We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We ...

In this paper we show that a “locally Lipschitz” locally compact transformation group acting continuously and effectively on a connected paracompact locally Euclidean topological manifold is a Lie group. This is a contribution to the proof of the Hilbert-Smith conjecture. It generalizes the classical Bochner-Montgomery-Kuranishi Theorem[1, 9] and also the Repovš-Ščepin Theorem [17] which holds ...

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications...

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M, \omega, L, h)$ are sequences measures $\{\mu_k^z\}_{k=1}^{\infty}$ parametrized by points $z \in M$. For each $z$ in open orbit, we prove central limit theorem for $\mu_k^z$. The center mass $\mu_k^z$ is image under moment map; after re-centering at $0$ and dilating $\sqrt{k}$, re-normalized measure tends centered Gaus...

The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. results include regularity assumptions on perturbation and involve excluding also constructs response a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing coefficients, alternative for nonlinear equations) exte...

In the paper [SY80] it was shown that a Kähler manifold with strictly positive bisectional curvature was biholomorphic to CP. In this paper, we use the techniques developed by [SY80], to prove that a compact Kähler manifold with positive orthogonal bisectional curvature is biholomorphic to CP, a condition strictly weaker than positive bisectional curvature. This gives a direct elliptic proof of...

We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...

A famous theorem due to Nash ([3]) assures that every Riemannian manifold can be embedded isometrically into some Euclidean space E. An interesting question is whether for a complete manifold M we can find a closed isometric embedding. This note gives the affirmative answer to this question asked to the author by Paolo Piccione. In his famous 1956 article John Nash proved that every Riemannian ...

According to a result of one of the authors [7] there are at most a countable number of inequivalent differentiable actions of a compact Lie group on a compact differentiable manifold. The results above show that both the compactness of the manifold and the differentiability of the action are necessary assumptions. In the course of our proof we also prove the following theorem, which is an elem...