Volume 23, Issue 1

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant ε goes to zero. If the drift of the diffusion is given by a Morse–Smale vector field b , the limits of the eigenfunctions concentrate on the recurrent set of b . A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set. We consider generalized Morse–Smale vector fields, the recurrent set of which is composed of hyperbolic critical points, limit cycles and two dimensional torii. Under some compatibility conditions between the flow of b and the Riemannian metric g along each of these components, we prove that the support of a limit lies on those recurrent components of maximal dimension, where the topological pressure is achieved. Also, the restriction of the limit measure to either a cycle or a torus is absolutely continuous with respect to the unique invariant probability measure of the restriction of b to the cycle or the torus. When the torii are not charged, the restriction of the limit measure is absolutely continuous with respect to the arclength on the cycle and we have determined the corresponding density. Finally, the support of the limit measures and the support of the measures selected by the variational formulation of the topological pressure (TP) are identical.

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines.

We show that for soluble groups of type FP n , centralisers of finite subgroups need not be of type FP n .

This paper considers the problem of diameter maximization of a convex body in the Euclidean space and its connection with Valentine's notion of visibility. The so-called diametral spectrum, or set of critical lengths, is studied in depth from a topological point of view. As a side result, we give a new characterization of planar convex bodies of constant width. Our methodology emphasizes the role played by the concepts of critical pair and critical curve.

We study the mass equidistribution of holomorphic cuspidal Hecke eigenforms on the Siegel modular varieties, and show that equidistribution holds on average, by means of the Bergman kernel.

In this paper, we consider the parabolic Schrödinger differential operators ∂ t – Δ + V ( x ), where the nonnegative potential V belongs to the reverse Hölder class B q for q ≥ n /2. The L p estimates for these operators (∂ t – Δ + V ) iγ ( γ ∈ ℝ) and ∇(∂ t – Δ + V ) –1/2 are obtained.

In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schrödinger equation. As applications, we prove the Strauss' conjecture with a kind of mild rough data for 2 ≤ n ≤ 4, and a result of global well-posedness with small data for some nonlinear Schrödinger equation with L 2 -subcritical nonlinearity.

In this paper, we classify all smooth threefolds , for which the Grassmann secant variety G 1,2 ( X ) ⊂ 𝔾(1, N ) (i.e. the closure of the set of lines contained in the span of 3 independent points of X ) has not the expected dimension.