## Notes of Davide Barilari’s lecture Nr 2

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 ${p_t(x,y)}$ when ${y}$ lies in the cutlocus of ${x}$.

1.1. Example : the round 2-sphere

If ${x}$ and ${y}$ are antipodal,

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{3/2}}e^{-d^2(x,y)/4t}, \end{array}$

where the exponent ${3/2}$ replaces the usual ${1=2/2}$.

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 1 Assume further a non degeneracy condition (conjugate locus of ${x}$ does not osculate the cutlocus at its endpoint ${y}$). Then

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{5/4}}e^{-d^2(x,y)/4t}. \end{array}$

In non generic cases, exponents like ${\frac{n}{2}-\frac{r}{3}}$ arise for all integer ${r\geq 3}$.

2. Sub-Riemannian cut and conjugate loci

In the sub-Riemannian case, distance${^2}$ is not smooth at ${x}$, it is smooth on an open and dense subset ${\Sigma(x)}$, the complement of the cut and conjugate loci (Agrachev). One can define an exponential map on the cotangent space at ${x}$, 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 ${x}$ does not osculate the cutlocus at its endpoint ${y}$). Then

$\displaystyle \begin{array}{rcl} \lim_{t\rightarrow 0}4t\log p_t(x,y)-d^2(x,y). \end{array}$

Theorem 3 (Bénarous) If ${y\notin\Sigma(x)}$,

$\displaystyle \begin{array}{rcl} p_t(x,y)\sim\frac{1}{t^{n/2}}e^{-d^2(x,y)/4t}. \end{array}$

Along the diagonal,

$\displaystyle \begin{array}{rcl} p_t(x,x)\sim\frac{1}{t^{Q/2}}, \end{array}$

where ${Q}$ is the homogeneous dimension à ${x}$.

Theorem 4 (Barilar, Boscain, Neel) If ${y\in\Sigma(x)}$ and every optimal geodesic joining ${x}$ to ${y}$ is strongly normal. Then

$\displaystyle \begin{array}{rcl} \frac{1}{t^{n/2}}e^{-d^2(x,y)/4t}\leq p_t(x,y)\leq\frac{1}{t^{n-1/2}}e^{-d^2(x,y)/4t}. \end{array}$

If ${y}$ is conjugate to ${x}$ or order ${r}$ (rank of differential of exponential map is ${n-r}$) along every optimal geodesic from ${x}$ to ${y}$, then

$\displaystyle \begin{array}{rcl} p_t(x,x)\sim\frac{1}{t^{(n-r)/2}}, \end{array}$

where ${Q}$ is the homogeneous dimension à ${x}$.

3.1. Proof

Use semi-group property and apply Bénarous’ estimate at ${t/2}$. Then

${y}$ is conjugate to ${x}$ along ${\gamma}$ ${\Leftrightarrow}$ ${Hess (d^2)}$ at midpoint is degenerate.

Furthermore, if there exist coordinates such that

$\displaystyle \begin{array}{rcl} h_{x,y}(z)=\frac{1}{4}d^2(x,y)+z_1^{m_1}+\cdots+z_k^{m_k}+o(|z|^{m_k}), \end{array}$

we get an expansion whose leading term in ${t}$ is ${t^{-n+\sum\frac{1}{2m_j}}}$.

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 ${\pi:E\rightarrow M}$, ${E}$ 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 ${\leq 5}$.

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

4.2. Generic Riemannian results

Theorem 6 In dimension ${\leq 5}$, fix a point. For a generic Riemannian metric, the singularities of the exponential map are generic Lagrangian singularities.

Theorem 7 In dimension ${\leq 5}$, fix a point. For a generic Riemannian metric, the singularities of the exponential map are of type ${A_3}$ or ${A_5}$. ${A_3}$ arise only in dimension ${\geq 2}$, ${A_5}$ only in dimension ${\geq 4}$.

Corollary 8 Fof generic metrics in dimension ${n\leq 5}$, heat kernel asymptotics have exponent ${\frac{n}{2}+\frac{1}{4}}$ in case of an ${A_3}$ singularity, and ${\frac{n}{2}+\frac{1}{6}}$ in case of an ${A_5}$ 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 9 If no optimal geodesic from ${x}$ to ${y}$ is conjugate, then exponent is ${3/2}$.

If at least one optimal geodesic from ${x}$ to ${y}$ is conjugate, then genericly, exponent is ${7/4}$.