Generalized Riccati difference and differential equations

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three-dimensional contact sub-Riemannian manifolds with symmetry.

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Bishop and Laplacian Comparison Theorems on Three-Dimensional Contact Sub-Riemannian Manifolds with Symmetry

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure ω, possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V ∈ L 2 loc (M). The main merit of this paper is the identification of an intrinsic quantity, the effective potential V eff , which allows to formulate simple criteria for quantum confinement. Let δ be the distance from the possibly non-compact metric boundary of M. A simplified version of the main result guarantees quantum completeness if V ≥ −cδ 2 far from the metric boundary and V eff + V ≥ 3 4δ 2 − κ δ , close to the metric boundary. These criteria allow us to: (i) obtain sharp quantum confinement results for measures with degeneracies or singularities near the metric boundary of M ; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [9].

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Quantum confinement on non-complete Riemannian manifolds

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a Sasakian manifold equipped with a natural sub-Riemannian distance to satisfy these properties. Moreover, the sufficient conditions are defined by the Tanaka-Webster curvature. This generalizes the earlier work in [2] for the three dimensional case and in [19] for the Heisenberg group. To obtain our results we use the intrinsic Jacobi equations along sub-Riemannian extremals, coming from the theory of canonical moving frames for curves in Lagrangian Grassmannians [24,25]. The crucial new tool here is a certain decoupling of the corresponding matrix Riccati equation. It is also worth pointing out that our
method leads to exact formulas for the measure contraction in the case of the corresponding homogeneous models in the considered class of sub-Riemannian structures.

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Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds

Displacement interpolations from a Hamiltonian point of view

On deduit une inegalite matricielle a partir d'une inegalite differentielle. On utilise ce resultat pour deduire des inegalites pour les solutions d'equations de Riccati matricielles

H. L. Royden

Papers

Comparison theorems for the matrix riccati equation