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 of the distribution. Then
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 with the inner product induced by the Lie bracket, which is canonically defined surjective linear map. Pick a complement to in , 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 is defined by where 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 of Hausdorff spherical measure with respect to Popp volume is continuous.
It is constant in dimension , or in dimension and corank .
In higher dimensions, corank , it is not smooth (not ).
Gauthier showed that is generically in dimensions .
is the volume of unit ball in the nilpotent approximation. In low dimensions, the nilpotent approximation is constant. In dimension , 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,
By definition, integrand vanishes at conjugate time.
3. Heat equation
3.1. Classical results
Hörmander : is hypo-elliptic. Therefore, heat flow is well defined with a smooth kernel .
Theorem 3 (Léandre)
This asymptotic can be refined at points where the distance is smooth.
Theorem 4 (Agrachev) Squared distance to a point is smooth away from the cut-locus, i.e. on the open and dense set of points joined to by a unique normal non-conjugate minimizer. On that set, the differential of is the normal extremal of that minimizer (a covector) at the endpoint.
Theorem 5 (Bénarous) If ,
3.2. Expansion at cut points
At cut points, one can still say something. The idea is to use the semi-group property,
and to plug in Bénarous’ asymptotics, leading to
is the hinge energy function. For the asymptotics when tends to , what matters is the behaviour of in a neighborhood of its minima.
3.3. From estimates on hinge energy to asymptotics of
Lemma 6 achieves its minimum along the set of mid-points of minimal geodesics from to .
Theorem 7 Assume that is a strongly normal minimizer, i.e. all sub-arcs are strictly normal. Let be its mid-point. Then
- is conjugate to along is degenerate.
Theorem 8 Assume that all minimizers from to are strongly normal. Then
where is a neighborhood of the set of mid-points.
In particular, if there exist coordinates such that
we get an expansion whose leading term in is .
For instance, if has Morse minima, one recovers the asymptotic in .
Corollary 9 Up to constants,
We can improve this last result when there is a 1-parameter family of minimizers.
Example 1 Heisenberg group.
Using the explicit formula for , one sees that along the vertical axis, . The exponent (instead of at generic points) accounts for the fact that there is a -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 is a neighborhood, which leads to
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.
Is the density always ?
Expansion of diagonal heat kernel ? We have the expansion
in dimension . can be interpreted as curvature (vanishes for Heisenberg group). What about higher dimensions ?
Abnormal geodesics ? Not accessible by our technique.