共查询到20条相似文献,搜索用时 0 毫秒
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We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian
distance. Under appropriate assumptions, we generalize Brenier–McCann’s theorem proving existence and uniqueness of the optimal
transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the
optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under
some weak assumptions on the measures the differentiability a.e.), which allows us to write a weak form of the Monge–Ampère
equation. 相似文献
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We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot–Carathéodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives). 相似文献
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Potential Analysis - Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup $P_{t}$ , we give two different stochastic... 相似文献
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Doklady Mathematics - The structure of the intersection of the sub-Riemannian sphere on the Cartan group with a 3-dimensional invariant manifold of main symmetries is described. 相似文献
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Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable Lp-functional calculi,or may be of holomorphic Lp-type for a given p 2. HolomorphicLp-type means that every Lp-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L2-spectrum of L. This can in factonly arise if the group algebra L1(G) is non-symmetric. Assume that p 2. For a point in the dual g* of the Lie algebrag of G, denote by ()=Ad*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicLp-type, provided that there exists a point g* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L1(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL1 (or, more generally, Lp) type on G if and only if there existsa point g* whose orbit is closed and which satisfies Boidol'scondition. 相似文献
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In this paper,we introduce a special class of nilpotent Lie groups defined by hermitianmaps,which includes all the groups of affine holomorphic automorphisims of Siegel domains oftype Ⅱ,in particular,the Heisenberg group.And we study harmonic analysis on these groupsas spectral theory of the associated Sub-Laplacian instead of the group representation theoryin usual way. 相似文献
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Andrea Tommasoli 《Potential Analysis》2010,32(1):57-66
The aim of this paper is to show the strong connection between regularity of bounded open set boundary points and quasi-boundedness, on the same set, of the fundamental solution of stratified Lie group sub-Laplacians. In the euclidean case the theorem was
proved by Kuran (J Lond Math Soc 2(19):301–311, 1979). We later give two examples using some direct consequences of main theorem. 相似文献
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We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem. 相似文献
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Potential Analysis - We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that,... 相似文献
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We compare different notions of curvature on contact sub-Riemannian manifolds. In particular, we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet–Myers theorem that applies to any contact manifold. 相似文献
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Der-Chen Chang Irina Markina Alexander Vasil’ev 《Complex Analysis and Operator Theory》2009,3(2):361-377
The unit sphere can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations
for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics
on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.
The first author is partially supported by a research grant from the United State Army Research Office and a Hong Kong RGC
competitive earmarked research grant #600607. The second and the third authors have been supported by the grant of the Norwegian
Research Council #177355/V30, and by the European Science Foundation Research Networking Programme HCAA. 相似文献
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We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide. 相似文献
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The goal of this paper is to consider a step 2 sub-Riemannian manifold where the connectivity bynormal geodesics between distant points fails. 相似文献
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Doklady Mathematics - This paper considers the left-invariant sub-Riemannian problem on the Engel group. The stratification of the cut locus and the structure of optimal synthesis are described. 相似文献
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V. N. Berestovskii 《Siberian Mathematical Journal》2018,59(1):31-42
Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p. 相似文献
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We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg. 相似文献
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Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals. 相似文献
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