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Sergiu Moroianu

Publications and source records attributed to Sergiu Moroianu.

At least 37 records · Page 2Linked to original sources
arXiv2011

On the Singularities of the Zeta and Eta functions of an Elliptic Operator

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.

math.DG↗
arXiv2011

Uniformization of $S^2$ and flat singular surfaces

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization theorem for the sphere.

math.DG↗
arXiv2010

Bergman and Calderón projectors for Dirac operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of 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 to the complete conformal metric $g=\bar{g}/ρ^2$ where $ρ$ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$. We show that $({\rm Id}+\tilde{S}(0))/2$ is the orthogonal Calderón projector, where $\tilde{S}(λ)$ is the holomorphic family in $\{\Re(λ)\geq 0\}$ of normalized scattering operators constructed in our previous work, which are classical pseudo-differential of order $2λ$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any spin manifold.

math.DG↗
arXiv2009

Quasi-Fuchsian manifolds with particles

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 $π$: 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.

math.DG↗
arXiv2009

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

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X$ and prove that $η(D)=\frac{1}{πi}\log Z_{Γ,Σ}^{\rm o}(0)$, 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.

math.SP↗
arXiv2009

Regularity of the eta function on manifolds with cusps

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.

math.DG↗
arXiv2008

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

We describe the spectrum of the $k$-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the $k$ and $k-1$ 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 $V$ such that the Schrödinger operator has compact resolvent, although $V$ tends to $-\infty$ in most of the infinity. 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 four whose cusps are rational homology spheres.

math.SP↗
arXiv2007

The Dirac spectrum on manifolds with gradient conformal vector fields

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.

math.DG↗
arXiv2007

Spectral analysis of magnetic Laplacians on conformally cusp manifolds

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.

math.SP↗
arXiv2006

On Carvalho's $K$-theoretic formulation of the cobordism invariance of the index

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 noncommutative residues inside the calculus of cusp pseudodifferential operators of Melrose.

math.AP↗
arXiv2006

Fibered cusp versus $d$- index theory

We prove that the indices of fibered-cusp and $d$-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics.

math.DG↗
arXiv2005

$L^2$-index of the Dirac operator of generalized Euclidean Taub-NUT metrics

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.

math-ph↗
arXiv2005

On the structure of quantum permutation groups

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.

math.QA↗
arXiv2005

The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds

We consider open manifolds which are interiors of a compact manifold with boundary, and Riemannian metrics asymptotic to a conformally cylindrical metric near the boundary. We show that the essential spectrum of the Laplace operator on functions vanishes under the presence of a magnetic field which does not define an integral relative cohomology class. It follows that the essential spectrum is not stable by perturbation even by a compactly supported magnetic field. We also treat magnetic operators perturbed with electric fields. In the same context we describe the essential spectrum of the $k$-form Laplacian. This is shown to vanish precisely when the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. In all the cases when we have pure-point spectrum we give Weyl-type asymptotics for the eigenvalue-counting function. In the other cases we describe the essential spectrum.

math.DG↗
arXiv2004

Gravitational and axial anomalies for generalized Euclidean Taub-NUT metrics

The gravitational anomalies are investigated for generalized Euclidean Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz vector of the Kepler-type problem. In order to evaluate the axial anomalies, the index of the Dirac operator for these metrics with the APS boundary condition is computed. The role of the Killing-Yano tensors is discussed for these two types of quantum anomalies.

math-ph↗
arXiv2004

Cusp geometry and the cobordism invariance of the index

The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies that the sum of the indices of the induced operators on the hyperfaces is null.

math.DG↗
arXiv2004

Adiabatic limits of eta and zeta functions of elliptic operators

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 $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ 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 $η(δ_t,s)$ and $tζ(δ_t,s)$ 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.

math.DG↗