- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 15:20:53
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=8&iss=1&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2006
1
Riemannian geometries on spaces of plane curves
Peter
Michor
Universität Wien, WIEN, AUSTRIA
David
Mumford
Brown University, PROVIDENCE, UNITED STATES
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative.
Global analysis, analysis on manifolds
General
1
48
10.4171/JEMS/37
http://www.ems-ph.org/doi/10.4171/JEMS/37
Stopping Markov processes and first path on graphs
Giacomo
Aletti
Università degli Studi di Milano, MILANO, ITALY
Ely
Merzbach
Bar-Ilan University, RAMAT-GAN, ISRAEL
Markov chains, stopping rules, directed graph
Given a strongly stationary Markov chain (discrete or continuous) and a finite set of stopping rules, we show a non combinatorial method to compute the law of stopping. Several applied examples are presented. The problem of embedding a graph into a larger but minimal graph under some constraints is studied. Given a connected graph, we show a non combinatorial manner to compute the law of a first given path among a set of stopping paths. We prove the existence of a minimal Markov chain without oversized information.
Probability theory and stochastic processes
General
49
75
10.4171/JEMS/38
http://www.ems-ph.org/doi/10.4171/JEMS/38
Submanifold averaging in riemannian and symplectic geometry
Marco
Zambon
University of California, BERKELEY, UNITED STATES
averaging, isotropic, Lagrangian, Legendrian, parallel tubes, shape operators
We give a construction to obtain canonically an \"isotropic average\" of given $C1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply \"Moser's trick\". We also present an application to Hamiltonian group actions.
Global analysis, analysis on manifolds
General
77
122
10.4171/JEMS/39
http://www.ems-ph.org/doi/10.4171/JEMS/39
The p-Laplace eigenvalue problem as p goes to infinity in a Finsler metric
M.
Belloni
Università di Parma, PARMA, ITALY
Petri
Juutinen
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Bernd
Kawohl
Universität Köln, KÖLN, GERMANY
p-Laplace, eigenfunction, Finsler metric
We consider the $p$--Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite $p$ and investigate the limit problem as $p\to\infty$.
Partial differential equations
Calculus of variations and optimal control; optimization
General
123
138
10.4171/JEMS/40
http://www.ems-ph.org/doi/10.4171/JEMS/40
A symmetry problem in the calculus of variations
Graziano
Crasta
Università di Roma La Sapienza, ROMA, ITALY
Minimizers of integral functionals, Distance function, Euler equation
We consider a class of integral functionals defined in a Sobolev space of functions vanishing at the boundary of a nonempty bounded connected open n-dimensional set. We prove that, if the functional admits a minimizer depending only on the distance from the boundary, then that set must be a ball.
Calculus of variations and optimal control; optimization
Differential geometry
General
139
154
10.4171/JEMS/41
http://www.ems-ph.org/doi/10.4171/JEMS/41