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1. Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Author

Henri Guenancia, Mihai Păun, Junyan Cao, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), University of Illinois [Chicago] (UIC), University of Illinois System, Universität Bayreuth, Stony Brook University [SUNY] (SBU), State University of New York (SUNY), Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Pure mathematics, Mathematics::Complex Variables, Mathematics - Complex Variables, Applied Mathematics, General Mathematics, 010102 general mathematics, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], 01 natural sciences, Mathematics - Algebraic Geometry, symbols.namesake, Variation (linguistics), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 0103 physical sciences, symbols, Kodaira dimension, Mathematics::Differential Geometry, [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 010307 mathematical physics, 0101 mathematics, Einstein, Mathematics::Symplectic Geometry, Mathematics
Abstract

Given a K\"ahler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular K\"ahler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted K\"ahler-Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiber-wise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension)., Comment: v2: Title changed as the initial text is now split into two separate articles (authored by the same persons). The main results we obtain here are less general than their respective counterparts in the previous version, due to a gap in our arguments

Published
2021
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2. Small $C^1$ actions of semidirect products on compact manifolds

Author

Christian Bonatti, Thomas Koberda, Sang-hyun Kim, Michele Triestino, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Centre National de la Recherche Scientifique (CNRS), School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), University of Virginia [Charlottesville], and ANR-19-CE40-0007,Gromeov,Groupes d'homéomorphismes de variétés(2019)

Subjects
Pure mathematics, 37D30, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Cyclic group, Dynamical Systems (math.DS), Group Theory (math.GR), 01 natural sciences, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 57M60, $C^1$–close to the identity, Mathematics - Geometric Topology, Primary 37C85. Secondary 20E22, 57K32, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, Mapping torus, FOS: Mathematics, 57R35, 20E22, 0101 mathematics, Abelian group, Mathematics - Dynamical Systems, Mathematics, 37C85, 010102 general mathematics, Geometric Topology (math.GT), groups acting on manifolds, Riemannian manifold, Surface (topology), 57M50, fibered $3$–manifold, hyperbolic dynamics, Unit circle, Monodromy, 010307 mathematical physics, Geometry and Topology, Finitely generated group, Mathematics - Group Theory
Abstract

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $\pi_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds., Comment: 11 pages; final version to appear in Algebraic & Geometric Topology

Published
2019
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3. The unexpectedly large dust and gas content of quiescent galaxies at $z$ > 1.4

Author

M. S. Yun, Alvio Renzini, Georgios E. Magdis, Marie Martig, Grant W. Wilson, Itziar Aretxaga, Francesco Valentino, Veronica Strazzullo, Emanuele Daddi, R. Gobat, Ho Seong Hwang, Frédéric Bournaud, Alexis Finoguenov, Matthieu Béthermin, Mark Sargent, Shuowen Jin, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), Astrophysique Interprétation Modélisation (AIM (UMR_7158 / UMR_E_9005 / UM_112)), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris Diderot - Paris 7 (UPD7), National Observatory of Athens (NOA), Dark Cosmology Centre (DARK), Niels Bohr Institute [Copenhagen] (NBI), Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (KU)-University of Copenhagen = Københavns Universitet (KU)-Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (KU)-University of Copenhagen = Københavns Universitet (KU), Astronomy Centre, University of Sussex, Liverpool John Moores University (LJMU), Max-Planck-Institut für Astronomie (MPIA), Max-Planck-Gesellschaft, Laboratoire d'Astrophysique de Marseille (LAM), Aix Marseille Université (AMU)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS), INAF - Osservatorio Astronomico di Padova (OAPD), Istituto Nazionale di Astrofisica (INAF), University of Massachusetts [Amherst] (UMass Amherst), University of Massachusetts System (UMASS), Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Ludwig-Maximilians-Universität München (LMU), Gobat, R., Daddi, E., Magdis, G., Bournaud, F., Sargent, M., Martig, M., Jin, S., Finoguenov, A., Béthermin, M., Hwang, H. S., Renzini, A., Wilson, G. W., Aretxaga, I., Yun, M., Strazzullo, V., Valentino, F., Astrophysique Interprétation Modélisation (AIM (UMR7158 / UMR_E_9005 / UM_112)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH)-Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Aix Marseille Université (AMU)-Centre National d'Études Spatiales [Toulouse] (CNES), and Department of Physics

Subjects
COSMIC EVOLUTION, Active galactic nucleus, Stellar mass, [SDU.ASTR.CO]Sciences of the Universe [physics]/Astrophysics [astro-ph]/Cosmology and Extra-Galactic Astrophysics [astro-ph.CO], Milky Way, MU-M, POPULATION SYNTHESIS, RADIO PROPERTIES, Astrophysics, Astrophysics::Cosmology and Extragalactic Astrophysics, 01 natural sciences, GMASS ULTRADEEP SPECTROSCOPY, STAR-FORMATION, ELLIPTIC GALAXIES, 0103 physical sciences, Astrophysics::Solar and Stellar Astrophysics, 010303 astronomy & astrophysics, QC, Astrophysics::Galaxy Astrophysics, QB, Physics, Spiral galaxy, 010308 nuclear & particles physics, Star formation, MASSIVE GALAXIES, Astronomy and Astrophysics, ATLAS(3D) PROJECT, 115 Astronomy, Space science, Astrophysics - Astrophysics of Galaxies, MOLECULAR GAS, Galaxy, Redshift, Interstellar medium, Astrophysics::Earth and Planetary Astrophysics, [SDU.ASTR.GA]Sciences of the Universe [physics]/Astrophysics [astro-ph]/Galactic Astrophysics [astro-ph.GA]
Abstract

Early type galaxies (ETG) contain most of the stars present in the local Universe and, above a stellar mass of ~5e10 Msun, vastly outnumber spiral galaxies like the Milky Way. These massive spheroidal galaxies have, in the present day, very little gas or dust, and their stellar populations have been evolving passively for over 10 billion years. The physical mechanisms that led to the termination of star formation in these galaxies and depletion of their interstellar medium remain largely conjectural. In particular, there are currently no direct measurements of the amount of residual gas that might be still present in newly quiescent spheroids at high redshift. Here we show that quiescent ETGs at z~1.8, close to their epoch of quenching, contained 2-3 orders of magnitude more dust at fixed stellar mass than local ETGs. This implies the presence of substantial amounts of gas (5-10%), which was however consumed less efficiently than in more active galaxies, probably due to their spheroidal morphology, and consistently with our simulations. This lower star formation efficiency, and an extended hot gas halo possibly maintained by persistent feedback from an active galactic nucleus (AGN), combine to keep ETGs mostly passive throughout cosmic time., Comment: 16 pages, 4 main figures main, 7 supplementary figures, 2 supplementary tables; accepted for publication in Nature Astronomy

Published
2018
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4. Dressed diffusion and friction coefficients in inhomogeneous multicomponent self-gravitating systems

Author

Jean Heyvaerts, Christophe Pichon, Jean-Baptiste Fouvry, Pierre-Henri Chavanis, Observatoire astronomique de Strasbourg (ObAS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS), Institute for Advanced Study [Princeton] (IAS), Physique Statistique des Systèmes Complexes (LPT) (PhyStat), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), Institut d'Astrophysique de Paris (IAP), Centre National de la Recherche Scientifique (CNRS)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Université de Strasbourg (UNISTRA)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Physics, Statistical Mechanics (cond-mat.stat-mech), Secular evolution, FOS: Physical sciences, Astronomy and Astrophysics, 01 natural sciences, Astrophysics - Astrophysics of Galaxies, Gravitation, Formalism (philosophy of mathematics), Classical mechanics, [PHYS.ASTR.GA]Physics [physics]/Astrophysics [astro-ph]/Galactic Astrophysics [astro-ph.GA], Space and Planetary Science, Astrophysics of Galaxies (astro-ph.GA), 0103 physical sciences, Mass segregation, Dynamical friction, Statistical physics, 010306 general physics, 010303 astronomy & astrophysics, Condensed Matter - Statistical Mechanics, ComputingMilieux_MISCELLANEOUS
Abstract

General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstrates their equivalence. The corresponding stochastic Langevin equations are presented: they can be a practical alternative to numerically solving the inhomogeneous Fokker-Planck and Balescu-Lenard equations. The present formalism allows for a self-consistent description of the secular evolution of different populations covering a spectrum of masses, with a proper accounting of the induced secular mass segregation, which should be of interest to various astrophysical contexts, from galactic centers to protostellar discs., 27 pages, 1 figure

Published
2017
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5. Conic singularities metrics with prescribed Ricci curvature: General cone angles along normal crossing divisors

Author

Mihai Păun, Henri Guenancia, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), H.G. is partially supported by the French A.N.R project MACK., ANR-10-BLAN-0104,MACK,Equations de Monge-Ampère complexes et Géométrie kählerienne.(2010), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Divisor, Kähler manifold, 01 natural sciences, Kähler-Einstein metrics, klt pairs, conic singularities, 0103 physical sciences, [MATH]Mathematics [math], 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Ricci curvature, Mathematics, orbifold tensors, Algebra and Number Theory, 010102 general mathematics, 2000 Mathematics Subject Classification : 32Q05, 32Q10, 32Q15, 32Q20, 32U05, 32U15, Order (ring theory), [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Monge-Ampère equations, Cone (topology), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Conic section, Gravitational singularity, 010307 mathematical physics, Geometry and Topology, Laplace operator, Analysis
Abstract

Let $X$ be a non-singular compact Kähler manifold, endowed with an effective divisor $D=\sum{(1-\beta_k) Y_k}$ having simple normal crossing support, and satisfying $\beta_k \in (0, 1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X,D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work concerning the Monge-Ampère equations on $(X,D)$ by establishing Laplacian and $\mathscr{C}^{2,\alpha,\beta}$ estimates for the solution of these equations regardless of the size of the coefficients $0 \lt \beta_k \lt 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kähler–Einstein metrics with conic singularities along a normal crossing divisor.

Published
2016
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6. Convex real projective structures and Hilbert metrics

Author

Inkang Kim, Athanase Papadopoulos, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), A. Papadopoulos and M. Troyanov, and ANR FINSLER,ANR FINSLER

Subjects
51M10, 57S25, 010102 general mathematics, 01 natural sciences, Busemann cocycle, hyperbolic structure, Convex real projective structure, deformation space, geodesic flow, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], volume entropy, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], 0103 physical sciences, topological entropy, geodesic current, Patterson-Sullivan measure, 010307 mathematical physics, 0101 mathematics, [MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]
Abstract

International audience; We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng-Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel-Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.

Published
2014
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7. Bergman and Calderón projectors for Dirac operators

Author

Colin Guillarmou, Jinsung Park, Sergiu Moroianu, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), and Guillarmou, Colin

Subjects
Spinor, 010102 general mathematics, Holomorphic function, Order (ring theory), Boundary (topology), Riemannian manifold, Dirac operator, 01 natural sciences, 58J32, 35P25, symbols.namesake, Differential geometry, [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], Quantum mechanics, 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, Geometry and Topology, 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematical physics, Spin-½, Mathematics
Abstract

For a Dirac operator \(D_{\bar{g}}\) over a spin compact Riemannian manifold with boundary \((\bar{X},\bar{g})\), we give a new construction of the Calderon projector on \(\partial\bar{X}\) and of the associated Bergman projector on the space of L2 harmonic spinors on \(\bar{X}\), and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of \(D_{\bar{g}}\) and the scattering theory for the Dirac operator associated with the complete conformal metric \(g=\bar{g}/\rho^{2}\) where ρ is a smooth function on \(\bar{X}\) which equals the distance to the boundary near \(\partial\bar{X}\). We show that \(\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))\) is the orthogonal Calderon projector, where \(\tilde{S}(\lambda)\) is the holomorphic family in {ℜ(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.

Published
2014

8. Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

Author

Sergiu Moroianu, Jinsung Park, Colin Guillarmou, Guillarmou, Colin, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, School of Mathematics (KIAS Séoul), Korea Institute for Advanced Study (KIAS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Mathematics - Differential Geometry, Eta invariants, Mathematics(all), Dirac operator, General Mathematics, 01 natural sciences, Relatively hyperbolic group, Mathematics - Spectral Theory, symbols.namesake, Eta invariant, 0103 physical sciences, [MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP], FOS: Mathematics, 0101 mathematics, Spectral Theory (math.SP), Meromorphic function, Mathematics, Mathematical physics, 010102 general mathematics, Mathematical analysis, Hyperbolic function, Hyperbolic manifold, Mathematics::Geometric Topology, Selberg zeta function, Signature operator, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], symbols, 58J52, 37C30, 11M36,11F72, 010307 mathematical physics, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\Gamma,\Sigma}^{\rm o}(\lambda)$ associated to the spinor bundle $\Sigma$ on odd dimensional convex co-compact hyperbolic manifolds $X:=\Gamma\backslash\hh^{2n+1}$. We define a natural eta invariant $\eta(D)$ associated to the Dirac operator $D$ on $X$ and prove that $\eta(D)=\frac{1}{\pi i}\log Z_{\Gamma,\Sigma}^{\rm o}(0)$, thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the Schottky space of 3-dimensional Schottky hyperbolic manifolds and relate it to Zograf factorization formula., Comment: 36 pages

Published
2009
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