Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

Positively Curved Manifolds with Almost Maximal Symmetry Rank

The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n–1)/2] (n 6, 7).     

Extremal Almost-Kähler Metrics and Seiberg–Witten Theory

We show that there exist compact non-Kähler almost-Kähler4-manifolds whose metrics minimize L ²-norm of(2/3) s + 2w among all metrics compatible with a fixeddecomposition H ²(M, = H ⁺ H ^–, where s is the scalar curvature and w is the lowest eigenvalue of self-dual Weyl curvature at each point. In particular, the moduli space of such metrics modulo diffeomorphisms is infinite dimensional. This example also shows that LeBrun's estimate of L ²-norm of (1 – )s + · 6won a compact oriented Riemannian4-manifold with a nontrivial Seiberg–Witten invariant cannot beextended over = 1/3.     

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature

We give a sufficient condition for a higher dimensional Kleinian group to be convex cocompact in terms of the critical exponent of . As a consequence, we see that the fundamental group of a compact conformally flat manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally flat manifolds with positive scalar curvature.

     

Letter: A Cylindrically Symmetric Solution in Einstein-Maxwell-Dilaton Gravity

We consider the existence of Einstein-Maxwell-dilaton plus fluid systems for the case of stationary cylindrically symmetric spacetimes. An exact inhomogeneous -order solution is found, where the parameter parametrizes the non-minimally coupled electromagnetic field. Some its physical attributes are investigated and a connection with the already known Gödel-type solution is given. It is shown that our solution also survives in the string-inspired charged gravity framework. We find that a magnetic field has positive influence on the chronology violation unlike the dilaton influence.     

The Renormalization Group and Its Finite Lattice Approximations

We investigate finite lattice approximations to the Wilson renormalization group in models of unconstrained spins. We discuss first the properties of the renormalization group transformation (RGT) that control the accuracy of this type of approximation and explain different methods and techniques to practically identify them. We also discuss how to determine the anomalous dimension of the field. We apply our considerations to a linear sigma model in two dimensions in the domain of attraction of the Ising fixed point using a Bell–Wilson RGT. We are able to identify optimal RGTs which allow accurate computations of quantities such as critical exponents, fixed-point couplings and eigenvectors with modest statistics. We finally discuss the advantages and limitations of this type of approach.     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then