Imperial College London

ProfessorGustavHolzegel

Faculty of Natural SciencesDepartment of Mathematics

Professor of Pure Mathematics
 
 
 
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625Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

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26 results found

Holzegel G, Luk J, Smulevici J, Warnick Cet al., 2019, Asymptotic properties of linear field equations in anti-de sitter space, Communications in Mathematical Physics, Vol: 374, Pages: 1125-1178, ISSN: 0010-3616

We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

Journal article

Holzegel G, Dafermos M, Rodnianski I, 2019, Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case |a|≪M, Annals of PDE, Vol: 5, Pages: 1-118, ISSN: 2524-5317

We prove boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equation on a Kerr exterior background with parameters satisfying |a|≪M . The bounds are obtained by introducing generalisations of the higher order quantities P and P–– used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters |a|<M . As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

Journal article

Dafermos M, Holzegel G, Rodnianski I, 2019, The linear stability of the Schwarzschild solution to gravitational perturbations, Acta Mathematica, Vol: 222, ISSN: 1871-2509

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.

Journal article

Holzegel G, Shao A, 2017, Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries, Communications in Partial Differential Equations, Vol: 42, Pages: 1871-1922, ISSN: 0360-5302

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □gϕ+σϕ = (ϕ,∂ϕ) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary.

Journal article

Holzegel G, Shao A, 2016, Unique continuation from infinity in asymptotically anti-de Sitter spacetimes, Communications in Mathematical Physics, Vol: 347, Pages: 723-775, ISSN: 1432-0916

We consider the unique continuation properties of asymptoticallyAnti-de Sitter spacetimes by studying Klein-Gordon-type equations gφ +σφ = G(φ, ∂φ), σ ∈ R, on a large class of such spacetimes. Our main resultestablishes that if φ vanishes to sufficiently high order (depending on σ) ona sufficiently long time interval along the conformal boundary I, then thesolution necessarily vanishes in a neighborhood of I. In particular, in theσ-range where Dirichlet and Neumann conditions are possible on I for theforward problem, we prove uniqueness if both these conditions are imposed.The length of the time interval can be related to the refocusing time of nullgeodesics on these backgrounds and is expected to be sharp. Some globalapplications as well a uniqueness result for gravitational perturbations arealso discussed. The proof is based on novel Carleman estimates established inthis setting.

Journal article

Holzegel G, 2016, Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric, Classical and Quantum Gravity, Vol: 33, ISSN: 1361-6382

We derive an energy conservation law for the system of gravitational perturbations on the Schwarzschild spacetime expressed in a double null gauge. The resulting identity involves only first derivatives of the metric perturbation. Exploiting the gauge invariance up to boundary terms of the fluxes that appear, we are able to establish positivity of the flux on any outgoing null hypersurface to the future of the initial data. This allows us to bound the total energy flux through any such hypersurface, including the event horizon, in terms of initial data. We similarly bound the total energy radiated to null infinity. Our estimates provide a direct approach to a weak form of stability, thereby complementing the proof of the full linear stability of the Schwarzschild solution recently obtained in Dafermos et al (2016 The linear stability of the Schwarzschild solution to gravitational perturbations arXiv:1601.06467).

Journal article

Holzegel G, Klainerman S, Speck J, Wong Wet al., 2016, Small-data shock formation in solutions to 3D quasilinear wave equations: An overview, Journal of Hyperbolic Differential Equations, Vol: 13, ISSN: 1793-6993

This 2007 monograph, D. Christodoulou proved a remarkable result giving a detailed description of shock formation, for small Hs-initial conditions (s sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body ofprior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by F. John in the mid 1970’s and continued by S. Klainerman, T. Sideris, L. H ̈ormander, H. Lindblad, S. 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 J. Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations inthree space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail the classic null condition.

Journal article

Holzegel G, Warnick CM, 2015, The Einstein-Klein-Gordon-AdS system for general boundary conditions, Journal of Hyperbolic Differential Equations, Vol: 12, Pages: 293-342, ISSN: 1793-6993

We construct unique local solutions for the spherically-symmetric Einstein–Klein–Gordon–anti-de Sitter (AdS) system subject to a large class of initial and boundary conditions including some considered in the context of the AdS-CFT correspondence. The proof relies on estimates developed for the linear wave equation by the second author and involves a careful renormalization of the dynamical variables, including a renormalization of the well-known Hawking mass. For some of the boundary conditions considered this system is expected to exhibit rich global dynamics, including the existence of hairy black holes. This paper furnishes a starting point for such global investigations.

Journal article

Holzegel G, Dafermos M, Rodnianski I, 2014, A scattering theory construction of dynamical vacuum black holes, Journal of Differential Geometry, ISSN: 1945-743X

We construct a large class of dynamical vacuum black hole spacetimes whoseexterior geometry asymptotically settles down to a fixed Schwarzschild or Kerrmetric. The construction proceeds by solving a backwards scattering problem forthe Einstein vacuum equations with characteristic data prescribed on the eventhorizon and (in the limit) at null infinity. The class admits the full “functional”degrees of freedom for the vacuum equations, and thus our solutions will in generalpossess no geometric or algebraic symmetries. It is essential, however, for theconstruction that the scattering data (and the resulting solution spacetime) convergeto stationarity exponentially fast, in advanced and retarded time, their rateof decay intimately related to the surface gravity of the event horizon. This canbe traced back to the celebrated redshift effect, which in the context of backwardsevolution is seen as a blueshift.

Journal article

Holzegel G, Smulevici J, 2014, Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes, Analysis and PDE, Vol: 7, Pages: 1057-1090, ISSN: 1948-206X

We construct quasimodes for the Klein–Gordon equation on the black hole exterior of Kerr–AdS (anti- de Sitter) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein–Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semiclassical nonlinear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semiclassical parameter. Our construction results in exponentially small errors in the semiclassical parameter. This implies that general solutions to the Klein Gordon equation on Kerr–AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

Journal article

Holzegel GH, Warnick CM, 2014, Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes, JOURNAL OF FUNCTIONAL ANALYSIS, Vol: 266, Pages: 2436-2485, ISSN: 0022-1236

Journal article

Holzegel G, 2014, EXISTENCE OF DYNAMICAL VACUUM BLACK HOLES, 17th International Congress on Mathematical Physics, Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD, Pages: 382-382

Conference paper

Holzegel G, Smulevici J, 2013, Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes, Communications on Pure and Applied Mathematics, Vol: 66, Pages: 1751-1802, ISSN: 0010-3640

This paper investigates the decay properties of solutions to the massivelinear wave equation $\Box_g \psi + \frac{{\alpha}}{l^2} \psi =0$ for $g$ themetric of a Kerr-AdS spacetime satisfying $|a|l<r_+^2$ and $\alpha<9/4$satisfying the Breitenlohner Freedman bound. We prove that the non-degenerateenergy of $\psi$ with respect to an appropriate foliation of spacelike slicesdecays like $(\log t^\star)^{-2}$. Our estimates are expected to be sharp fromheuristic and numerical arguments in the physics literature suggesting thatgeneral solutions will only decay logarithmically. The underlying reason forthe slow decay rate can be traced back to a stable trapping phenomenon forasymptotically anti de Sitter black holes which is in turn a consequence of thereflecting boundary conditions for $\psi$ at null-infinity.

Journal article

Holzegel G, Smulevici J, 2013, Stability of Schwarzschild-AdS for the Spherically Symmetric Einstein-Klein-Gordon System, COMMUNICATIONS IN MATHEMATICAL PHYSICS, Vol: 317, Pages: 205-251, ISSN: 0010-3616

Journal article

Holzegel G, 2012, WELL-POSEDNESS FOR THE MASSIVE WAVE EQUATION ON ASYMPTOTICALLY ANTI-DE SITTER SPACETIMES, JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS, Vol: 9, Pages: 239-261, ISSN: 0219-8916

Journal article

Holzegel G, Smulevici J, 2012, Self-Gravitating Klein-Gordon Fields in Asymptotically Anti-de-Sitter Spacetimes, ANNALES HENRI POINCARE, Vol: 13, Pages: 991-1038, ISSN: 1424-0637

Journal article

Holzegel G, 2010, Stability and decay rates for the five-dimensional Schwarzschild metric under biaxial perturbations, ADVANCES IN THEORETICAL AND MATHEMATICAL PHYSICS, Vol: 14, Pages: 1245-1372, ISSN: 1095-0761

Journal article

Holzegel G, 2010, On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes, COMMUNICATIONS IN MATHEMATICAL PHYSICS, Vol: 294, Pages: 169-197, ISSN: 0010-3616

Journal article

Holzegel G, Schmelzer T, Warnick C, 2007, Ricci flows connecting Taub–Bolt and Taub–NUT metrics, Classical and Quantum Gravity, Vol: 24, Pages: 6201-6217, ISSN: 1361-6382

We use the Ricci flow with surgery to study four-dimensional SU(2) × U(1)-symmetric metrics on a manifold with fixed boundary given by a squashed 3-sphere. Depending on the initial metric we show that the flow converges to either the Taub–Bolt or the Taub–NUT metric, the latter case potentially requiring surgery at some point in the evolution. The Ricci flow allows us to explore the Euclidean action landscape within this symmetry class. This work extends the recent work of Headrick and Wiseman (2006 Class. Quantum Grav. 23 6683) to more interesting topologies.

Journal article

Holzegel G, Schmelzer T, Warnick C, 2007, Ricci Flow of Biaxial Bianchi IX Metrics, Classical and Quantum Gravity, ISSN: 1361-6382

We use the Ricci flow with surgery to study four-dimensional SU(2) xU(1)-symmetric metrics on a manifold with fixed boundary given by a squashed3-sphere. Depending on the initial metric we show that the flow converges toeither the Taub-Bolt or the Taub-NUT metric, the latter case potentiallyrequiring surgery at some point in the evolution. The Ricci flow allows us toexplore the Euclidean action landscape within this symmetry class. This workextends the recent work of Headrick and Wiseman to more interesting topologies.

Journal article

Dafermos M, Holzegel G, 2006, On the nonlinear stability of higher dimensional triaxial Bianchi-{IX} black holes, Advances in Theoretical and Mathematical Physics, Vol: 10, Pages: 503-523, ISSN: 1095-0761

Journal article

Gibbons GW, Holzegel G, 2006, The positive mass and isoperimetric inequalities for axisymmetric black holes in four and five dimensions, CLASSICAL AND QUANTUM GRAVITY, Vol: 23, Pages: 6459-6478, ISSN: 0264-9381

Journal article

Holzegel G, 2006, On the instability of Lorentzian Taub-NUT space, CLASSICAL AND QUANTUM GRAVITY, Vol: 23, Pages: 3951-3962, ISSN: 0264-9381

Journal article

Holzegel G, Stability and decay-rates for the five-dimensional Schwarzschild metric under biaxial perturbations

In this paper we prove the non-linear asymptotic stability of thefive-dimensional Schwarzschild metric under biaxial vacuum perturbations. Thisis the statement that the evolution of (SU(2) x U(1))-symmetric vacuumperturbations of initial data for the five-dimensional Schwarzschild metricfinally converges in a suitable sense to a member of the Schwarzschild family.It constitutes the first result proving the existence of non-stationary vacuumblack holes arising from asymptotically flat initial data dynamicallyapproaching a stationary solution. In fact, we show quantitative rates ofapproach. The proof relies on vectorfield multiplier estimates, which are usedin conjunction with a bootstrap argument to establish polynomial decay ratesfor the radiation on the perturbed spacetime. Despite being applied here in afive-dimensional context, the techniques are quite robust and may admitapplications to various four-dimensional stability problems.

Journal article

Holzegel G, Ultimately Schwarzschildean Spacetimes and the Black Hole Stability Problem

In this paper, we introduce a class of spacetimes$\left(\mathcal{M},g\right)$ which satisfy the vacuum Einstein equations anddynamically approach a Schwarzschild solution of mass $M$, a class we shallcall \emph{ultimately Schwarzschildean spacetimes}. The approach is captured interms of boundedness and decay assumptions on appropriate spacetime-norms ofthe Ricci-coefficients and spacetime curvature. Given such assumptions at thelevel of $k$ derivatives of the Ricci-coefficients (and hence $k-1$ derivativesof curvature), we prove boundedness and decay estimates for $k$ derivatives of\emph{curvature}. The proof employs the framework of vectorfield multipliersand commutators for the Bel-Robinson tensor, pioneered byChristodoulou-Klainerman in the context of the stability of the Minkowskispace. We provide multiplier analogues capturing the essential decay mechanisms(which have been identified previously for the scalar wave equation on blackhole backgrounds) for the Bianchi equations. In particular, a formulation ofthe redshift-effect near the horizon is obtained. Morever, we identify acertain hierarchy in the Bianchi equations, which leads to the control ofstrongly $r$-weighted spacetime curvature-norms near infinity. This allows toavoid the use the classical conformal Morawetz multiplier $K$, therbygeneralizing recent work of Dafermos and Rodnianski in the context of the waveequation. Finally, the proof requires a detailed understanding of the structureof the error-terms in the interior. This is particularly intricate in view ofboth the phenomenon of trapped orbits and the fact that, unlike in thestability of Minkowski space, not all curvature components decay to zero.

Journal article

Giulini D, Holzegel G, Corvino's construction using Brill waves

For two-black-hole time-symmetric initial data we consider the Corvinoconstruction of gluing an exact Schwarzschild end. We propose to do this byusing Brill waves. We address the question of whether this method can be usedto reduce the overall energy, which seems to relate to the question of whetherit can reduce the amount of `spurious' gravitational radiation. We find apositive answer at first order in the inverse gluing radius.

Journal article

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