Conformal Geometry on Four Manifolds – Noether Lecture
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SOME PROGRESS in CONFORMAL GEOMETRY Sun-Yung A. Chang, Jie Qing and Paul Yang Dedicated to the Memory of Thomas Branson 1. Confo
SOME PROGRESS IN CONFORMAL GEOMETRY Sun-Yung A. Chang, Jie Qing and Paul Yang Dedicated to the memory of Thomas Branson Abstract. In this paper we describe our current research in the theory of partial differential equations in conformal geometry. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes. 1. Conformal Gap and finiteness Theorem for a class of closed 4-manifolds 1.1 Introduction Suppose that (M 4; g) is a closed 4-manifold. It follows from the positive mass theorem that, for a 4-manifold with positive Yamabe constant, σ dv 16π2 2 ≤ ZM and equality holds if and only if (M 4; g) is conformally equivalent to the standard 4-sphere, where 1 1 σ [g] = R2 E 2; 2 24 − 2j j R is the scalar curvature of g and E is the traceless Ricci curvature of g. This is an interesting fact in conformal geometry because the above integral is a conformal invariant like the Yamabe constant. Typeset by AMS-TEX 1 2 CONFORMAL GEOMETRY One may ask, whether there is a constant 0 > 0 such that a closed 4-manifold M 4 has to be diffeomorphic to S4 if it admits a metric g with positive Yamabe constant and σ [g]dv (1 )16π2: 2 g ≥ − ZM for some < 0? Notice that here the Yamabe invariant for such [g] is automatically close to that for the round 4-sphere. -
Non-Linear Elliptic Equations in Conformal Geometry, by S.-Y
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 2, April 2007, Pages 323–330 S 0273-0979(07)01136-6 Article electronically published on January 5, 2007 Non-linear elliptic equations in conformal geometry, by S.-Y. Alice Chang, Z¨urich Lectures in Advanced Mathematics, European Mathematical Society, Z¨urich, 2004, viii+92 pp., US$28.00, ISBN 3-03719-006-X A conformal transformation is a diffeomorphism which preserves angles; the differential at each point is the composition of a rotation and a dilation. In its original sense, conformal geometry is the study of those geometric properties pre- served under transformations of this type. This subject is deeply intertwined with complex analysis for the simple reason that any holomorphic function f(z)ofone complex variable is conformal (away from points where f (z) = 0), and conversely, any conformal transformation from one neighbourhood of the plane to another is either holomorphic or antiholomorphic. This provides a rich supply of local conformal transformations in two dimensions. More globally, the (orientation pre- serving) conformal diffeomorphisms of the Riemann sphere S2 are its holomorphic automorphisms, and these in turn constitute the non-compact group of linear frac- tional transformations. By contrast, the group of conformal diffeomorphisms of any other compact Riemann surface is always compact (and finite when the genus is greater than 1). Implicit here is the notion that a Riemann surface is a smooth two-dimensional surface together with a conformal structure, i.e. a fixed way to measure angles on each tangent space. There is a nice finite dimensional structure on the set of all inequivalent conformal structures on a fixed compact surface; this is the starting point of Teichm¨uller theory. -
WHAT IS Q-CURVATURE? 1. Introduction Throughout His Distinguished Research Career, Tom Branson Was Fascinated by Con- Formal
WHAT IS Q-CURVATURE? S.-Y. ALICE CHANG, MICHAEL EASTWOOD, BENT ØRSTED, AND PAUL C. YANG In memory of Thomas P. Branson (1953–2006). Abstract. Branson’s Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties. 1. Introduction Throughout his distinguished research career, Tom Branson was fascinated by con- formal differential geometry and made several substantial contributions to this field. There is no doubt, however, that his favorite was the notion of Q-curvature. In this article we outline the construction and basic properties of Branson’s Q-curvature. As a Riemannian invariant, defined on even-dimensional manifolds, there is apparently nothing special about Q. On a surface Q is essentially the Gaussian curvature. In 4 dimensions there is a simple but unrevealing formula (4.1) for Q. In 6 dimensions an explicit formula is already quite difficult. What is truly remarkable, however, is how Q interacts with conformal, i.e. angle-preserving, transformations. We shall suppose that the reader is familiar with the basics of Riemannian differ- ential geometry but a few remarks on notation are in order. Sometimes, we shall write gab for a metric and ∇a for the corresponding connection. Let us write Rab and R for the Ricci and scalar curvatures, respectively. We shall use the metric to ‘raise and lower’ indices in the usual fashion and adopt the summation convention whereby one implicitly sums over repeated indices. Using these conventions, the Laplacian is ab a the differential operator ∆ ≡ g ∇a∇b = ∇ ∇a. A conformal structure on a smooth manifold is a metric defined only up to smoothly varying scale. -
The Wave Equation on Asymptotically Anti-De Sitter Spaces
THE WAVE EQUATION ON ASYMPTOTICALLY ANTI-DE SITTER SPACES ANDRAS´ VASY Abstract. In this paper we describe the behavior of solutions of the Klein- ◦ Gordon equation, (g + λ)u = f, on Lorentzian manifolds (X ,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian ana- logues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metricg ˆ on X, where X has a non-trivial boundary, in the sense that g = x−2gˆ, with x a boundary defining function. The boundary is confor- mally time-like for these spaces, unlike asymptotically de Sitter spaces studied in [38, 6], which are similar but with the boundary being conformally space- like. Here we show local well-posedness for the Klein-Gordon equation, and also global well-posedness under global assumptions on the (null)bicharacteristic flow, for λ below the Breitenlohner-Freedman bound, (n − 1)2/4. These have been known under additional assumptions, [8, 9, 18]. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at ∂X) of regular solutions. We also define the scattering operator, which in this case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is shown that below the Breitenlohner-Freedman bound, the Klein-Gordon equation behaves much like it would for the conformally related metric,g ˆ, with Dirichlet boundary conditions, for which propagation of singularities was shown by Melrose, Sj¨ostrand and Taylor [25, 26, 31, 28], though the precise form of the asymptotics is different. -
Math 865, Topics in Riemannian Geometry
Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor . -
Towards Physically Motivated Proofs of the Poincaré and Geometrization Conjectures
Towards physically motivated proofs of the Poincar´eand geometrization conjectures Arkady L.Kholodenko 375 H.L.Hunter Laboratories, Clemson University, Clemson, SC 29634-0973, USA. e-mail: [email protected] Although the Poincare′ and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow pre- viously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan (Ann.Physics 163 (1985), 318-419). In his work the renormalization group flow for a nonlinear sigma model in 2 + ε dimensions was obtained and studied. For ε =0, by ap- proximating the β function for such a flow by the lowest order terms in the sigma model coupling− constant, the equations for Ricci flow are obtained. In view of such an approximation, the existence of this type of flow in nature is questionable. In this work, we find totally independent justification for the exis- tence of Ricci flows in nature. This is achieved by developing a new formalism extending the results of two dimensional conformal field theories (CFT’s) to three and higher dimensions. Equations describing critical dynamics of these CFT’s are examples of the Yamabe and Ricci flows realizable in nature. Al- though in the original works by Perelman some physically motivated arguments can be found, their role in his proof remain rather obscure. In this paper, steps are made toward making these arguments more explicit thus creating an op- portunity for developing alternative, more physically motivated, proofs of the Poincare′ and geometrization conjectures. -
The Yamabe Problem
The Yamabe Problem Selim Tawfik Abstract The Yamabe problem asks if any Riemannian metric g on a compact smooth man- ifold M of dimension n ≥ 3 is conformal to a metric with constant scalar curvature. The problem can be seen as that of generalizing the uniformization theorem to higher dimensions, since in dimension 2 scalar and Gaussian curvature are, up to a factor of 2, equal. 1 Introduction In 1960, Yamabe claimed to have found a solution to the problem that would later come to bear his name. Eight years following, however, Neil Trudinger discovered a serious error in Yamabe’s proof. Trudinger was able to salvage some of Yamabe’s work but only by introducing further assumptions on the manifold M. In fact, Trudinger showed that there is a positive constant α(M) such that the result is true when λ(M) < α(M) (λ(M) is the Yamabe invariant, to be defined later). In particular, if λ(M) ≤ 0, the question is resolved. In 1976, Aubin improved on Trudinger’s work by showing that α(M) = λ(Sn), where the n-sphere is equipped with its standard metric. Moreover, Aubin showed that if M has dimension n ≥ 6 and is not locally conformally flat, then λ(M) < λ(Sn). The remaining cases proved to be more difficult and it was not until 1984 that they had been resolved by Richard Schoen, thereby completing the solution to the Yamabe problem. In order to gain an appreciation for these developments and the final solution, it is necessary to look at Yamabe’s approach. -
CONFORMAL SYMMETRY BREAKING OPERATORS for ANTI-DE SITTER SPACES 1. Introduction Let X Be a Manifold Endowed with a Pseudo-Rieman
CONFORMAL SYMMETRY BREAKING OPERATORS FOR ANTI-DE SITTER SPACES TOSHIYUKI KOBAYASHI, TOSHIHISA KUBO, AND MICHAEL PEVZNER Abstract. For a pseudo-Riemannian manifold X and a totally geodesic hypersur- face Y , we consider the problem of constructing and classifying all linear differential operators Ei(X) !Ej(Y ) between the spaces of differential forms that intertwine multiplier representations of the Lie algebra of conformal vector fields. Extending the recent results in the Riemannian setting by Kobayashi{Kubo{Pevzner [Lecture Notes in Math. 2170, (2016)], we construct such differential operators and give a classification of them in the pseudo-Riemannian setting where both X and Y are of constant sectional curvature, illustrated by the examples of anti-de Sitter spaces and hyperbolic spaces. Key words and phrases: conformal geometry, conformal group, symmetry breaking operator, branching law, holography, space form, pseudo-Riemannian geometry, hyperbolic manifold. 1. Introduction Let X be a manifold endowed with a pseudo-Riemannian metric g. A vector field Z on X is called conformal if there exists ρ(Z; ·) 2 C1(X)(conformal factor) such that LZ g = ρ(Z; ·)g; where LZ stands for the Lie derivative with respect to the vector field Z. We denote by conf(X) the Lie algebra of conformal vector fields on X. Let E i(X) be the space of (complex-valued) smooth i-forms on X. We define a family of multiplier representations of the Lie algebra conf(X) on E i(X) (0 ≤ i ≤ dim X) with parameter u 2 C by 1 (1.1) Π(i)(Z)α := L α + uρ(Z; ·)α for α 2 E i(X): u Z 2 i (i) i For simplicity, we write E (X)u for the representation Πu of conf(X) on E (X). -
Computational Conformal Geometry: Theories and Applications
ComputationalComputationalComputational ConformalConformalConformal Geometry:Geometry:Geometry: TheoriesTheoriesTheories andandand ApplicationsApplicationsApplications XianfengXianfeng GuGu,, Shing Shing-Tung-Tung Yau Yau UFL,UFL, HarvardHarvard UniversityUniversity AnAnAn importantimportantimportant goalgoalgoal ofofof classicalclassicalclassical geometrygeometrygeometry isisis tototo givegivegive aaa descriptiondescriptiondescription ofofof geometricgeometricgeometric figuresfiguresfigures thatthatthat wewewe seeseesee ininin nature.nature.nature. HighHighHigh powerpowerpower computercomputercomputer computationcomputationcomputation andandand threethreethree dimensiondimensiondimension cameracameracamera allowsallowsallows ususus tototo applyapplyapply classicalclassicalclassical andandand modernmodernmodern geometricgeometricgeometric technologytechnologytechnology tototo realrealreal featuresfeaturesfeatures ofofof objectsobjectsobjects thatthatthat wewewe seeseesee ininin daydayday tototo daydayday life.life.life. ForForFor aaa givengivengiven surfacesurfacesurface ininin threethreethree space,space,space, ititit isisis importantimportantimportant tototo realizerealizerealize thatthatthat theretherethere areareare severalseveralseveral geometricgeometricgeometric structuresstructuresstructures thatthatthat areareare inheritinheritinherit ininin theirtheirtheir description.description.description. TheTheThe mostmostmost importantimportantimportant oneoneone isisis thethethe conformalconformalconformal structure.structure.structure. -
The S1-Equivariant Yamabe Invariant of 3-Manifolds
THE S1-EQUIVARIANT YAMABE INVARIANT OF 3-MANIFOLDS BERND AMMANN, FARID MADANI, AND MIHAELA PILCA Abstract. We show that the S1-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe in- variant of the 3-sphere. More generally, we establish a topological upper bound for the S1-equivariant Yamabe invariant of any closed oriented 3-manifold en- dowed with an S1-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold. 1. Introduction Before we can describe the main results of the article in Section 1.2 we will give a summary on known results about the non-equivariant (= classical) Yamabe invariant in Section 1.1. 1.1. Overview of the classical Yamabe invariant. The Yamabe constant µ(M; [g]) of an n-dimensional conformal compact manifold (M; [g]) is the infimum of the re- striction to the conformal class [g] of the Einstein{Hilbert functional defined on the set of all Riemannian metrics as R M Scalh dvh h 7−! n−2 : vol(M; h) n Aubin [10] proved that the Yamabe constant of (M; [g]) is bounded above by the n Yamabe constant of the sphere, i.e. µ(M; [g]) ≤ µ(S ; [gst]). The Yamabe invariant σ(M) of a compact manifold M is defined as σ(M) := sup µ(M; [g]); [g]2C(M) where C(M) is the set of all conformal classes on M. It follows that σ(M) ≤ n n σ(S ) = µ(S ; [gst]). -
Space-Time Philosophy Reconstructed Via Massive Nordstr\" Om Scalar Gravities? Laws Vs. Geometry, Conventionality, and Underdetermination
Space-time Philosophy Reconstructed via Massive Nordstr¨om Scalar Gravities? Laws vs. Geometry, Conventionality, and Underdetermination October 15, 2018 J. Brian Pitts Faculty of Philosophy, University of Cambridge John Templeton Foundation grant #38761 [email protected] arXiv:1509.03303v1 [physics.hist-ph] 9 Sep 2015 1 Abstract What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a “graviton mass term” for gravity, which is algebraic in the potential. Unlike Nordstr¨om’s “massless” theory, massive scalar grav- ity is strictly special relativistic in the sense of being invariant under the Poincar´e group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein’s principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity(s): matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric (barely or on long distances) in the mass term. What is the ‘true’ geometry, one might wonder, in line with Poincar´e’s modal conventionality argument? Infinitely many theories exhibit this bimetric ‘geometry,’ all with the total stress-energy’s trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. -
The Relative Yamabe Invariant
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 10, Number 5, 935-969, 2002 The Relative Yamabe Invariant KAZUO AKUTAGAWA1 AND BORIS BOTVINNIK2 We define the relative Yamabe invariant of a compact smooth man- ifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the Yamabe invari- ant. We develop approximation techniques which lead to gluing theorems of two manifolds along their boundaries for the relative Yamabe invariant. We show that there are many examples of man- ifolds with both positive and non-positive relative Yamabe invari- ants. In particular, we construct families of four-manifolds with strictly negative relative Yamabe invariant and give an exact com- putation of the invariant. 1. Introduction. 1.1. General setting. Let W be a compact smooth manifold with bound- ary dW = M ^ 0, and n = dim W > 3. Let TZiem(W) be the space of all Riemannian metrics on W. For a metric g G 7^iem(V^) we denote by Hg the mean curvature along the boundary dW = M, and g = g\M- We also denote by [g] and [g] the corresponding conformal classes, and by C(M) and C{W) the space of conformal classes on M and W respectively. Let C and C be conformal classes of metrics on W and M respectively. We write dC = C if C\M_ = C. Let C(W, M) be the space of pairs ((7, C) with dC = C. De- note C0 = {g ^ C \ Hg — 0}. We call C0 C C the normalized conformal class. Let C0(W,M) be the space of pairs (C0,C) such that C0 C C and (<7, C) 6 C(W,M).