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Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick (2013)

Analysis and Geometry in Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...

Dirac and Plateau billiards in domains with corners

Misha Gromov (2014)

Open Mathematics

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry,...

Dirac operators on hypersurfaces

Jarolím Bureš (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper some relation among the Dirac operator on a Riemannian spin-manifold N , its projection on some embedded hypersurface M and the Dirac operator on M with respect to the induced (called standard) spin structure are given.

Direct approach to mean-curvature flow with topological changes

Petr Pauš, Michal Beneš (2009)

Kybernetika

This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves Γ ( t ) : S → ℝ 2 , t ≧ 0 . The curves are driven by the normal velocity v which is the function of curvature κ and the position. The evolution law reads as: v = - κ + F . The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved...

Discrétisation de zeta-déterminants d’opérateurs de Schrödinger sur le tore

Laurent Chaumard (2006)

Bulletin de la Société Mathématique de France

Nous donnons ici deux résultats sur le déterminant ζ -régularisé det ζ A d’un opérateur de Schrödinger A = Δ g + V sur une variété compacte ℳ . Nous construisons, pour ℳ = S 1 × S 1 , une suite ( G n , ρ n , Δ n ) où G n est un graphe fini qui se plonge dans ℳ via ρ n de telle manière que ρ n ( G n ) soit une triangulation de ℳ et où  Δ n est un laplacien discret sur G n tel que pour tout potentiel V sur ℳ , la suite de réels det ( Δ n + V ) converge après renormalisation vers det ζ ( Δ g + V ) . Enfin, nous donnons sur toute variété riemannienne compacte ( ℳ , g ) de dimension inférieure ou égale à 3 ...

Distinguished connections on ( J 2 = ± 1 ) -metric manifolds

Fernando Etayo, Rafael Santamaría (2016)

Archivum Mathematicum

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of ( J 2 = ± 1 ) -metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted...

Distinguished geodesics and jacobi fields on first order jet spaces

Vladimir Balan, Nicoleta Voicu (2004)

Open Mathematics

In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.

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