Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth

In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Some Remarks on Quantum Limits on Zoll Manifolds

We study the concentration of eigenfunctions of the Laplace–Beltrami operator on manifolds all whose geodesics are closed (the so-called Zoll manifolds). Some results on the structure of the set of invariant semiclassical measures associated to sequences of eigenfunctions are given. Among these, we show that any probability measure on the unit tangent bundle of a compact rank-one symmetric space that is invariant by the geodesic flow may be realized as the semiclassical measure of a sequence of eigenfunctions of the Laplacian. This extends a previous result of Jakobson and Zelditch on spheres.     

THE GIBBS PHENOMENON,THE PINSKY PHENOMENON,AND VARIANTS FOR EIGENFUNCTION EXPANSIONS

ABSTRACT

We examine analogues of the Gibbs phenomenon for eigenfunction expansions of functions with singularities across a smooth surface, though of a more general nature than a simple jump. The Gibbs phenomena that arise still have a universal form, but a more general class of “fractional sine integrals” arises, and we study these functions. We also make a uniform analysis of eigenfunction expansions in the presence of the Pinsky phenomenon, and see an analogue of the Gibbs phenomenon there. These analyses are done on three classes of manifolds: strongly scattering manifolds, including Euclidean space; compact manifolds without strongly focusing geodesic flows, including flat tori and quotients of hyperbolic space, and compact manifolds with periodic geodesic flow; including spheres and Zoll surfaces. Finally, we uncover a new divergence phenomenon for eigenfunction expansions of characteristic functions of balls, for a perturbation of the Laplace operator on a sphere of dimension ≥5.     

Global solutions for nonlinear Klein-Gordon equations in infinite homogeneous waveguides

In this paper we prove a global existence result for nonlinear Klein-Gordon equations in infinite homogeneous waveguides, R×M, with smooth small data, where M=(M,g) is a Zoll manifold, or a compact revolution hypersurface. The method is based on normal forms, eigenfunction expansion and the special distribution of eigenvalues of the Laplace-Beltrami on such manifolds.     

The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds

The subject of this paper is the relationships among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms. Research at MSRI supported in part by NSF grant DMS-9022140. Research at MSRI and Texas Tech supported in part by NSF grant DMS-9409209.     

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.     

Harmonic transforms of complete Riemannian manifolds

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.     

On the mean curvatures sharp estimates of hypersurfaces

Sharp estimates for the mean curvatures of hypersurfaces in Riemannian manifolds are known from the works of Jorge-Xavier ^[3], Markvorsen ^[6] and Vlachos ^[11]. We first give a simplified proof of these estimates. This proof shows that a similar original result holds for hypersurfaces in Einstein manifolds which are warped product of by Ricci-flat manifolds.     

An abstract Nash–Moser theorem with parameters and applications to PDEs

We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.     

A note on the almost-one-half holomorphic pinching

Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem on Kähler manifolds with almost-one-half pinched holomophic sectional curvature. The proof is motivated by the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.     

Cobordism invariance of the index of Callias-type operators

We introduce a notion of cobordism of Callias-type operators overcomplete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application, we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two noncompact but topologically simpler manifolds. As another application, we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.     

De Rham theorem for Schwartz functions on Nash manifolds

In [2] the authors proved the de Rham theorem for Schwartz functions on affine Nash manifolds. Here we simplify the proof and generalize their result to the case of non-affine Nash manifolds.     

Forms and currents defining generalized <Emphasis Type="Italic">p</Emphasis>-Kähler structures

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant role than “closed” forms. In this setting, we state the appropriate characterization theorems and give some interesting applications.     

The partial quotients of Moment-angle manifolds over a polygon

In this paper, we generalize the conception of characteristic function in toric topology and construct many new smooth manifolds by using it. As an application, we classify the Moment-Angle manifolds and the partial quotients manifolds of them over a polygon. In the appendix we give a simple new proof for Orlik–Raymond's theorem in terms of characteristic function which gives the classification for quasitoric manifolds of dimension 4.     

Conjugate and conformally conjugate parallelisms on Finsler manifolds

In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijyō: the class of generalized Berwald manifolds is closed under conformal change.     

The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.     

De Rham decomposition of affinely connected manifolds

This article contains an extension of the de Rham decomposition theorem to affinely connected manifolds which may have torsion. Our proof is based on a geometric homotopy lemma, which allows a simple and comparatively short proof of this result by means of the Cartan-Ambrose-Hicks theorem.     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.