## Notes of Davide Barilari’s lecture

Geometry and heat equation on sub-Riemannian manifolds

Which is the right Laplace operator associated to a sub-Riemannian metric ? Which of its properties can be read from the Carnot-Carathéodory distance ?

1. Intrinsic volume and sub-Laplacian

These were defined by Brockett (1982) in dimension 3, later by Popp for step 2 equiregular distributions, by Montgomery in general in 2002. Let us start with volume.

In dimension 3, pick an orthonormal basis ${(X_1,X_2)}$ of the distribution. Then

$\displaystyle (X_1,X_2,[X_1,X_2])$

is a basis. Take the dual basis of differential 1-forms, wedge them. The result does not depend on the choice of basis (Brackett).

For a 2-step equiregular distribution, equip ${TM/\Delta}$ with the inner product induced by the Lie bracket, which is canonically defined surjective linear map. Pick a complement to ${\Delta}$ in ${TM}$, get a Riemannian metric. It turns out that its volume element does not depend on the choice of complement (Popp).

Once one has an intrinsic volume, one gets an intrinsic divergence operator.

Definition 1 (Montgomery) The sub-Laplacian ${L}$ is defined by ${L=div \circ grad}$ where ${grad}$ is the horizontal gradient.

2. Intrinsic volume versus Hausdorff measure

2.1. How does volume relate to Hausdorff measure ?

Theorem 2 (Agrachev, Barilari, Boscain) The density ${f}$ of Hausdorff spherical measure with respect to Popp volume is continuous.

It is constant in dimension ${\leq 4}$, or in dimension ${5}$ and corank ${1}$.

In higher dimensions, corank ${1}$, it is not smooth (not ${C^5}$).

Gauthier showed that ${f}$ is generically ${C^1}$ in dimensions ${(4,6)}$.

2.2. Proof

${f}$ is the volume of unit ball in the nilpotent approximation. In low dimensions, the nilpotent approximation is constant. In dimension ${\geq 5}$, the unit ball, specifically, the cut time, is not a smooth function of the Lie algebra structure.

If cut time coincides with conjugate time, we gain a bit of regularity. Indeed,

$\displaystyle \begin{array}{rcl} vol(B)=\int_{\{\lambda\}}\int_{0}^{cut~time(\lambda)}|det(d \exp)|\,dt\,d\lambda. \end{array}$

By definition, integrand ${det(d \exp)}$ vanishes at conjugate time.

3. Heat equation

3.1. Classical results

Hörmander : ${L}$ is hypo-elliptic. Therefore, heat flow is well defined with a smooth kernel ${p_t}$.

Theorem 3 (Léandre)

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

This asymptotic can be refined at points where the distance is smooth.

Theorem 4 (Agrachev) Squared distance to a point ${x_0}$ is smooth away from the cut-locus, i.e. on the open and dense set ${\Sigma(x_0)}$ of points joined to ${x_0}$ by a unique normal non-conjugate minimizer. On that set, the differential of ${d^2/2}$ is the normal extremal of that minimizer (a covector) at the endpoint.

Theorem 5 (Bénarous) If ${y\in \Sigma(x)}$,

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

3.2. Expansion at cut points

At cut points, one can still say something. The idea is to use the semi-group property,

$\displaystyle \begin{array}{rcl} p_{t}(x,y)=\int_{M}p_{t/2}(x,z)p_{t/2}(z,y)\,dz, \end{array}$

and to plug in Bénarous’ asymptotics, leading to

$\displaystyle \begin{array}{rcl} p_{t}(x,y)&\sim&t^{-n}\int_{M}e^{-\frac{d(x,z)^2}{2t}}e^{-\frac{d(z,y)^2}{2t}}\,dz\\ &=&t^{-n}\int_{M}e^{-\frac{h_{x,y}(z)}{2t}}\,dz, \end{array}$

where

$\displaystyle h_{x,y}(z)=\frac{1}{2}(d(x,z)^2+d(y,z)^2).$

is the hinge energy function. For the asymptotics when ${t}$ tends to ${0}$, what matters is the behaviour of ${h_{x,y}}$ in a neighborhood of its minima.

3.3. From estimates on hinge energy to asymptotics of ${p_t}$

Lemma 6 ${h_{x,y}}$ achieves its minimum along the set of mid-points of minimal geodesics from ${x}$ to ${y}$.

Theorem 7 Assume that ${\gamma}$ is a strongly normal minimizer, i.e. all sub-arcs are strictly normal. Let ${z_0}$ be its mid-point. Then

1. ${y}$ is conjugate to ${x}$ along ${\gamma}$ ${\Leftrightarrow}$ ${Hess_{z_0}h_{x,y}}$ is degenerate.

Theorem 8 Assume that all minimizers from ${x}$ to ${y}$ are strongly normal. Then

$\displaystyle \begin{array}{rcl} p_t(x,y)=t^{-n}\int_{N}(c_{x,y}(z)+O(t))e^{-\frac{h_{x,y}(z)}{2t}}, \end{array}$

where ${N}$ is a neighborhood of the set of mid-points.

In particular, 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 ${t^{-n+\sum\frac{1}{m_j}}}$.

For instance, if ${h}$ has Morse minima, one recovers the asymptotic in ${t^{-n/2}}$.

Corollary 9 Up to constants,

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

We can improve this last result when there is a 1-parameter family of minimizers.

3.4. Examples

Example 1 Heisenberg group.

Using the explicit formula for ${p_t}$, one sees that along the vertical axis, ${p_t(x,y)\sim t^{-2}\exp(-\pi z/t)}$. The exponent ${2}$ (instead of ${3/2}$ at generic points) accounts for the fact that there is a ${1}$-parameter family of minimizers.

Example 2 Grushin plane.

One can compute the cut and conjugate loci of ordinary points. We have studied in detail the cut-conjugate points (there is only one up to symmetry). We have an expansion of ${h}$ is a neighborhood, which leads to

$\displaystyle \begin{array}{rcl} p_t \sim t^{-5/4}\exp(-\pi^2/t). \end{array}$

We have been able to compute the Grushin plane merely because its geodesic flow is integrable in terms of trigonometric functions. The fact the metric degenerates along a line is not essential. We expect the result to generalize to all generic Riemannian metrics.

4. Questions

Is the density ${f}$ always ${C^1}$ ?

Expansion of diagonal heat kernel ? We have the expansion

$\displaystyle \begin{array}{rcl} p_t (x,x)\sim t^{-2}(1+\kappa(x)t +O(t^2)) \end{array}$

in dimension ${3}$. ${\kappa}$ can be interpreted as curvature (vanishes for Heisenberg group). What about higher dimensions ?

Abnormal geodesics ? Not accessible by our technique.