Integrable Hamiltonian systems with positive topological entropy

In this paper we provide a class of integrable Hamiltonian systems on a three-dimensional Riemannian manifold whose flows have a positive topological entropy on almost all compact energy surfaces. As our knowledge, these are the first examples of C^∞ Liouvillian integrable Hamiltonian flows with potential energy on a Riemannian manifold which has a positive topological entropy.     

Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is an addendum to the recent Cambridge Tract “Harmonic maps between Riemannian polyhedra”, by J. Eells and the present author. H?lder continuity of locally energy minimizing maps from an admissible Riemannian polyhedron X to a complete geodesic space Y is established here in two cases: (1) Y is simply connected and has curvature (in the sense of A.D. Alexandrov), or (2) Y is locally compact and has curvature , say, and is contained in a convex ball in Y satisfying bi-point uniqueness and of radius (best possible). With Y a Riemannian polyhedron, and in case (2), this was established in the book mentioned above, though with H?lder continuity taken in a weaker, pointwise sense. For X a Riemannian manifold the stated results are due to N.J. Korevaar and R.M. Schoen, resp. T. Serbinowski. Received: 10 October 2001 / Accepted: 20 November 2001 / Published online: 6 August 2002     

Partial regularity for weak heat flows into spheres

In this paper we show that a weak heat flow of harmonic maps from a compact Riemannian manifold (possibly with boundary) into a sphere, satisfying the monotonicity inequality and the energy inequality, is regular off a closed set of m-dimensional Hausdorff measure zero.     

Local Existence for Inhomogeneous Schrödinger Flow into Kähler Manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schr?dinger flow for maps from a compact Riemannian manifold M with dim(M) ≤ 3 into a compact K?hler manifold (N, J) with nonpositive Riemannian sectional curvature Received November 1, 1999, Revised January 14, 2000, Accepted March 29, 2000     

Conformally Einstein products and nearly Kähler manifolds

In the first part of this note we study compact Riemannian manifolds (M, g) whose Riemannian product with is conformally Einstein. We then consider 6-dimensional almost Hermitian manifolds of type W ₁ + W ₄ in the Gray–Hervella classification admitting a parallel vector field and show that (under some mild assumption) they are obtained as Riemannian cylinders over compact Sasaki–Einstein 5-dimensional manifolds.      

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M ^G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M ^G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.     

Unilateral problems for the wave equation with degenerate and localized nonlinear damping: well‐posedness and non‐stability results

Unilateral problems related to the wave model subject to degenerate and localized nonlinear damping on a compact Riemannian manifold are considered. Our results are new and concern two main issues: (a) to prove the global well‐posedness of the variational problem; (b) to establish that the corresponding energy functional is not (uniformly) stable to equilibrium in general, namely, the energy does not converge to zero on the trajectory of every solution, even if a full linear damping is taken in place.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

Random Čech complexes on Riemannian manifolds

In this paper we study the homology of a random ?ech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of 7 , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.     

Partial Hölder continuity for Q-valued energy minimizing maps

We consider multivalued maps between Ω ? ?^N open (N ≥ 2) and a smooth, compact Riemannian manifold 𝒩 locally minimizing the Dirichlet energy. An interior partial Hölder regularity results in the spirit of R. Schoen and K. Uhlenbeck is presented. Consequently a minimizer is Hölder continuous outside a set of Hausdorff dimension at most N ? 3. Almgren's original theory includes a global interior Hölder continuity result if the minimizers are valued into some ?^m. It cannot hold in general if the target is changed into a Riemannian manifold, since it already fails for “classical” single valued harmonic maps.     

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Long-Time Behaviour of Langevin Algorithms with Time-dependent Energy Function

We study the Langevin algorithm on C^∞ n-dimensional compact connected Riemannian manifolds and on IRⁿ, allowing the energy function U to vary with time. We find conditions under which the distribution of the process at hand becomes indistinguishable as t → ∞ from the “instantaneous” equilibrium distribution. Such conditions do not necessarily imply that U(t) converges pointwise as t → ∞.     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

A topological minorization for the volume of vector fields on 5-manifolds

A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional.For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.Received: 29 April 2004     

Optimal transportation on non-compact manifolds

In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type d ^r , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.     

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25