Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

We investigate the finiteness structure of a complete non-compact n-dimensional Riemannian manifold M whose radial curvature at a base point of M is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact set contains all critical points for any Busemann function on M. As corollaries by the main theorem, M has finite topological type, and the isometry group of M is compact.     

Busemann Functions in a Harmonic Manifold

For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces.     

Random data Cauchy theory for supercritical wave equations I: local theory

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H ^s (M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H ^s (M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary. Mathematics Subject Classification (2000)  35Q55, 35BXX, 37K05, 37L50, 81Q20     

On wedge extendability of CR-meromorphic functions

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a -smooth embedded Cauchy-Riemann generic manifold M (CR functions on ) and more generally, we consider holomorphic functions defined in wedgelike domains attached to . Our main result establishes the wedge- and the L¹-removability of E under the hypothesis that the -dimensional Hausdorff volume of E is zero and that M and are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everywhere locally minimal M to which every CR-meromorphic function on M extends meromorphically. Received: 7 September 2000; in final form: 20 December 2001 / Published online: 6 August 2002     

Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds

Consider a Riemannian manifold M which is a Galois covering of a compact manifold, with nilpotent deck transformation group G. For the Laplace operator on M, we prove a precise estimate for the gradient of the heat kernel, and show that the Riesz transforms are bounded in L^p(M), 1 < p < . We also obtain estimates for discrete oscillations of the heat kernel, and boundedness of discrete Riesz transform operators, which are defined using the action of G on M.Mathematics Subject Classification (2000): 58J35, 35B65, 42B20in final form: 8 August 2003     

关于Ma(n)é集与Aubry集的研究

J. Mather and A. Fathi defined Ma(n)ié set and Aubry set, which are the important invariant sets in positive definite Lagrangian system, in different ways.They use variational principle and Weak KAM theory respectively. In this paper we provide a proof of the equivalence between the two kinds of definitions, and generalize A. Fathi's definition. In the end of the paper, we calculate the Ma(n)é set and Aubry set for a single pendulum system.     

Closing Aubry Sets II

Given a Tonelli Hamiltonian H:T*M → ? of class C^k, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution that is of class C^{k + 1} in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then there exists a potential V : M → ? of class C^k?1, small in the C² topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) For every ? > 0 there exists a potential V : M → ? of class C^k?2, with for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. The latter result solves in the affirmative the Mañé density conjecture in the C¹ topology. © 2015 Wiley Periodicals, Inc.     

Partial regularity for the Landau-Lifshitz system

The aim of this work is to consider the partial regularity for the stationary weak solutions to the Landau-Lifshitz system of Ferromagnetic spin chain from a m-dimensional manifold M into the unit sphere S² of R³. The Landau-Lifshitz system is in appearance very similar to the heat flows of harmonic maps into sphere. However the monotonicity inequality, which plays an important role in getting partial regularity, does not hold in this case. This becomes a large barrier to regularity. In the present paper we get a generalized monotonicity inequality, and find the singular set of the stationary weak solutions of Landau-Lifshitz system.Received: 23 December 2002, Accepted: 10 July 2003, Published online: 22 September 2003Mathematics Subject Classification (1991): 58E20, 58Z05Project 10071013 supported by NSFC.     

Locally free actions on Lorentz manifolds

If a topological group G acts on a topological space M, then we say that the action is orbit nonproper provided that, for some ,the orbit map is nonproper. In this paper we characterize the connected, simply connected Lie groups that admit a locally free, orbit nonproper action by isometries of a connected Lorentz manifold. We also consider a number of variants on this question. Submitted: November 1998, Revised version: April 1999, Final version: April 2000.     

Discs in Stein manifolds containing given discrete sets

Denote by the open unit disc in . We prove that given a discrete subset S of a connected Stein manifold M there is a proper holomorphic map such that ; if the map f can be chosen to be an embedding. In addition we prove that we can prescribe higher order contacts of with given one dimensional submanifolds in M. Received: 19 June 2000; in final form: 29 November 2000 / Published online: 19 October 2001     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

On the Energy of Distributions, with Application to the Quaternionic Hopf Fibrations

 The energy of an oriented q-distribution ? in a compact oriented manifold M is defined to be the energy of the section of the Grassmannian manifold of oriented q-planes in M induced by ?. In the Grassmannian, the Sasaki metric is considered. We show here a condition for a distribution to be a critical point of the energy functional. In the spheres, we see that Hopf fibrations are critical points. Later, we prove the instability for these fibrations. (Received 30 December 2000; in revised form 11 April 2001)     

Hyperbolicity and exponential convergence of the Lax-Oleinik semigroup

For a convex superlinear Lagrangian on a compact manifold M it is known that there is a unique number c such that the Lax-Oleinik semigroup has a fixed point. Moreover for any u∈C(M,R) the uniform limit exists.In this paper we assume that the Aubry set consists in a finite number of periodic orbits or critical points and study the relation of the hyperbolicity of the Aubry set to the exponential rate of convergence of the Lax-Oleinik semigroup.     

Lagrangian intersections, critical points and Qcategory

For a manifold M, we prove that any function defined on a vector bundle of basis M and quadratic at infinity has at least Qcat(M)+1 critical points. Here Qcat(M) is a homotopically stable version of the LS-category defined by Scheerer, Stanley and Tanré [27]. The key homotopical result is that Qcat(M) can be identified with the relative LS-category of Fadell and Husseini [9] of the pair (M×D ⁿ⁺¹ ,M×S ⁿ ) for n big enough. Combining this result with the work of Laudenbach and Sikorav [19], we obtain that if M is closed, for any hamiltonian diffeomorphism with compact support of T ^* M, #((M)M)Qcat(M)+1, which improves all previously known homotopical estimates of this intersection number. Mathematics Subject Classification (2000):53D12, 55M30, 57R70.     

Well-posed infinite horizon variational problems on a compact manifold

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i.e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics.     

On the holonomy of connections with skew-symmetric torsion

We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits ²-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.Mathematics Subject Classification (2000):53 C 25, 81 T 30We thank Andrzej Trautman for drawing our attention to these papers by Cartan – see [27].     

Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust L^p estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L¹ contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.