Scientific Reports
The preprint collection of the Department of Mathematics.
Full-text generally not available for preprints prior to may 2006.
Found 868 products
-
QDD203 - 04/08/2015
Catino, G.; Mastrolia, P.; Monticelli, D.D.; Rigoli, M.
On the geometry of gradient Einstein-type manifolds | Abstract | | In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold (M,g), unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen. |
-
QDD202 - 03/26/2015
Catino, G.; Mantegazza, C.; Mazzieri, L.
Locally Conformally Flat Ancient Ricci Flows | Abstract | | We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature. |
-
QDD201 - 03/26/2015
Catino, G.; Mazzieri, L.
Gradient Einstein solitons | Abstract | | In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon. We show that these structures are more rigid then standard Ricci solitons. In particular, we prove that there is only one complete three-dimensional, positively curved, Riemannian manifold satisfying $$ Ric -frac{1}{2} R g + nabla2 f = 0 $$, for some smooth function $f$. This solution is rotationally symmetric and asymptotically cylindrical and it represents the analogue of the Hamilton's cigar in dimension three. The key ingredient in the proof is the rectifiability of the potential function $f$. It turns out that this property holds also in the Lorentzian setting and for a more general class of structures which includes some gravitational theories. |
-
QDD200 - 03/24/2015
Barchielli, A.; Vacchini, B.
Quantum Langevin equations for optomechanical systems | Abstract | | We provide a fully quantum description of a mechanical oscillator in the presence of thermal environmental noise by means of a quantum Langevin formulation based on quantum stochastic calculus. The system dynamics is determined by symmetry requirements and equipartition at equilibrium, while the environment is described by quantum Bose fields in a suitable non-Fock representation which allows for the introduction of temperature. A generic spectral density of the environment can be described by introducing its state trough a suitable $P$-representation. Including interaction of the mechanical oscillator with a cavity mode via radiation pressure we obtain a description of a simple optomechanical system in which, besides the Langevin equations for the system, one has the exact input-output relations for the quantum noises. The whole theory is valid at arbitrarily low temperature. This allows the exact calculation of the stationary value of the mean energy of the mechanical oscillator, as well as both homodyne and heterodyne spectra. The present analysis allows in particular to study possible cooling scenarios and to obtain the exact connection between observed spectra and fluctuation spectra of the position of the mechanical oscillator. |
-
QDD199 - 03/13/2015
Catino, G.; Mantegazza, C.; Mazzieri, L.
A note on Codazzi tensors | Abstract | | We discuss a gap in Besse's book, recently pointed out by Merton, which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons. |
-
QDD198 - 03/13/2015
Catino, G.; Mazzieri, L.; Mongodi, S.
Rigidity of gradient Einstein shrinkers | Abstract | | In this paper we consider a perturbation of the Ricci solitons equation proposed in [Bourguignon] and studied in [Catino-Mazzieri] and we classify noncompact gradient shrinkers with bounded nonnegative sectional curvature. |
-
QDD196 - 02/26/2015
Cao, H.-D.; Catino, G.; Chen, Q.; Mantegazza, C.; Mazzieri, L.
Bach-flat gradient steady Ricci solitons | Abstract | | In this paper we prove that any n-dimensional (n>3) complete Bach-flat gradient steady Ricci solitons with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. |
-
QDD197 - 02/26/2015
Catino, G.
Critical metrics of the $L^2$-norm of the scalar curvature | Abstract | | In this paper we investigate complete critical metrics of the $L^2$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four. |
|
