Search results for "Mathematical physics"
showing 10 items of 2687 documents
Pseudodifferential operators on manifolds with a Lie structure at infinity
2003
to appear in Anal. Math.; Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at …
Algebraic models of the Euclidean plane
2018
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.
Symplectic Applicability of Lagrangian Surfaces
2009
We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equa- tions. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.
A sharp quantitative version of Alexandrov's theorem via the method of moving planes
2015
We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and $r_e -r_i \leq C \, osc(H)$, with $C$ depending only on $n$ and upper bounds on the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based on a…
Manifolds with vectorial torsion
2015
Abstract The present note deals with the properties of metric connections ∇ with vectorial torsion V on semi-Riemannian manifolds ( M n , g ) . We show that the ∇-curvature is symmetric if and only if V ♭ is closed, and that V ⊥ then defines an ( n − 1 ) -dimensional integrable distribution on M n . If the vector field V is exact, we show that the V-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature—for example, a V-Ricci-flat connection with vectorial torsion in di…
Mass transportation on sub-Riemannian structures of rank two in dimension four
2017
International audience; This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.
Prescribing the behaviour of geodesics in negative curvature
2010
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon desc…
Nonexistence of Quasiconformal Maps Between Certain Metric Measure Spaces
2013
We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application, we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.
Extending an example by Colding and Minicozzi
2018
Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $\Sigma_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $\Sigma_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.
Some perturbation results for quasi-bases and other sequences of vectors
2023
We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space $\Hil$ and producing new sequences which share, with the original ones, { reconstruction formulas on a dense subspace of $\Hil$ or on the whole space}. We also propose some preliminary results on the same issue, but in a distributional settings.