Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

An Unsolved Problem of Fenchel

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in R^n>+1 aregiven by C³ n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRⁿ⁺¹.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

Holomorphic Maps of Compact Generalized Hopf Manifolds

We investigate holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kähler manifolds with parallel Lee form). We show that they preserve the canonical foliations. Moreover, we study compact complex submanifolds of g.H. manifolds and holomorphic submersions from compact g.H. manifolds.     

Multiple attractors in a Leslie–Gower competition system with Allee effects

We investigate asymptotic dynamics of the classical Leslie–Gower competition model when both competing populations are subject to Allee effects. The system may possess four interior steady states. It is proved that for certain parameter regimes both competing populations may either go extinct, coexist or one population drives the other population to extinction depending on initial conditions.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.     

Isolated complex singularities and their CR links

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

In a given Kähler manifold (M,J) we introduce the notion of Kähler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kähler immersions of M into an ambient Kähler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .