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Regular projectively Anosov flows with compact leaves

Takeo Noda (2004)

Annales de l’institut Fourier

This paper concerns projectively Anosov flows φ t with smooth stable and unstable foliations ℱ s and ℱ u on a Seifert manifold M . We show that if the foliation ℱ s or ℱ u contains a compact leaf, then the flow φ t is decomposed into a finite union of models which are defined on T 2 × I and bounded by compact leaves, and therefore the manifold M is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible...

Régularité Lipschitzienne des Géodésiques Minimisantes pour Quelques Distributions Affines

Bensalem, Naceurdine (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 49J15, 49J30, 53B50.In the context of sub-Riemannian geometry and the Lipschitzian regularity of minimizers in control theory, we investigate some properties of minimizing geodesics for certain affine distributions. In particular, we consider the case of a generalized H2-strong affine distribution and the case of an affine Plaff system of maximal class.

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Andrea C. G. Mennucci (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H ( x , D u ( x ) ) = 0 on a n dimensional manifold M , with assumptions of convexity of H ( x , · ) and regularity of H (locally in a neighborhood of { H = 0 } in T * M ); we define the “min solution” u , a generalized solution; to this end, we view T * M as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about u ; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number...

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Andrea C.G. Mennucci (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where...

Regularity of optimal transport maps on multiple products of spheres

Alessio Figalli, Young-Heon Kim, Robert McCann (2013)

Journal of the European Mathematical Society

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps....

Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry

Marco Castelpietra, Ludovic Rifford (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...

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