43 results on '"Sergiu Moroianu"'
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2. Cauchy spinors on $3$-manifolds
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Brice Flamencourt and Sergiu Moroianu
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Mathematics - Differential Geometry ,53C27 ,Differential Geometry (math.DG) ,FOS: Mathematics ,Geometry and Topology - Abstract
Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a non-elliptic, non-linear equation in terms of a symmetric $2$-tensor on $M$. When $M$ is compact and has positive scalar curvature, the linearized equation has finite dimensional kernel. Four families of solutions are known on the round $3$-sphere $\mathbb{S}^3$. We study the linearized equation in the vicinity of these solutions and we construct as a byproduct an incomplete hyperk\"ahler metric on $\mathbb{S}^3\times \mathbb{R}$ closely related to the Euclidean Taub-NUT metric on $\mathbb{R}^4$. On $\mathbb{S}^3$ there do not exist other solutions which either are constant in a left (or right) invariant frame, have three distinct constant eigenvalues, or are invariant in the direction of a left (or right)-invariant eigenvector. We deduce from this last result an extension of Liebmann's sphere rigidity theorem., Comment: 26 pages, references updates, minor changes
- Published
- 2021
3. Renormalized volume on the Teichmueller space of punctured surfaces
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Sergiu Moroianu, Frédéric Rochon, and Colin Guillarmou
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Teichmüller space ,Mathematics (miscellaneous) ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Interpolation space ,010307 mathematical physics ,0101 mathematics ,Lp space ,01 natural sciences ,Theoretical Computer Science ,Volume (compression) ,Mathematics - Published
- 2017
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4. THE RENORMALIZED VOLUME AND UNIFORMIZATION OF CONFORMAL STRUCTURES
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Jean-Marc Schlenker, Sergiu Moroianu, and Colin Guillarmou
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General Mathematics ,010102 general mathematics ,Conformal map ,Cotangent space ,Submanifold ,01 natural sciences ,Combinatorics ,Maxima and minima ,symbols.namesake ,Differential geometry ,Global analysis ,Reciprocity (electromagnetism) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Einstein ,Mathematics - Abstract
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\unicode[STIX]{x2202}M$ has dimension $n$ even. Its definition depends on the choice of metric $h_{0}$ on $\unicode[STIX]{x2202}M$ in the conformal class at infinity determined by $g$, we denote it by $\text{Vol}_{R}(M,g;h_{0})$. We show that $\text{Vol}_{R}(M,g;\cdot )$ is a functional admitting a ‘Polyakov type’ formula in the conformal class $[h_{0}]$ and we describe the critical points as solutions of some non-linear equation $v_{n}(h_{0})=\text{constant}$, satisfied in particular by Einstein metrics. When $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while if $n=4$ this amounts to solving the $\unicode[STIX]{x1D70E}_{2}$-Yamabe problem. Next, we consider the variation of $\text{Vol}_{R}(M,\cdot ;\cdot )$ along a curve of AHE metrics $g^{t}$ with boundary metric $h_{0}^{t}$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_{n}(h)=\text{constant}$ and $\text{Vol}(\unicode[STIX]{x2202}M,h)=1$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to the identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space ${\mathcal{T}}(\unicode[STIX]{x2202}M)$ of conformal structures on $\unicode[STIX]{x2202}M$. We obtain, as a consequence, a higher-dimensional version of McMullen’s quasi-Fuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.
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- 2016
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5. THE RENORMALIZED VOLUME AND UNIFORMISATION OF CONFORMAL STRUCTURES
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Colin Guillarmou, Sergiu Moroianu, Jean-Marc Schlenker, 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, 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), ANR-10-BLAN-0105,ACG,Aspects Conformes de la Géométrie(2010), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-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), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Paris (ENS Paris)
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Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,FOS: Mathematics ,Mathematics [G03] [Physical, chemical, mathematical & earth Sciences] ,FOS: Physical sciences ,Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre] ,Mathematical Physics (math-ph) ,Mathematical Physics - Abstract
We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the conformal class at infinity determined by $g$, we denote it by ${\rm Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional admitting a "Polyakov type" formula in the conformal class $[h_0]$ and we describe the critical points as solutions of some non-linear equation $v_n(h_0)={\rm const}$, satisfied in particular by Einstein metrics. In dimension $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while in dimension $n=4$ this amounts to solving the $\sigma_2$-Yamabe problem. Next, we consider the variation of ${\rm Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric $h_0^t$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\pl M}v_n(h){\rm dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to Identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space $\mc{T}(\pl M)$ of conformal structures on $\pl M$. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling., Comment: 58 pages, 2 figures
- Published
- 2018
6. Odd Pfaffian forms
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Sergiu Moroianu and Daniel Cibotaru
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Mathematics - Differential Geometry ,Riemann curvature tensor ,Pure mathematics ,Invariant polynomial ,General Mathematics ,Fibration ,Boundary (topology) ,Fibered knot ,Pfaffian ,Riemannian manifold ,Volume form ,symbols.namesake ,Differential Geometry (math.DG) ,symbols ,FOS: Mathematics ,Mathematics::Differential Geometry ,58A10, 53C05 (Primary), 57R18 (Secondary) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet formula for incomplete edge singularities in terms of the odd Pfaffian on the fibers of the boundary fibration. The formula holds for product-type model edge metrics where the degeneration is of conical type in each fiber, but also for general classes of perturbations of the model metrics. The same method produces a Chern- Gauss-Bonnet formula for complete, non-compact manifolds with fibered boundaries in the sense of Mazzeo-Melrose and perturbations thereof, involving the odd Pfaffian of the base of the fibration. We deduce the rationality of the usual Pfaffian form on Riemannian orbifolds, and exhibit obstructions for certain metrics on a fibration to be realized as the model at infinity of a flat metric with conical, edge or fibered boundary singularities., Comment: This second version corrects a statement about the degenerate metric on the blow-up of a submanifold, a few typos and includes new references
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- 2018
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7. Locally conformally K\'ahler manifolds with holomorphic Lee field
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Sergiu Moroianu, Liviu Ornea, and Andrei Moroianu
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Mathematics - Differential Geometry ,Pure mathematics ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,Manifold ,Computational Theory and Mathematics ,Norm (mathematics) ,0103 physical sciences ,Vector field ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman., Comment: 6 pages
- Published
- 2017
8. Singularities of the eta function of first order differential operators
- Author
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Paul Loya and Sergiu Moroianu
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Physics ,Pure mathematics ,symbols.namesake ,Spectral asymmetry ,symbols ,General Materials Science ,Gravitational singularity ,Operator theory ,Riemannian geometry ,Differential operator ,First order ,Fourier integral operator - Abstract
We report on a particular case of the paper [7], joint with Raphael Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n− 1, n− 3, n− 5, . . .. Version francaise abregee Soit D la classe des operateurs differentiels elliptiques symmetriques d’ordre 1 sur une variete Riemannienne fermee M , agissant sur les sections d’un fibre vectoriel Hermitien E. Il est connu que pour un operateur D ∈ D , le spectre de D en tant qu’operateur non-borne dans L(M,E) est discret. Les fonctions eta et zeta associees a D sont defines par les series
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- 2012
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9. The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds
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Sylvain Golénia and Sergiu Moroianu
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Pure mathematics ,Operator (computer programming) ,Conjecture ,Applied Mathematics ,General Mathematics ,Hodge theory ,Essential spectrum ,De Rham cohomology ,Homology (mathematics) ,Laplace operator ,Resolvent ,Mathematics - Abstract
We describe the spectrum of the k k -form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k k and k − 1 k-1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V V such that the Schrödinger operator has compact resolvent, although in most directions the potential V V tends to − ∞ -\infty . We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.
- Published
- 2011
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10. The Dirac Operator on Generalized Taub-NUT Spaces
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Andrei Moroianu and Sergiu Moroianu
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Mathematics - Differential Geometry ,Pure mathematics ,Conjecture ,Spinor ,FOS: Physical sciences ,58J50, 58J20 ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Base (topology) ,Dirac operator ,Manifold ,Euclidean distance ,General Relativity and Quantum Cosmology ,symbols.namesake ,Differential Geometry (math.DG) ,Line bundle ,Cone (topology) ,FOS: Mathematics ,symbols ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics - Abstract
We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author., Comment: Final version, 16 pages
- Published
- 2011
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11. Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ℝ)
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Sergiu Moroianu, Jinsung Park, and Paul Loya
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Mathematics - Differential Geometry ,Algebra and Number Theory ,Trace (linear algebra) ,Riemann surface ,Dirac operator ,Mathematics::Geometric Topology ,Mathematics - Spectral Theory ,symbols.namesake ,Eta invariant ,Differential Geometry (math.DG) ,Selberg trace formula ,FOS: Mathematics ,symbols ,Mathematics::Differential Geometry ,Limit (mathematics) ,58J28, 58J50, 11F72, 22E46 ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Quotient ,Mathematics ,Mathematical physics - Abstract
We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends., 25 pages
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- 2008
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12. The Dirac spectrum on manifolds with gradient conformal vector fields
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Andrei Moroianu and Sergiu Moroianu
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Mathematics - Differential Geometry ,Curl (mathematics) ,Gradient conformal vector fields ,Primary field ,Vector operator ,Dirac operator ,Mathematical analysis ,58J50, 58J20 ,Clifford analysis ,Dirac spectrum ,Continuous spectrum ,symbols.namesake ,Differential Geometry (math.DG) ,FOS: Mathematics ,symbols ,Hyperbolic manifolds ,Vector field ,Mathematics::Differential Geometry ,Analysis ,Mathematics ,Mathematical physics ,Vector potential - Abstract
We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing., Comment: 12 pages
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- 2007
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13. Weyl laws on open manifolds
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Sergiu Moroianu
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Mathematics - Differential Geometry ,Curvature of Riemannian manifolds ,58G50 ,General Mathematics ,Spectrum (functional analysis) ,Spectral geometry ,Clifford analysis ,Riemannian geometry ,Dirac operator ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Laplace–Beltrami operator ,Law ,Ricci-flat manifold ,FOS: Mathematics ,symbols ,Mathematics::Differential Geometry ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume., 24 pages
- Published
- 2007
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14. Boundaries of locally conformally flat manifolds in dimensions $4k$
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Sergiu Moroianu
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Differential Geometry (math.DG) ,58J28, 53A30 ,General Mathematics ,Dimension (graph theory) ,FOS: Mathematics ,Mathematics - Abstract
We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension $4k$ in terms of integrality of eta invariants., 10 pages
- Published
- 2015
15. On the structure of quantum permutation groups
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Sergiu Moroianu and Teodor Banica
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Quantum group ,Generator (category theory) ,Applied Mathematics ,General Mathematics ,Clifford algebra ,Structure (category theory) ,Permutation group ,Hopf algebra ,Combinatorics ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation (mathematics) ,Quantum ,Mathematics - Abstract
The quantum permutation group of the set $X_n=\{1,..., n\}$ corresponds to the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $\cc (S_n)$ for $n\leq 3$, and to be infinite dimensional for $n\geq 4$. In this paper we find an explicit representation of the algebra $A_{aut}(X_n)$, related to Clifford algebras. For $n=4$ the representation is faithful in the discrete quantum group sense., 9 pages
- Published
- 2006
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16. On Carvalho’s $K$-theoretic formulation of the cobordism invariance of the index
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Sergiu Moroianu
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Cusp (singularity) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Residue theorem ,Mathematical analysis ,Boundary (topology) ,Cobordism ,K-theory ,Manifold ,Elliptic operator ,Mathematics::K-Theory and Homology ,Mathematics ,Analytic proof - Abstract
We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a K-theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.
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- 2006
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17. Finiteness of theL2-index of the Dirac operator of generalized Euclidean Taub–NUT metrics
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Sergiu Moroianu and Mihai Visinescu
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Nut ,Index (economics) ,Essential spectrum ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dirac operator ,High Energy Physics::Theory ,General Relativity and Quantum Cosmology ,symbols.namesake ,Euclidean geometry ,Metric (mathematics) ,symbols ,Anomaly (physics) ,Mathematics::Symplectic Geometry ,Real line ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
We compute the axial anomaly for the Taub-NUT metric on $R^4$. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line.
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- 2006
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18. On the Lpindex of spin Dirac operators on conical manifolds
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André Legrand and Sergiu Moroianu
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General Mathematics ,Mathematical analysis ,Dirac (software) ,Clifford analysis ,Riemannian manifold ,Dirac operator ,symbols.namesake ,Eta invariant ,symbols ,Gravitational singularity ,Atiyah–Singer index theorem ,Mathematical physics ,Mathematics ,Spin-½ - Abstract
We compute the index of the Dirac operator on a spin Riemannian manifold with conical singularities, acting from Lp(_+) to Lq(_-) with p, q > 1. When 1+n/p-n/q > 0 we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at (n + 1)/2 - n/q instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the L2 index. For 1+n/p-n/q _ 0 the index formula contains an extra term related to the Calderon projector.
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- 2006
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19. Quantum anomalies for generalized Euclidean Taub–NUT metrics
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Sergiu Moroianu, Mihai Visinescu, and Ion I. Cotaescu
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Physics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dirac operator ,Domain (mathematical analysis) ,Gravitation ,High Energy Physics::Theory ,General Relativity and Quantum Cosmology ,symbols.namesake ,Tensor product ,Metric (mathematics) ,Euclidean geometry ,symbols ,Boundary value problem ,Anomaly (physics) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematical physics - Abstract
The generalized Taub–NUT metrics exhibit in general gravitational anomalies. This is in contrast with the fact that the original Taub–NUT metric does not exhibit gravitational anomalies, which is a consequence of the fact that it admits Killing–Yano tensors forming Stackel–Killing tensors as products. We have found that for axial anomalies, interpreted as the index of the Dirac operator, the presence of Killing–Yano tensors is irrelevant. In order to evaluate the axial anomalies, we compute the index of the Dirac operator with the APS boundary condition on balls and on annular domains. The result is an explicit number-theoretic quantity depending on the radii of the domain. This quantity is 0 for metrics close to the original Taub–NUT metric but it does not vanish in general.
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- 2005
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20. Adiabatic limits of eta and zeta functions of elliptic operators
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Sergiu Moroianu
- Subjects
Mathematics - Differential Geometry ,58J28, 58J52 ,General Mathematics ,Holonomy ,Differential operator ,law.invention ,Riemann zeta function ,Elliptic operator ,symbols.namesake ,Invertible matrix ,Differential Geometry (math.DG) ,law ,FOS: Mathematics ,symbols ,Adiabatic process ,Mathematics ,Meromorphic function ,Mathematical physics - Abstract
We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and $t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms., Comment: 32 pages, final version
- Published
- 2004
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21. HEAT KERNEL ASYMPTOTICS FOR ROOTS OF GENERALIZED LAPLACIANS
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Christian Bär and Sergiu Moroianu
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Pointwise ,Closed manifold ,Laplace transform ,General Mathematics ,Mathematical analysis ,Institut für Mathematik ,Heat kernel ,Mathematics - Abstract
We describe the heat kernel asymptotics for roots of a Laplace type operator Δ on a closed manifold. A previously known relation between the Wodzicki residue of Δ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.
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- 2003
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22. Homology of pseudodifferential operators on manifolds with fibered cusps
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Sergiu Moroianu and Robert Lauter
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Computer Science::Machine Learning ,Hochschild homology ,Applied Mathematics ,General Mathematics ,Fibered knot ,Homology (mathematics) ,Computer Science::Digital Libraries ,Cohomology ,Manifold ,Algebra ,Statistics::Machine Learning ,Elliptic operator ,Eta invariant ,Mathematics::K-Theory and Homology ,Spectral sequence ,Computer Science::Mathematical Software ,Mathematics - Abstract
The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the 0 0 -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
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- 2003
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23. K-Theory of Suspended Pseudo-Differential Operators
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Sergiu Moroianu
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Semi-elliptic operator ,Elliptic operator ,Parametrix ,General Mathematics ,Mathematical analysis ,Operator theory ,Operator norm ,Pseudo-differential operator ,Symbol of a differential operator ,Mathematics ,Quasinormal operator - Published
- 2003
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24. [Untitled]
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Sergiu Moroianu and Robert Lauter
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Sobolev space ,Differential geometry ,Mathematical analysis ,Geometry and Topology ,Codimension ,Differential operator ,Analysis ,Manifold ,Mathematics - Abstract
We recall a Fredholm criterion for fully elliptic cusp(pseudo)differential operators on a compact manifold with corners ofarbitrary codimension, acting on suitable Sobolev spaces. Then we give aformula for the index in terms of regularized `trace' functionalssimilar to the residue trace of Wodzicki and Guillemin.
- Published
- 2002
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25. Sur la limite adiabatique des fonctions êta et zêta
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Sergiu Moroianu
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General Medicine ,Humanities ,Mathematics - Abstract
Resume Dans cette Note, on demontre l'existence de la limite adiabatique de la fonction η ( s ) d'un operateur sur l'espace total d'une fibration au dessus de S 1 , construit a partir d'une famille d'operateurs differentiels inversibles d'ordre 1. Nous identifions cette limite a l'holonomie d'une famille meromorphe de connexions dans le fibre trivial. Dans le meme contexte, la fonction ζ diverge. On donne une formule pour les deux premiers coefficients du developpement asymptotique. Le premier resultat reste vrai pour une famille non-inversible si on se restreint a s =0. Dans le cas d'une famille d'operateurs de Dirac, on retrouve la formule d'holonomie de Bismut–Freed. Pour citer cet article : S. Moroianu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 131–134
- Published
- 2002
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26. Positivity of the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds
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Corina Ciobotaru and Sergiu Moroianu
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Hyperbolic manifold ,Relatively hyperbolic group ,Mathematics::Geometric Topology ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics ,Volume (compression) - Abstract
We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds., 10 pages, to appear in Proceedings AMS
- Published
- 2014
27. FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES
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Sergiu Moroianu and Robert Lauter
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Pure mathematics ,Exact sequence ,Applied Mathematics ,Mathematical analysis ,Fibration ,Fredholm integral equation ,Operator theory ,Fredholm theory ,Manifold ,Sobolev space ,symbols.namesake ,Mathematics::K-Theory and Homology ,Bounded function ,symbols ,Analysis ,Mathematics - Abstract
We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in ...
- Published
- 2001
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28. Chern–Simons line bundle on Teichmüller space
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Sergiu Moroianu, Colin Guillarmou, 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, École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
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Teichmüller space ,Pure mathematics ,hyperbolic manifolds ,Mathematical analysis ,Boundary (topology) ,Hyperbolic manifold ,Orthonormal frame ,Mathematics::Geometric Topology ,Mapping class group ,Manifold ,Chern–Simons invariants ,High Energy Physics::Theory ,Line bundle ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,32G15 ,58J28 ,Mathematics::Differential Geometry ,Compact Riemann surface ,58J28, 32G15 ,Geometry and Topology ,Mathematics::Symplectic Geometry ,renormalized volume ,Mathematics - Abstract
36 pages. Minor modifications in the introduction.; International audience; Let $X$ be a non-compact geometrically finite hyperbolic $3$-manifold without cusps of rank $1$. The deformation space $\mc{H}$ of $X$ can be identified with the Teichmüller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a Kähler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.
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- 2014
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29. Bergman and Calderón projectors for Dirac operators
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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
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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.
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- 2014
30. On Pluricanonical Locally Conformally Kähler Manifolds
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Andrei Moroianu, Sergiu Moroianu, Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), 'Simion Stoilow' Institute of Mathematics (IMAR), and Romanian Academy of Sciences
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Mathematics - Differential Geometry ,Tangent bundle ,Pure mathematics ,Endomorphism ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Harmonic map ,01 natural sciences ,Manifold ,Covariant derivative ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Exterior derivative ,Hermitian manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,[MATH]Mathematics [math] ,0101 mathematics ,Complex manifold ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We give a short proof of the fact that compact pluricanonical locally conformally K\"ahler manifolds have parallel Lee form., Comment: 6 pages, to appear in IMRN
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- 2016
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31. Ricci surfaces
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Andrei Moroianu, Sergiu Moroianu, and Juppin, Carole
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Mathematics - Differential Geometry ,Mathematics (miscellaneous) ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG] ,Theoretical Computer Science - Abstract
A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point $x$ of a Ricci surface has a neighborhood which embeds isometrically in $\mathbb{R}^3$ as a minimal surface, provided $K(x), Comment: 27 pages; final version, to appear in Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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- 2012
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32. On the Singularities of the Zeta and Eta functions of an Elliptic Operator
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Raphael Ponge, Sergiu Moroianu, and Paul Loya
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Mathematics - Differential Geometry ,Pure mathematics ,Parametrix ,General Mathematics ,Mathematical analysis ,Operator theory ,Dirac operator ,Differential operator ,Semi-elliptic operator ,Elliptic operator ,symbols.namesake ,Operator (computer programming) ,Spectral asymmetry ,Differential Geometry (math.DG) ,symbols ,FOS: Mathematics ,Primary 58J50, 58J42, Secondary 58J40 ,Mathematics - Abstract
Let P be a selfadjoint elliptic operator of order m>0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form s=k/m, where k ranges over all non-zero integers less than or equal to n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions be singular at all the points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, selfadjoint first-order differential operators, and selfadjoint elliptic pseudodifferential operators. As a result, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a new proof of a well known result of Branson-Gilkey, which was obtained by means of Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of the third author's paper [Po1]. Corrections to that statement are given in Appendix B., Final version. To appear in International Journal of Mathematics. 20 pages
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- 2010
33. Regularity of the eta function on manifolds with cusps
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Jinsung Park, Paul Loya, and Sergiu Moroianu
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Mathematics - Differential Geometry ,Finite volume method ,General Mathematics ,58J28, 58J50 ,Vector bundle ,Conformal map ,Function (mathematics) ,Dirac operator ,Mathematics::Geometric Topology ,Manifold ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Simple (abstract algebra) ,symbols ,FOS: Mathematics ,Mathematics::Symplectic Geometry ,Spin-½ ,Mathematics ,Mathematical physics ,Analysis of PDEs (math.AP) - Abstract
On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire., Comment: 22 pages, 2 figures
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- 2009
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34. Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds
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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)
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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
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- 2009
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35. Quasi-Fuchsian manifolds with particles
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Jean-Marc Schlenker, Sergiu Moroianu, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, 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), 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)
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Mathematics - Differential Geometry ,Infinitesimal ,Conformal map ,Geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Rigidity (electromagnetism) ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,53C80 (53A30 57M50) ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematical analysis ,Regular polygon ,Infinitesimal deformation ,Geometric Topology (math.GT) ,Graph ,Differential Geometry (math.DG) ,Mathematics [G03] [Physical, chemical, mathematical & earth Sciences] ,Gravitational singularity ,010307 mathematical physics ,Geometry and Topology ,Mathématiques [G03] [Physique, chimie, mathématiques & sciences de la terre] ,Analysis - Abstract
We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $\pi$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure., Comment: Now 48 pages, no figure. v2: new title, various corrections, results extended to include graph singularities ("interacting particles"). v3: various corrections/improvements, in particular thanks to comments by an anonymous referee
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- 2009
36. Spectral analysis of magnetic Laplacians on conformally cusp manifolds
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Sergiu Moroianu and Sylvain Golénia
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Nuclear and High Energy Physics ,Singular perturbation ,Continuous spectrum ,Essential spectrum ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Manifold ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,Weyl law ,FOS: Mathematics ,Gauge theory ,Laplace operator ,Spectral Theory (math.SP) ,35P20, 46N50, 47A10, 47A40, 81Q10 ,Mathematical Physics ,Mathematics ,Vector potential ,Mathematical physics ,Analysis of PDEs (math.AP) - Abstract
We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When $B$ is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When $B$ is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect. In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limit absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential., 52 pages. Revised version: references added. To appear in Annales Henri Poincar\'e
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- 2007
37. Homology and residues of adiabatic pseudodifferential operators
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Sergiu Moroianu
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Discrete mathematics ,010308 nuclear & particles physics ,Pseudodifferential operators ,Mathematics::Operator Algebras ,General Mathematics ,Cellular homology ,010102 general mathematics ,Homology (mathematics) ,Mathematics::Spectral Theory ,01 natural sciences ,Mathematics::Algebraic Topology ,58J42 ,Mathematics::K-Theory and Homology ,0103 physical sciences ,58J28 ,0101 mathematics ,Adiabatic process ,Mathematics::Symplectic Geometry ,47G30 ,47L80 ,Mathematics ,Relative homology - Abstract
We compute the Hochschild homology groups of the adiabatic algebra Ψa(X), a deformation of the algebra of pseudodifferential operators Ψ(X) when X is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on Ψa(X) and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is S1, these functionals are related to the η function of Atiyah-Patodi-Singer.
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- 2004
38. Homology of pseudo-differential operators on manifolds with fibered boundaries
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Robert Lauter and Sergiu Moroianu
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Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Fibered knot ,Operator theory ,Homology (mathematics) ,Differential operator ,Mathematics - Published
- 2002
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39. An index formula on manifolds with fibered cusp ends
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Sergiu Moroianu and Robert Lauter
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Mathematics - Differential Geometry ,Cusp (singularity) ,Pure mathematics ,58J40, 58J20, 58J28 ,Boundary (topology) ,Fibered knot ,Cohomology ,Manifold ,Eta invariant ,Operator (computer programming) ,Differential Geometry (math.DG) ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Fiber bundle ,Geometry and Topology ,Mathematics - Abstract
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of the Atiyah-Singer form in the interior minus the adiabatic limit of the eta invariant of the restriction of the operator to the boundary., Comment: 22 pages. Prepublication du Laboratoire Emile Picard n.253. See also http://picard.ups-tlse.fr
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- 2002
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40. Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ?).
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Paul Loya, Sergiu Moroianu, and Jinsung Park
- Subjects
- *
ADIABATIC invariants , *DIRAC equation , *MATHEMATICAL analysis , *EIGENVALUES , *INVARIANTS (Mathematics) , *RIEMANN surfaces - Abstract
AbstractThe eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,?) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the S1-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer. [ABSTRACT FROM AUTHOR]
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- 2008
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41. On the structure of quantum permutation groups.
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Teodor Banica and Sergiu Moroianu
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- 2006
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42. Quantum anomalies for generalized Euclidean TaubNUT metrics.
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Ion I Cotaescu and Sergiu Moroianu and Mihai Visinescu
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- 2005
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43. Homology of pseudodifferential operators on manifolds with fibered cusps.
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Robert Lauter and Sergiu Moroianu
- Subjects
- *
PSEUDODIFFERENTIAL operators , *HOMOLOGY theory - Abstract
The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the $0$-dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case. [ABSTRACT FROM AUTHOR]
- Published
- 2003
- Full Text
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