374 research outputs found
Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions
In this paper we study the sub-Finsler geometry as a time-optimal control
problem. In particular, we consider non-smooth and non-strictly convex
sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet
distributions. Motivated by problems in geometric group theory, we characterize
extremal curves, discuss their optimality, and calculate the metric spheres,
proving their Euclidean rectifiability.Comment: 24 pages, 17 figure
Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions
In this paper, we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability. © 2016, Springer Science+Business Media New York
Extremal -invariant eigenvalues of the Laplacian of -invariant metrics
The study of extremal properties of the spectrum often involves restricting
the metrics under consideration. Motivated by the work of Abreu and Freitas in
the case of the sphere endowed with -invariant metrics, we consider
the subsequence of the spectrum of a Riemannian manifold
which corresponds to metrics and functions invariant under the action of a
compact Lie group . If has dimension at least 1, we show that the
functional admits no extremal metric under volume-preserving
-invariant deformations. If, moreover, has dimension at least three,
then the functional is unbounded when restricted to any conformal
class of -invariant metrics of fixed volume. As a special case of this, we
can consider the standard O(n)-action on ; however, if we also require the
metric to be induced by an embedding of in , we get an
optimal upper bound on .Comment: To appear in Mathematische Zeitschrif
Il problema isoperimetrico in spazi di Carnot-Carathéodory
We present some recent results obtained on the isoperimetric problem in a class of Carnot-Carathéodory spaces, related to the Heisenberg group. This is the framework of Pansu’s conjecture about the shape of isoperimetric sets. Two different approaches are considered. On one hand we describe the isoperimetric problem in Grushin spaces, under a symmetry assumption that depends on the dimension and we provide a classification of isoperimetric sets for special dimensions. On the other hand, we present some results about the isoperimetric problem in a family of Riemannian manifolds approximating the Heisenberg group. In this context we study constant mean curvature surfaces. Inspired by Abresch and Rosenberg techniques on holomorphic quadratic differentials, we classify isoperimetric sets under a topological assumption.Presentiamo risultati recenti ottenuti sul problema isoperimetrico in spazi di Carnot-Carathéodory legati al gruppo di Heisenberg. Questo è il contesto della congettura di Pansu sulla forma degli insiemi isoperimetrici. Presentiamo due diversi approcci. Da un lato, descriviamo il problema isoperimetrico negli spazi di Grushin, sotto un’ipotesi di simmetria che dipende dalla dimensione. In questo contesto, forniamo una classificazione degli insiemi isoperimetrici, valida per specifiche dimensioni. Dall’altro lato, presentiamo alcuni risultati sul problema isoperimetrico in una famiglia di varietà Riemanniane che approssimano il gruppo di Heisenberg. In questo contesto, studiamo superfici a curvatura media costante. Ispirati dalle tecniche di Abresch and Rosenberg sui differenziali quadratici olomorfi, classifichiamo gli insiemi isoperimetrici sotto un’ipotesi topologica
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