angular momenta. This reveals the existence of a rich family of bounded geodesic motions in even dimension $n \geq 4$. This in particular allows us to conclude that the generalization of the swirling and shear flows of Sideris to even dimensions $n \geq 4$ are in fact dynamically unstable., 3 figures; report from SURIEM 2020 Summer REU program. v2: journal accepted manuscript

We prove that there does not exist global-in-time axisymmetric solutions to the time-like minimal submanifold system in Minkowski space. We further analyze the limiting geometry as the maximal time of existence is approached.

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation., Comment: 16 pages; expository with some (hopefully) new material. v2: fixed typos in the Strichartz exponent, sharpened interpolation to remove epsilon loss, updated reference list. v4: added Remark 23 showing how to recover standard L^1-L^\infty decay estimates from the weighted L^2-L^\infty argument for Schrodinger

In a 2004 paper, Lindblad demonstrated that the minimal surface equation on R 1 , 1 \mathbb {R}^{1,1} describing graphical timelike minimal surfaces embedded in R 1 , 2 \mathbb {R}^{1,2} enjoy small data global existence for compactly supported initial data, using Christodoulou’s conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity.

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking–Galloway–Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein–Euler equations whose spatial geometry is generally not isotropic.

In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present., Comment: This version is very close to the one that will soon be published. Based in part on the comments of referees, we have made the following changes: 1) Change of title; 2) We have provided additional context for our work in the text; 3) Addition of a few new references; 4) Corrections of typos and minor errors

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H[superscript s]-initial conditions (with ss sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition., National Science Foundation (U.S.) (Grant DMS-1162211), Solomon Buchsbaum AT&T Research Fund

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain symmetry class the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid., Comment: 59 pages, 2 figures ; v3 fixed some minor typos and problems with cross references within the document, submitted; v4 added some additional references and a minor clarification in Appendix A; v5 incorporated comments from the referees

We introduce a suitable concept of weak evolution in the context of the radial quintic focussing semilinear wave equation on $\mathbb{R}^{3+1}$, that is adapted to continuation past type II singularities. We show that the weak extension leads to type I singularity formation for initial data corresponding to: (i) the Kenig-Merle blow-up solutions with initial energy below the ground state and (ii) the Krieger-Nakanishi-Schlag blow-up solutions sitting initially near and "above" the ground state static solution., Comment: 11 pages. Version 3 incorporated suggestions from the referee. The main argument remains unchanged, but the presentation is revised and clarified. Theorem 3.4 (which is an immediately corollary from the method of proof in the original manuscript) is new. Version 4 improved literature review

We show that a stationary asymptotically flat electro-vacuum solution of Einstein's equations that is everywhere locally "almost isometric" to a Kerr-Newman solution cannot admit more than one event horizon. Axial symmetry is not assumed. In particular this implies that the assumption of a single event horizon in Alexakis-Ionescu-Klainerman's proof of perturbative uniqueness of Kerr black holes is in fact unnecessary., Version 2: improved presentation; no changes to the result. Version 3: corrected an oversight in the historical review. Version 4: version accepted for publication

We consider classical static Skyrmion-anti-Skyrmion and Skyrmion-Skyrmion configurations, symmetric with respect to a reflection plane, or symmetric up to a $G$-parity transformation respectively. We show that the stress tensor component completely normal to the reflection plane, and hence its integral over the plane, is negative definite or positive definite respectively. Classical Skyrmions always repel classical Skyrmions and classical Skyrmions always attract classical anti-Skyrmions and thus no static equilibrium, whether stable or unstable, is possible in either case. No other symmetry assumption is made and so our results also apply to multi-Skyrmion configurations. Our results are consistent with existing analyses of Skyrmion forces at large separation, and with numerical results on Skymion-anti-Skyrmion configurations in the literature which admit a different reflection symmetry. They also hold for the massive Skyrme model. We also point out that reflection symmetric self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with self-gravitating anti-Skyrmions., Comment: v2 Typos corrected, refs added. v3 Journal version

The goal of the present paper is three-fold. First is to clarify the connection between the dominant energy condition and hyperbolicity properties of Lagrangian field theories. Second is to provide further analysis on the breakdown of hyperbolicity for the Skyrme model, sharpening the results of Crutchfield and Bell and comparing against a result of Gibbons, and provide a local well-posedness result for the dynamical problem in the Skyrme model. Third is to provide a short summary of the framework of regular hyperbolicity of Christodoulou for the relativity community. In the process, a general theorem about dominant energy conditions for Lagrangian theories of maps is proved, as well as several results concerning hyperbolicity of those maps., Comment: Version 2: 26 pages. No figures. Subsumes the content previously announced in http://arxiv.org/abs/0909.4706 Small corrections + some additional remarks

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite order around a point, then they must coincide, up to a local diffeomorphism, in a neighborhood of the point. The novelty of our method lies in the use of a Carleman inequality and thus circumventing the use of analyticity; thus the method is robust under certain non-analytic perturbations. As an example, we also show the strong unique continuation property for the Riemannian Einstein-scalar-field system with cosmological constant., 12 pages; some minor errors are fixed in revision, some clarifications are made

In the present paper, the characterization of the Kerr metric found by Marc Mars is extended to the Kerr-Newman family. A simultaneous alignment of the Maxwell field, the Ernst two-form of the pseudo-stationary Killing vector field, and the Weyl curvature of the metric is shown to imply that the space-time is locally isometric to domains in the Kerr-Newman metric. The paper also presents an extension of Ionescu and Klainerman's null tetrad formalism to explicitly include Ricci curvature terms., (current version) 29 pages. Incorporated changes suggested by the anonymous referee. Resubmitted to Annales Henri Poincare

Ejection of the three electrons from lithium in a single electron collision has been observed for the first time. Triply charged lithium was observed in an ion time-of-flight spectrum following electron impact on a sample of ultracold, trapped lithium. The higher signal/background afforded by the trap environment made the observation of ${\mathrm{L}\mathrm{i}}^{3+}$ possible. We measured the ratios of triple-to-double and double-to-single ionization at an impact energy of 1000 eV. The $3+/2+$ ratio is $\ensuremath{\approx}0.001$, a value 2 orders of magnitude lower than semiempirical predictions. We present a simple method that uses photoionization data combined with sum-rule analysis to predict the asymptotic charge-state ratios. The sum-rule predictions compare reasonably with experiment and shake calculations, but disagree sharply with the semiempirical estimates.

Under mild assumptions, we remove all traces of the axiom of choice from the construction of the maximal globally hyperbolic Cauchy development in general relativity. The construction relies on the notion of direct union manifolds, which we review. The construction given is very general: any physical theory with a suitable geometric representation (in particular all classical fields), and such that a strong notion of "local existence and uniqueness" of solutions for the corresponding initial value problem is available, is amenable to the same treatment., Comment: Version 2: 9 (+epsilon; depending on compiler) pages; updated references. Version 3: switched to revtex, 6 pages, version accepted for publication