** Heat diffusion for generic Riemannian and sub-Riemannian manifolds **

With Boscain, Charlot, Jendrej, Neel.

**1. Motivation **

Understand interplay between analysis (heat) and geometry (distance, geodesics, curvature). In particular, asymptotics of when lies in the cutlocus of .

** 1.1. Example : the round 2-sphere **

If and are antipodal,

where the exponent replaces the usual .

** 1.2. Surfaces of revolution **

For ellipsoids of revolution close to round, the cutlocus is a segment. This holds more generally for surfaces of revolution symmetric w.r.t. the equator.

Theorem 1Assume further a non degeneracy condition (conjugate locus of does not osculate the cutlocus at its endpoint ). Then

In non generic cases, exponents like arise for all integer .

**2. Sub-Riemannian cut and conjugate loci **

In the sub-Riemannian case, distance is not smooth at , it is smooth on an open and dense subset , the complement of the cut and conjugate loci (Agrachev). One can define an exponential map on the cotangent space at , whence the notion of conjugate points.

**3. General results **

The Laplacian is defined once a smooth volume is defined.

Theorem 2 (Léandre)Assume further a non degeneracy condition (conjugate locus of does not osculate the cutlocus at its endpoint ). Then

Theorem 3 (Bénarous)If ,Along the diagonal,

where is the homogeneous dimension à .

Theorem 4 (Barilar, Boscain, Neel)If and every optimal geodesic joining to is strongly normal. ThenIf is conjugate to or order (rank of differential of exponential map is ) along every optimal geodesic from to , then

where is the homogeneous dimension à .

** 3.1. Proof **

Use semi-group property and apply Bénarous’ estimate at . Then

* is conjugate to along at midpoint is degenerate.*

Furthermore, if there exist coordinates such that

we get an expansion whose leading term in is .

This applies when kernel of Hessian is 1-dimensional (Gromoll-Meyer), and in particular in the Heisenberg group.

**4. Generic results **

** 4.1. Exponential map as a Lagrangian map **

A map , symplectic, is Lagrangian if the fibers are Lagrangian. Exponential map does, since at each point it is a composition of projection with an hamiltonian flow.

Theorem 5 (Arnold’s school)Classification of generic Lagrangian singularities (i.e. Lagrangian maps up to symplectic coordinate changes of the domain and coordinate changes of the range) in dimensions .

For instance, the A3 singularity in dimension 2 is the usual fronce.

** 4.2. Generic Riemannian results **

Theorem 6In dimension , fix a point. For a generic Riemannian metric, the singularities of the exponential map are generic Lagrangian singularities.

Theorem 7In dimension , fix a point. For a generic Riemannian metric, the singularities of the exponential map are of type or . arise only in dimension , only in dimension .

Corollary 8Fof generic metrics in dimension , heat kernel asymptotics have exponent in case of an singularity, and in case of an singularity.

** 4.3. Generic Sub-Riemannian results **

Stick to 3-dimensional contact structures. A generic points, the cut locus is a surface made of two opposite horned triangles.

Theorem 9If no optimal geodesic from to is conjugate, then exponent is .

If at least one optimal geodesic from to is conjugate, then genericly, exponent is .