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Willie Wai-Yeung Wong

Publications and source records attributed to Willie Wai-Yeung Wong.

10 recordsLinked to original sources
arXiv2025

Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman

Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schrödinger operators of the form $-Δ_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves strengthened versions of the same facts.

math.AP↗
arXiv2014

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.

math.DG↗
arXiv2013

A comment on the construction of the maximal globally hyperbolic Cauchy development

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.

gr-qc↗
arXiv2013

Non-existence of multiple-black-hole solutions close to Kerr-Newman

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.

gr-qc↗
arXiv2012

A positive mass theorem for two spatial dimensions

We observe that an analogue of the Positive Mass Theorem in the time-symmetric case for three-space-time-dimensional general relativity follows trivially from the Gauss-Bonnet theorem. In this case we also have that the spatial slice is diffeomorphic to $\Real^2$.

gr-qc↗
arXiv2011

Regular hyperbolicity, dominant energy condition and causality for Lagrangian theory of maps

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.

math-ph↗
arXiv2009

On strong unique continuation of coupled Einstein metrics

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.

math.AP↗
arXiv2009

A space-time characterization of the Kerr-Newman metric

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.

gr-qc↗