Searcharxiv⌕ Search

arXiv subjects

John M. Lee

Publications and source records attributed to John M. Lee.

12 recordsLinked to original sources
arXiv2022

Sobolev-class asymptotically hyperbolic manifolds and the Yamabe problem

We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity, and introduce several technical tools for studying PDEs on such manifolds. Our results employ two novel families of function spaces suitable for metrics potentially having a large amount of interior differentiability, but whose Sobolev regularity implies only a Hölder continuous conformal structure at infinity. We establish Fredholm theorems for elliptic operators arising from metrics in these families. To demonstrate the utility of our methods, we solve the Yamabe problem in this category. As a special case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.

math.DG↗
arXiv2019

Asymptotic gluing of shear-free hyperboloidal initial data sets

We present a procedure for asymptotic gluing of hyperboloidal initial data sets that preserves the shear-free condition. Our construction is modeled on a previous gluing construction by the last three named authors, but with significant modifications that incorporate the shear-free condition. We rely on the special Hölder spaces, and the corresponding theory for elliptic operators on weakly asymptotically hyperbolic manifolds, introduced by the authors and applied to the Einstein constraint equations in two previous papers.

math.DG↗
arXiv2016

Weakly asymptotically hyperbolic manifolds

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to $-1$ and are $C^0$, but are not necessarily $C^1$, conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.

math.DG↗
arXiv2015

The shear-free condition and constant-mean-curvature hyperboloidal initial data

We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conformal boundary at future null infinity. We work with initial data sets in a variety of regularity classes, primarily considering those data sets whose geometries are weakly asymptotically hyperbolic, as defined in [arXiv:1506.03399]. These metrics are $C^{1,1}$ conformally compact, but not necessarily $C^2$ conformally compact. In order to ensure that the data sets we construct are indeed shear-free, we make use of the conformally covariant traceless Hessian introduced in [arXiv:1506.03399]. We furthermore construct a class of initial data sets with weakly asymptotically hyerbolic metrics that may be only $C^{0,1}$ conformally compact; these data sets are insufficiently regular to make sense of the shear-free condition.

math.DG↗
arXiv2013

A Note on Flux Integrals over Smooth Regular Domains

We provide new bounds on a flux integral over the portion of the boundary of one regular domain contained inside a second regular domain, based on properties of the second domain rather than the first one. This bound is amenable to numerical computation of a flux through the boundary of a domain, for example, when there is a large variation in the normal vector near a point. We present applications of this result to occupational measures and two-dimensional differential equations, including a new proof that all minimal invariant sets in the plane are trivial.

math.DG↗
arXiv2008

Multiplicative functional for reflected Brownian motion via deterministic ODE

We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.

math.PR↗
arXiv2006

Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds

The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an elementary derivation of sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds.

math.DG↗
arXiv1994

The spectrum of an asymptotically hyperbolic Einstein manifold

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum. The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is non-negative, then there are no such eigenvalues. This generalizes results of R. Schoen, S.-T. Yau, and D. Sullivan for the case of hyperbolic manifolds.

dg-ga↗
arXiv1994

CR manifolds with noncompact connected automorphism groups

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater than 3, it has been pointed out by D. Burns that this result follows from known results on biholomorphism groups of complex manifolds with boundary and the fact that any such CR manifold M can be realized as the boundary of an analytic variety. When M is 3-dimensional, Burns's proof breaks down because abstract CR 3-manifolds are generically not realizable as boundaries. This paper provides an intrinsic proof of compactness that works in any dimension.

math.CV↗