Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds

It is known that ifH _m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH _m andg a left invariant metric, then (Γ/H _{m, g}) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H _{m, g α}) withg=g ₁ such that for every α≠1, (Γ/H _{m, g α})does admit a nontrivial isospectral deformation. For α≠1 the metricsg _α will not beH _m-left invariant, and the (Γ/H _m, gx_α) will not be nilmanifolds, but still solvmanifolds.     

Equivariant deformations of meromorphic actions on compact complex manifolds

Meromorphicity is the most basic property for holomorphic -actions on compact complex manifolds. We prove that the meromorphicity of -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space. Received January 25, 2000 / Published online October 30, 2000     

Cohomological constraint on deformations of compact Kähler manifolds

We prove that for every compact Kähler manifold X the cup product

     

The J-flow on toric manifolds

We study the long time behavior of J-flows on toric manifolds. By introducing the transition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.     

Pseudotoric structures on toric symplectic manifolds

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds.     

Algebraic deformations of compact Kähler surfaces II

In the first version of this paper, a short proof was given of Kodaira’s result that every compact Kähler surface is a deformation of an algebraic surface under the extra assumption that the infinitesimal deformations of the surface were unobstructed. In this paper, the extra assumption is removed.     

The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth

In this article, we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth. We prove that on each energy level E(x, v) = k with k c(L), where c(L) is Mane's critical value, the EulerLagrange flow has positive topological entropy. This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.     

Symmetries on compact pseudohermitian manifolds

Si caratterizzano le varietà pseudohermitiane compatte dotate di simmetrie rispetto al campo di vettori definito dalla struttura pseudoherminitana come ipersuperfici in fibrati lineari olomorfi su varietà simmetriche hermitiane.     

Relative K-stability and modified K-energy on toric manifolds

In this paper, we give a sufficient condition for both the relative K-stability and the properness of modified K-energy associated to Calabi's extremal metrics on toric manifolds. In addition, several examples of toric manifolds which satisfy the sufficient condition are presented.     

The Floer homotopy type of height functions on complex Grassmann manifolds

A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author's preprint The Floer homotopy type of complex Grassmann manifolds.

     

Continuous wavelets on compact manifolds

Let M be a smooth compact oriented Riemannian manifold, and let Δ _M be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb {R}^+)}\) , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ _M ). We show that K _t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t ²Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K _t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S ², and f (s)  =  se ^?s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K _t , one to be used when t is large, and one to be used when t is small.     

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.     

Conjugate variables in analytic number theory. Phase space and Lagrangian manifolds

For an arithmetic semigroup (G, ?), we define entropy as a function on a naturally defined continuous semigroup ? containing G. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup (G, ?).     

Minimal Morse flows on compact manifolds

In this paper we prove, using the Poincaré-Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.     

Fractional Cauchy problems on compact manifolds

We investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.     

环流形上的极值度量——存在性和K-稳定性

本文主要研究环流形上的极值度量的存在性和K-稳定性.本文将Donaldson关于环流形上有关常数量曲率度量的稳定性概念的约化推广到一般的极值度量的情形.通过这个约化,本文证明环流形上极值度量的存在性可以推出流形对于环形变的相对K-稳定性.在不知道是否存在极值度量的情形下,本文还给出环流形相对K-稳定的一个充分性条件.对环曲面的情形,基于Arrezo-Pacard-Singer的工作,本文证明任意一个环曲面上存在含有极值度量的Ka¨hler类,并给出一些环曲面上有不存在极值度量的K¨ahler类的例子.关于一般的环流形上的极值度量的存在性,本文用变分方法研究其弱解,证明在能量泛函逆紧性假设下,存在弱极小化子.