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$ℒ$ denotes a (compact, nonsingular) lamination by hyperbolic Riemann surfaces. We prove that a probability measure on $ℒ$ is harmonic if and only if it is the projection of a measure on the unit tangent bundle $T^{1} ℒ$ of $ℒ$ which is invariant under both the geodesic and the horocycle flows.

A characterization of harmonic sections and a Liouville theorem

Simão Stelmastchuk (2012)

Archivum Mathematicum

Let $P (M, G)$ be a principal fiber bundle and $E (M, N, G, P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection $π_{E}$ of $E$ into $M$ . Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of $π_{E}$ .

A class of nonlinear conservative elliptic equations in cylinders

Jean René Licois, Laurent Véron (1998)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

A classification of Monge-Ampère equations

V. V. Lychagin, V. N. Rubtsov, I. V. Chekalov (1993)

Annales scientifiques de l'École Normale Supérieure

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

For $n = 2 m \geq 4$ , let $Ω \in ℝ^{n}$ be a bounded smooth domain and $𝒩 \subset ℝ^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that ${u_{k}} \in W^{m, 2} (Ω, 𝒩)$ is a sequence of weak solutions in the critical dimension to the perturbed $m$ -polyharmonic maps $\frac{d}{d t} |_{t = 0} E_{m} (Π (u + t ξ)) = 0$ with $Φ_{k} \to 0$ in ${(W^{m, 2} (Ω, 𝒩))}^{*}$ and $u_{k} ⇀ u$ weakly in $W^{m, 2} (Ω, 𝒩)$ . Then $u$ is an $m$ -polyharmonic map. In particular, the space of $m$ -polyharmonic maps is sequentially compact for the weak- $W^{m, 2}$ topology.

A complement to the Fredholm theory of elliptic systems on bounded domains.

Rabier, Patrick J. (2009)

Boundary Value Problems [electronic only]

A complete differential formalism for stochastic calculus in manifolds

James R. Norris (1992)

Séminaire de probabilités de Strasbourg

A computation of the equivariant index of the Dirac operator

Nicole Berline, Michèle Vergne (1985)

Bulletin de la Société Mathématique de France

A conformally invariant sphere theorem in four dimensions

Sun-Yung A. Chang, Matthew J. Gursky, Paul C. Yang (2003)

Publications Mathématiques de l'IHÉS

A construction of non-flat non-homogeneous symmetric parabolic geometries

Jan Gregorovič, Lenka Zalabová (2016)

Archivum Mathematicum

We construct series of examples of non-flat non-homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.

A convergence result for finite volume schemes on Riemannian manifolds

Jan Giesselmann (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_{t} + \nabla_{g} \cdot f (x, u) = 0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}}$ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}} .$

A counterexample for the optimality of Kendall-Cranston coupling.

Kuwada, Kazumasa, Sturm, Karl-Theodor (2007)

Electronic Communications in Probability [electronic only]

A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas

Christopher Deninger, Wilhelm Singhof (2001)

Annales de l’institut Fourier

We construct a two dimensional foliation with dense leaves on the Heisenberg nilmanifold for which smooth leafwise Hodge decomposition does not hold. It is also shown that a certain type of dynamical trace formulas relating periodic orbits with traces on leafwise cohomologies does not hold for arbitrary flows.

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