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Habilitation Thesis Contents Abstract (in Czech) 2 1 Introduction 4 1.1 A motivation: symmetries of the Laplacian on Rn .........4 1.2 From Euclidean space to conformal geometry . .4 1.3 From conformal geometry to AHS structures . .5 1.4 Invariant quantization on AHS structures . .6 1.5 Prolongation of first BGG operators . .7 1.6 Summary: the author's contribution and further directions . .8 2 Almost Hermitean symmetric structures 9 2.1 1 {graded Lie algebras and first order structures . .9 2.2 Canonicalj j Cartan connections and AHS{structures . 10 2.3 Natural bundles, fundamental derivative and tractor connection . 11 2.4 Conformal geometry in the world of AHS structures . 13 3 Invariant quantization 16 3.1 A generic construction of invariant quantization . 16 3.2 Invariant quantization on conformal densities . 18 4 Symmetries of conformal powers of the Laplacian 20 4.1 Symmetries of Pr in the locally flat case . 21 4.2 An inroad to the curved case . 23 5 Prolongation of first BGG operators 24 5.1 Invariant prolongation connections on AHS manifolds . 24 5.2 Prolongation connection in conformal geometry . 26 6 Reprints of articles 31 Reference [13]: Equivariant quantizations for AHS-structures ...... 32 Reference [34]: Conformally invariant quantization { Towards the com- plete classification ........................... 50 Reference [20]: Higher symmetries of the conformal powers of the Lapla- cian on conformally flat manifolds ................. 65 Reference [30]: Second order symmetries of the conformal Laplacian .. 92 Reference [23]: On a new normalization for tractor covariant derivatives 118 Reference [24]: Invariant prolongation of overdetermined PDEs in pro- jective, conformal, and Grassmannian geometry .......... 143 1 Abstrakt Habilitaˇcn´ıpr´aceje souborem ˇcl´ank˚u[13, 34, 20, 30, 23, 24] publikovan´ych v mezi- n´arodn´ıch ˇcasopisech, kter´ejsou vˇsechny evidov´any v datab´az´ıch WoS nebo SCO- PUS. Vˇetˇsinatˇechto ˇcl´ank˚um´aspoluautory, jimiˇzjsou A. Rod Gover, Andreas Cap,ˇ Fabian Radoux, Jean-Philippe Michel, Matthias Hammerl, Petr Somberg a Vladim´ır Souˇcek. Pod´ıl vˇsech autor˚una spoleˇcn´ych ˇcl´anc´ıch je rovnocenn´y. Reprinty ˇcl´ank˚ujsou v Sekci 6. Oblast v´yzkumu tˇechto matematik˚use prot´ın´av konformn´ıgeometrii, kter´aje nejzn´amˇejˇs´ıstrukturou ve tˇr´ıdˇetzv. AHS struktur. V´yznamkonformn´ıgeometrie spoˇc´ıv´amj. v jej´ımbl´ızk´emvztahu k matematick´efyzice a tak´ek anal´yze.Pr´avˇe na pomez´ı tˇechto oblast´ı matematiky patˇr´ı probl´em, kter´ymotivuje v´ysledky shrnut´ev t´etohabilitaci: studium symetri´ı Laplaceova oper´atoru∆. To jsou oper´atory Σ takov´e,ˇze∆ Σ = Σ0 ∆ pro nˇejak´yoper´atorΣ0. Technick´afor- mulace probl´em˚u,jejichˇzvyˇreˇsen´ıje◦ ◦ nutn´epro ´upln´ypopis symmetri´ıLaplaceova oper´atoru, je v Sekci 1.1. Symetrie Laplaceova oper´atorujsou zn´am´ed´ıky ˇcl´anku[15] publikovan´em v Annals of Mathematics. C´ılt´etohabilitace je ovˇsemmnohem obecnˇejˇs´ı:prezen- tovat geometrick´en´astroje a postupy pro studium symetri´ıinvariantn´ıch oper´ator˚u ve tˇr´ıdˇeAHS geometrick´ych struktur. Pˇresnˇejˇs´ıformulace vyˇzaduj´ıjist´ymatema- tick´yapar´ata ˇcten´aˇrje najde v Sekci 1. Hlavn´ıv´ysledkyse t´ykaj´ı invariant- n´ıhokvantov´an´ı v Sekci 3, symetri´ı konformn´ıchmocnin Laplaceova oper´atoru a diskuze ploch´aversus kˇriv´ageometrie v Sekci 4 a prodluˇzov´an´ı prvn´ıchBGG oper´ator˚u v Sekci 5. 2 Preface The thesis is a collection of articles [13, 34, 20, 30, 23, 24]. All of them have been published in international journals indexed by WoS or SCOPUS. Results and some background theory are summarized in the survey part (Sections 1 { 5) on pages 4 { 27. Reprints of articles are in Section 6. Our main research interest lies in the theory (bi)linear invariant operators on manifolds with a geometrical structure (known as AHS structure). This provides a suitable geometrical framework for the specific problem which motivates this survey: to understand symmetries of the Laplace operator [15]. Such symmet- ries play an important role in analysis (in the study of separable solutions of PDE's) and in mathematical physics (where the terminology `higher' or `hidden' symmetries is often used). I hope this survey will be useful for researches on the borderline of these fields. Pronouncement Almost all papers included in this thesis have co-authors, namely A. Cap,ˇ A.R. Gover, J.-P. Michel, F. Radoux, M. Hammerl, P. Somberg and V. Souˇcek. The contributions of all authors were equivalent since the results were based on com- mon discussions. Formally, the author's contribution to the paper [34] was 100%, the author's contribution the papers [13, 20] was 50%, the author's contribution the paper [30] was 33% and the authors contribution to the papers [23, 24] was 25%. However, the developement of articels [23, 24] was rather specific. Main results of these articels were obtained more or less independently by M. Hammerl on one side and by P. Somberg, V. Souˇcekand the author of the habilition on the other side. After we found out that we work on the same problem, we decided to publish our results together. From this point of view, the author's contribution to [23, 24] can be also considered as 33%. Acknowledgement I wish to thank all the co-authors for their friendly and always very helpful col- laboration. I would like to express my gratitude to my colleague Prof. Jan Slov´ak for our numerous interesting discussions. 3 1 Introduction 1.1 A motivation: symmetries of the Laplacian on Rn The Laplacian operator is of prominent interest in differential geometry, mathe- matical physics and analysis and has many analogues in other mathematical fields. We shall start with the simplest version, i.e. the Laplacian on smooth functions n i n n on the Euclidean space R . This is the operator ∆ = i : C1(R ) C1(R ) r r ! where i = @=@xi and we have used the Einstein's summation convention. There r n n is an obvious notion of symmetries of ∆ : C1(R ) C1(R ) given by diffe- n n ! rential operators Σ : C1(R ) C1(R ) such that ∆Σ = Σ∆. We shall term such operators commuting symmetries! of ∆. As a slight reformulation, one can also introduce commuting symmetries as operators preserving eigenspaces of ∆. Another (and weaker) possibility is to study operators Σ which preserve the null space of ∆ and this will be the property of our interest. We say Σ is a conformal symmetry or just a symmetry of ∆ if n n ∆Σ = Σ0∆ for some Σ0 : C1(R ) C1(R ): (1) ! Note this also means that Σ0 preserves the range of ∆. A short computation reveals that operators Σ and Σ0 have the same symbol. The vector space of symmetries forms obviously an algebra which we denote by HS. Operators of the form Σ = T ∆ are always symmetries which we shall term trivial symmetries. Observe the space of trivial symmetries is a left ideal which we denote by (∆) HS. The main problem is to describe the quotient HS := HS=(∆). ⊆ What do we need to understand fully symmetries of ∆? First, we need to classify symmetries Σ which means to find out which tensor fields can appear as symbols of symmetries in the first place. Another question is to construct a preferred symmetry with a prescribed symbol. Then one should understand the algebra HS. All these problems are solved in the essential Eastwood's paper [15]. This result also shows the significance of conformal geometry. Although defined using the Euclidean metric, the Laplacian is in a suitable sense confor- mally invariant, cf. Section 1.2. (Note this follows already from knowledge of first order symmetries.) Thus the basic step in the study of symmetries should be to understand invariance of ∆. These questions motivate the main aim of this thesis: we explore to which extent and directions one can develop a general theory of symmetries of (suitably invariant) differential operators. 1.2 From Euclidean space to conformal geometry Conformal structure is the pair (M; [g]) where M is a smooth manifold of the 4 dimension n, g a (pseudo)Riemannian metric and the class [g] contains metrics obtained by rescaling g by a positive smooth function. This determines the class of ab corresponding Levi-Civita connections [ ]. Although the Laplacian ∆ = g a b is not conformally invariant, there is ar curvature modification known as ther con-r n 2 formal Laplacian (or Yamabe operator) ∆ := ∆ − Sc : [ n=2 + 1] Y − 4(n 1) E − ! [ n=2 1] where Sc is the scalar curvature of g and [−w] are density bundles, E − − E cf. Section 2.4 for details concerning our convention for conformal weights w R. 2 The conformal invariance means ∆Y = ∆Y for the operator ∆Y defined with respect to another choiceg ^ [g] with corresponding [ ]. 2 r 2 r Note the problem of symmetries of ∆Y bis much simpler in theb Euclidean case (where ∆Y = ∆ because Sc = 0, cf. Section 1.1)b than for general conformal structures (M; [g]). In fact, the full understanding of symmetries of ∆Y is available only on locally flat conformal structures (M; [g]), i.e. when g is locally isomorphic n to the Euclidean metric on R . Generally, first order symmetries of ∆Y come from infinitesimal symmetries of the structure (i.e. conformal Killing fields) but the classification of higher order symmetries is a highly nontrivial problem. Only the case of second order symmetries is solved and we shall discuss this result in Section 4.2. Above we started with the Laplacian ∆ and observed its conformal invariance. Taking a slightly different point of view, we can start with a given conformal struc- ture (M; [g]), choose an arbitrary invariant differential operator Φ and consider symmetries of Σ analogously as in (1).
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