Philippe Delanoë

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (Vn, g) be a C∞ compact Riemannian manifold. For a suitable function ϑ on Vn, let us consider the change of metric: g′ = g + Hess(ϑ), and the function, as a ratio of two determinants, M(ϑ) = ¦g′¦ ¦g¦−1. Using the method of continuity, we first solve in C∞ the problem: Log M(ϑ) = λϑ + ƒ, λ > 0, ƒ ϵ C∞. Then, under weak hypothesis on F, we solve the general equation: Log M(ϑ) = F(P, ϑ), F in C∞(Vn × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦▽ϑ¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ ⩽ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.

Partial decay on simple manifolds

Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnérowicz-York equation of General Relativity); Ricci curvature in a weighted Kähler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnerowicz-York equation of General Relativity); Ricci curvature in a weighted Kahler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.

Regularity of optimal transport on compact, locally nearly spherical, manifolds

Abstract Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge–Ampère type satisfied by the potential function u, such that G = exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u, which was done on the flat torus by the first author. The recent local C 2 estimate of Ma–Trudinger–Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image G(m) of a generic point m ∈ M, uniformly away from the cut-locus of m; he checked a fourth-order inequality satisfied by the squared distance cost function, proving its so-called (strict) regularity. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in C 2 norm—specifying and proving a conjecture stated by Trudinger.

Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Perturbing fully nonlinear second order elliptic equations

We present two types of perturbations with reverse effects on some scalar fully nonlinear second order elliptic differential operators: on the other hand, first order perturbations which destroy the global solvability of the Dirichlet problem, in smooth bounded domains of

Variational Heuristics for Optimal Transportation Maps on Compact Manifolds

\mathbb R^n

Second order variational heuristics for the Monge problem on compact manifolds

; on the other hand, an integral perturbation which restore the local solvability, on compact connected manifolds without boundary.

Differential Geometric Heuristics for Riemannian Optimal Mass Transportation

Abstract Variational derivation of the expression of the solution of Monge’s problem posed on compact manifolds (possibly with boundary), assuming all data are smooth, the solution is a diffeomorphism and the cost function satisfies a generating type condition.

Construction of spheres with handles and non-vanishing mean curvature in the euclidean 3-space

Abstract. We consider Monges optimal transport problem posed on compact manifolds (possibly with boundary) for a lower semi-continuous cost function c. When all data are smooth and the given measures positive, we restrict the total cost to diffeomorphisms. If a diffeomorphism is stationary for , we know that it admits a potential function. If it realizes a local minimum of , we prove that the c-Hessian of its potential function must be non-negative, positive if the cost function c is non-degenerate. If c is generating non-degenerate, we reduce the existence of a local minimizer of to that of an elliptic solution of the Monge–Ampère equation expressing the measure transport; moreover, the local minimizer is unique. It is global, thus solving Monges problem, provided c is superdifferentiable with respect to one of its arguments.

Analyzing non-degenerate 2-forms with riemannian metrics

We give an account on Otto’s geometrical heuristics for realizing, on a compact Riemannian manifold M, the L 2 Wasserstein distance restricted to smooth positive probability measures, as a Riemannian distance. The Hilbertian metric discovered by Otto is obtained as the base metric of a Riemannian submersion with total space, the group of diffeomorphisms of M equipped with the Arnol’d metric, and projection, the push-forward of a reference probability measure. The expression of the horizontal constant speed geodesics (time dependent optimal mass transportation maps) is derived using the Riemannian geometry of M as a guide.