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}.

Advertisements

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
This entry was posted in seminar and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s