Kahler manifolds and their relatives

Let M1 and M2 be two K¨ahler manifolds. We call M1 and M2 relatives if they share a non-trivial K¨ahler submanifold S, namely, if there exist two holomorphic and isometric immersions (K¨ahler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two K¨ahler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) K¨ahler manifolds S1 and S2 which admit two K¨ahler immersions into M1 and M2 respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain D ⊂ Cn with its Bergman metric and a projective K¨ahler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective K¨ahler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involve

Symplectic duality between complex domains

In this paper after extending the denition of symplectic duality (given in [3] for bounded symmetric domains ) to arbitrary complex domains of Cn centered at the origin we generalize some of the results proved in [3] and [4] to those domain

The bisymplectomorphism group of a bounded symmetric domain

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.Comment: 19 pages. Version 2: minor correction

Quasi-saddles as relevant points of the potential energy surface in the dynamics of supercooled liquids

The supercooled dynamics of a Lennard-Jones model liquid is numerically investigated studying relevant points of the potential energy surface, i.e. the minima of the square gradient of total potential energy $V$. The main findings are: ({\it i}) the number of negative curvatures $n$ of these sampled points appears to extrapolate to zero at the mode coupling critical temperature $T_c$; ({\it ii}) the temperature behavior of $n(T)$ has a close relationship with the temperature behavior of the diffusivity; ({\it iii}) the potential energy landscape shows an high regularity in the distances among the relevant points and in their energy location. Finally we discuss a model of the landscape, previously introduced by Madan and Keyes [J. Chem. Phys. {\bf 98}, 3342 (1993)], able to reproduce the previous findings.Comment: To be published in J. Chem. Phy

Surfaces in R4 with constant principal angles with respect to a plane

We study surfaces in R4 whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We classify all surfaces with one principal angle equal to $0$ and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in R4 with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector fiel

Global symplectic coordinates on gradient Kaehler-Ricci solitons

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,\omega)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional complete Kaehler manifolds with non-negative sectional curvature globally symplectomorphic to $R^{2n}$ through a symplectomorphism satisfying Ciriza's property.Comment: 8 page

Bugs for atopy: the Lactobacillus rhamnosus GG strategy for food allergy prevention and treatment in children

Food allergy (FA) is a major health issue for children living in Western countries. At this time the only proven treatment for FA is elimination of offender antigen from the diet. It is becoming clear that the development of gut microbiota exerts a profound influence on immune system maturation and tolerance acquisition. Increasing evidence suggests that perturbations in gut microbiota composition of infants are implicated in the pathogenesis of FA. These findings have unveiled new strategies to prevent and treat FA using probiotics bacteria or bacterial substance to limit T-helper (Th)/Th2 bias, which changes during the disease course. Selected probiotics administered during infancy may have a role in the prevention and treatment of FA. Lactobacillus rhamnosus GG (LGG) is the most studied probiotic in this field. Administration of LGG in early life have a role in FA prevention. Preliminary evidence shows that LGG accelerates oral tolerance acquisition in cow's milk allergic infants. We are understanding the mechanisms elicited by LGG and metabolites in influencing food allergen sensitization. A deeper definition of these mechanisms is opening the way to new immunotherapeutics for children affected by FA that can efficiently limit the disease burden

Cholesteatoma vs granulation tissue: a differential diagnosis by DWI-MRI apparent diffusion coefficient

To diagnose cholesteatoma when it is not visible through tympanic perforation, imaging techniques are necessary. Recently, the combination of computed tomography and magnetic resonance imaging has proven effective to diagnose middle ear cholesteatoma. In particular, diffusion weighted images have integrated the conventional imaging for the qualitative assessment of cholesteatoma. Accordingly, the aim of this study was to obtain a quantitative analysis of cholesteatoma calculating the apparent diffusion coefficient value. So, we investigated whether it could differentiate cholesteatoma from other inflammatory tissues both in a preoperative and in a postoperative study