Robert W. Neel

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ∆ω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement c to the sub-Riemannian distribution, and is denoted L c. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popps one P) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation: • On contact structures, for every volume ω, there exists a unique complement c such that ∆ω = L c. • On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ∆H = L c. However this complement is not unique in general. • For quasi-contact structures, in general, ∆P = L c for any choice of c. In particular, L c is not symmetric w.r.t. Popps measure. This is surprising especially in dimension 4 where, in a suitable sense, ∆P is the unique intrinsic macroscopic Laplacian. A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.

Brownian Motion and the Dirichlet Problem at Infinity on Two-dimensional Cartan-Hadamard Manifolds

After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we describe what is known under various curvature assumptions and the difference between the two-dimensional and the higher-dimensional cases. We discuss the probabilistic formulation of the problem in terms of the asymptotic behavior of the angular component of Brownian motion. We then introduce a new (and appealing) probabilistic approach that allows us to prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard manifold is solvable under the curvature condition K ≤ (1 + ε)/(r2 logr) outside of a compact set, for some ε > 0, in polar coordinates around some pole. This condition on the curvature is sharp, and improves upon the previously known case of quadratic curvature decay. Finally, we briefly discuss the issues which arise in trying to extend this method to higher dimensions.

A martingale approach to minimal surfaces

Abstract We provide a probabilistic approach to studying minimal surfaces in R 3 . After a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in minimal surface theory, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.

A stochastic target approach to Ricci flow on surfaces

We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi’s representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every Ck-norm. In the case of C0 and C1-convergence, we achieve this by coupling two particles. To get C2-convergence (in particular, convergence of the curvature), we use a coupling of three particles. This triple coupling is developed here only for the case of constant curvature metrics on surfaces, though we suspect that some variants of this idea are applicable in other situations and therefore be of independent interest. Finally, for k≥3, the Ck-convergence follows relatively easily using induction and coupling of two particles. None of these techniques appear in the Ricci flow literature and thus provide an alternative approach to the field.

Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.

Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes

— Abstract. We provide the small-time asymptotics of the heat kernel at the cut locus in three cases: generic Riemannian manifolds in dimension less or equal to 5, generic 3D contact and 4D quasi-contact sub-Riemannian manifolds (close to the starting point). As a byproduct we show that, for generic Riemannian manifolds of dimension less or equal to 5, the only possible singularities of the exponential map along a minimizing geodesic are A3 and A5. 1. Résultats connus en riemannien et sous-riemannien En géométrie riemannienne, le laplacien est généralement défini de la façon suivante. La divergence d’un champ de vecteur est définie à partir du volume riemannien par div(X)vol = LXvol. et le gradient riemannien d’une fonction par