## Curvature inequalities in sub-Riemannian geometry

An important part of my research from the past few years has been to try to understand the notion of Ricci curvature lower bound in sub-Riemannian geometry. On this problem, my point of view is more the one of an analyst or even a probabilist than a geometer: My main device is a second order differential operator, a sub-Laplacian, and my main tools to understand global geometric properties are heat flow techniques. In sub-Riemannian geometry, this point of view can be fruitful in some aspects that I will describe now.

Although during the last three decades there have been several advances in the study of sub Riemannian spaces and the closely connected theory of sub-elliptic pde’s, most of the developments to date are of a local nature. As a consequence, the theory presently lacks a body of results which, similarly to the Riemannian case, connect properties of solutions of the relevant pde’s to the geometry of the ambient manifold. To illustrate this point, let me discuss some of the most fundamental local estimates concerning the subelliptic Hörmander‘s type operators.

Let $(\mathbb M,g)$ be a smooth and connected Riemannian manifold and $\mu$ be a smooth measure on $\mathbb{M}$. Let us assume that there exists on $\mathbb{M}$ a family of vector fields $\{ X_1, \cdots X_d \}$ that satisfy the bracket generating condition. We are interested in the subelliptic operator $L=-\sum_{i=1}^d X_i^* X_i$ which is symmetric with respect to the measure $\mu$. In order to study $L$, the Riemannian distance $d_R$ of $\mathbb M$ is most of the times confined to the background. There is another distance on $\mathbb M$, that was introduced by Carathéodory, which plays a central role. A piecewise $C^1$ curve $\gamma:[0,T]\to \mathbb M$ is called subunitary at $x$ if for every $\xi\in T_x\mathbb M$ one has $g(\gamma'(t),\xi)^2 \le \sum_{i=1}^d g(X_i(\gamma(t)),\xi)^2.$ We define the subunit length of $\gamma$ as $L_s(\gamma) = T$. If we indicate with $S(x,y)$ the family of subunit curves such that $\gamma(0) = x$ and $\gamma(T) = y$, then thanks to the fundamental accessibility theorem of Chow-Rashevsky the connectedness of $\mathbb M$ implies that $S(x,y) \not= \varnothing$ for every $x,y\in \mathbb M$. This allows to define the sub-Riemannian distance on $\mathbb M$ as follows $d(x,y) = \inf \{L_s(\gamma)\mid \gamma\in S(x,y)\}.$
Another elementary consequence of the Chow-Rashevsky theorem is that the identity map $i:(\mathbb M,d) \hookrightarrow (\mathbb M,d_R)$ is continuous and thus, the topologies of $d_R$ and $d$ coincide. Several fundamental properties of the metric $d$ have been discussed in the seminal paper by Nagel, Stein and Wainger. In particular, the following result provides a uniform local control of the growth of the metric balls in $(\mathbb{M},d)$. It is known as the local doubling condition.

Theorem 1: For any $x\in \mathbb{M}$ there exist constants $C(x), R(x) > 0$ such that with $Q(x) = \log_2 C(x)$ one has
$\mu(B(x,tr)) \ge C(x)^{-1} t^{Q(x)} \mu(B(x,r)),\ \ \ 0\le t\le 1,\ 0 < r\le R(x).$
Moreover, given any compact set $K\subset \mathbb{M}$ one has
$\underset{x\in K}{\inf}\ C(x) >0,\ \ \ \underset{x\in K}{\inf}\ R(x) > 0.$

This remarkable theorem is a fundamental local property of the metric space $(\mathbb{M},d)$ and may be connected to several local estimates related to the operator $L$. We mention in particular the Poincaré inequality on balls:

Theorem 2: (D. Jerison) Let $x \in \mathbb{M}$ and $R > 0$. There exists a constant $C=C(x,R) > 0$ such that for every $0 < r < R$ and $f \in C^1 (B(x,r))$,
$\int_{B(x,r)} | f -f_{B(x,r)}|^ 2 d\mu \le C r^2 \int_{B(x,r)} \sum_{i=1}^d (X_if)^2 d\mu,$
where $f_{B(x,r)} =\frac{1}{\mu(B(x,r))} \int_{B(x,r)} f d\mu$.

We stress again that these theorems and the methods to prove them are local in nature. This naturally raises the following question:

Question 1: Can we find conditions on the operator $L$ ensuring that global versions of theorems like Theorems 1 and 2 hold true ?

By global versions of Theorems 1 and 2, we mean that the estimates should hold true on non necessarily compact manifolds and with constants uniformly controlled by the sub-Riemannian geometry of $\mathbb{M}$. To find a way to tackle this question, let us first discuss what could be a satisfying answer in the simplest non trivial case: The case where $L$ is elliptic, that is $\{ X_1,\cdots,X_d \}$ form a basis of the tangent space at each point. In this case, there is a definite satisfying answer to Question 1: Global versions of the Theorems 1 and 2 may be obtained under the assumption that $L$ satisfies a curvature dimension inequality.

The notion of curvature dimension inequality originates from the analysis of the Laplace-Beltrami operator on a Riemannian manifold and more precisely from the Bochner’s formula which states that if $\mathbb{M}$ is a Riemannian manifold with Laplacian $\Delta$, for any $f\in C^\infty(\mathbb{M})$ one has
$\Delta(|\nabla f|^2) = 2 ||\nabla^2 f||^2 + 2 (\nabla f, \nabla \Delta f) + 2 \text{Ric}(\nabla f,\nabla f).$

Using the Cauchy-Schwarz inequality, which gives $\| \nabla^2 f \|_2^2\ge \frac{1}{n} (\Delta f)^2$, we thus see that the assumption that the Riemannian Ricci tensor on $\mathbb{M}$ be bounded from below by $\rho_1 \in \mathbb{R}$ implies
$\frac{1}{2}\Delta(|\nabla f|^2) - (\nabla f, \nabla \Delta f) \ge \frac{1}{n} (\Delta f)^2 + \rho_1 \| \nabla f \|^2,\ \ \ \ f\in C^\infty(\mathbb{M}).$

What is remarkable is that this inequality perfectly captures the notion of Ricci lower bounds and dimensional upper bound: On a finite dimensional Riemannian manifold $\mathbb{M}$ the inequality is actually equivalent to Ric $\ge \rho_1$ and $\mathbf{dim}(\mathbb{M}) \le n$.

This observation leads Bakry to define the notion of curvature dimension inequality for arbitrary second-order differential operators. Let $L$ be, as before, a second order differential operator. Consider the following bilinear differential forms on functions $f, g \in C^\infty(\mathbb{M})$,
$\Gamma(f,g) =\frac{1}{2}(L(fg)-fL g-gL f),$
and
$\Gamma_{2}(f,g) = \frac{1}{2}\big[L \Gamma(f,g) - \Gamma(f, L g)-\Gamma (g,L f)\big].$
When $f=g$, we simply write $\Gamma(f) = \Gamma(f,f)$, $\Gamma_2(f) = \Gamma_2(f,f)$. A straightforward computation shows that if, in a local chart,
$L=\sum_{i,j=1}^n a_{ij} (x) \frac{\partial^2}{ \partial x_i \partial x_j} +\sum_{i=1}^n b_i (x)\frac{\partial}{\partial x_i},$
then, in the same chart
$\Gamma (f,g)=\sum_{i,j=1}^n a_{ij} (x) \frac{\partial f}{\partial x_i} \frac{\partial g}{\partial x_j}.$
As a consequence, for every smooth function $f$, $\Gamma(f) \ge 0.$ The bilinear form $\Gamma_2$ is of second order and has no sign in general. We observe that if $L$ is a Laplace-Beltrami operator, then Bochner’s formula writes
$\Gamma_2(f)= ||\nabla^2 f||^2 + \ \text{Ric}(\nabla f,\nabla f).$

Definition: The operator $L$ is said to satisfy the curvature dimension inequality CD$(\rho_1,n)$, $\rho_1 \in \mathbb{R}$, $n >0$, if for every function $f \in C^\infty(\mathbb{M})$:
$\Gamma_{2}(f) \ge \frac{1}{n} (L f)^2 + \rho_1\Gamma(f).$

As we just pointed it out, if $L$ is a Laplace-Beltrami operator, then it satisfies CD$(\rho_1,n)$ if and only if Ric $\ge \rho_1$ and $\mathbf{dim}(\mathbb{M}) \le n$. But, what is interesting here is that the curvature dimension inequality may be satisfied without $L$ necessarily being a Laplace-Beltrami operator. In that case, the curvature dimension inequality is equivalent to a lower bound on the Bakry-Emery tensor of $L$. We have then the following, a priori, satisfying answer to our Question 1:

Fact: If $L$ satisfies CD$(\rho_1,n)$, with $\rho_1 \ge 0$ and $n >0$, then we have global versions of Theorems 1 and 2.

More precisely, we have the following result:

Theorem: Let $L$ be a subelliptic second order operator that satisfies CD$(\rho_1,n)$, with $\rho_1 \ge 0$ and $n > 0$. Then, there exist constants $C_d, C_p > 0$, depending only on $\rho_1$ and $n$, for which one has for every $x\in \mathbb{M}$ and every $r > 0$:
$\mu(B(x,2r)) \le C_d\ \mu(B(x,r));$
and
$\int_{B(x,r)} |f - f_B|^2 d\mu \le C_p r^2 \int_{B(x,r)} \Gamma(f) d\mu,$
for every $f\in C^1( B(x,r))$.

Unfortunately, this fact only has a limited interest for us, since it is possible to prove that if a subelliptic operator $L$ satisfies CD$(\rho_1,n)$, with $\rho_1 \in \mathbb{R}$ and $n > 0$, then $L$ is actually elliptic ! With this in mind, the next question that naturally arises is:

Question 2: Is there an appropriate notion of curvature dimension inequality that applies to subelliptic operators ?

At this level of generality, this question certainly does not admit a definite satisfying answer. However with my co-author, N. Garofalo, we introduced a curvature dimension inequality that applies to a large class of subelliptic operators.

To introduce the relevant setting we consider a smooth, connected manifold $\mathbb{M}$ endowed with a smooth measure $\mu$ and a smooth, locally subelliptic, second order differential operator $L$ which is symmetric with respect to $\mu$. As above, we associate with $L$ the following symmetric, first-order, differential bilinear form:
$\Gamma(f,g) =\frac{1}{2}(L(fg)-fLg-gLf), \quad f,g \in C^\infty(\mathbb{M}).$
There is a genuine distance $d$ canonically associated with $L$ which is continuous and defines the topology of $\mathbb{M}$. It is given by
$d(x,y)=\sup \left\{ |f(x) -f(y) | \mid f \in C^\infty(\mathbb{M}) , \| \Gamma(f) \|_\infty \le 1 \right\},\ \ \ \ x,y \in \mathbb{M}.$

We always assume that this distance is finite and that the metric space $(\mathbb{M},d)$ is complete. This implies that $L$ is essentially self-adjoint.

We also suppose, and this is the main new ingredient, that we are given on $\mathbb{M}$ a symmetric, first-order differential bilinear form $\Gamma^Z$, satisfying
$\Gamma^Z(fg,h) = f\Gamma^Z(g,h) + g \Gamma^Z(f,h).$
The following hypothesis on $\Gamma^Z$ plays a pervasive role in the results that we will describe and is natural in several geometric situations: For every $f \in C^\infty(\mathbb{M})$, we have $\Gamma(f, \Gamma^Z(f))=\Gamma^Z( f, \Gamma(f)).$

Before we proceed with the discussion, we pause to stress that, in the generality in which we work the bilinear differential form $\Gamma^Z$, unlike $\Gamma$, is not a priori intrinsic. Whereas $\Gamma$ is determined once $L$ is assigned, the form $\Gamma^Z$ in general is not intrinsically associated with $L$. However, in several geometric examples the choice of $\Gamma^Z$ will be natural and even canonical, up to a constant. This is the case, for instance, for one of the important geometric examples covered by our analysis: The CR Sasakian manifolds. Roughly speaking, we can think of $\Gamma^Z$ as an orthogonal complement of $\Gamma$: the bilinear form $\Gamma$ represents the square of the length of the gradient in the horizontal directions, whereas $\Gamma^Z$ represents the square of the length of the gradient along the vertical directions.

Given the subelliptic operator $L$ and the first-order bilinear forms $\Gamma$ and $\Gamma^Z$ on $\mathbb{M}$, we introduce the following second-order differential bilinear forms:
$\Gamma_{2}(f,g) = \frac{1}{2}\big[L\Gamma(f,g) - \Gamma(f, Lg)-\Gamma (g,Lf)\big],$
and
$\Gamma^Z_{2}(f,g) = \frac{1}{2}\big[L\Gamma^Z (f,g) - \Gamma^Z(f, Lg)-\Gamma^Z (g,Lf)\big].$
Observe that if $\Gamma^Z\equiv 0$, then $\Gamma^Z_2 \equiv 0$ as well. As for $\Gamma$ and $\Gamma^Z$, we will use the notations $\Gamma_2(f) = \Gamma_2(f,f)$, $\Gamma_2^Z(f) = \Gamma^Z_2(f,f)$. The main definition and tool we proposed is the following:

Definition: We say that the subelliptic operator $L$ satisfies the generalized curvature-dimension inequality CD$(\rho_1,\rho_2,\kappa,d)$ if there exist constants $\rho_1 \in \mathbb{R}$, $\rho_2 > 0$, $\kappa \ge 0$, and $0 < d \le +\infty$ such that the inequality
$\Gamma_2(f) +\nu \Gamma_2^Z(f) \ge \frac{1}{d} (Lf)^2 +\left( \rho_1 -\frac{\kappa}{\nu} \right) \Gamma(f) +\rho_2 \Gamma^Z(f)$
holds for every $f\in C^\infty(\mathbb{M})$ and every $\nu > 0$.

The motivation behind this definition comes from examples arising in geometry.

Theorem: Let $(\mathbb{M},\theta)$ be a strictly pseudo convex CR manifold with real dimension $2n+1$ and vanishing Tanaka-Webster pseudohermitian torsion, i.e., a Sasakian manifold. The Tanaka-Webster Ricci tensor satisfies the lower bound
$\emph{Ric}_x(v,v)\ \ge \rho_1|v|^2,\ \ \ \text{for every horizontal vector}\ v\in \mathcal H_x=\mathbf{Ker} ( \theta_x),$
if and only if the CR sub-Laplacian of $\mathbb{M}$ satisfies the curvature-dimension inequality CD$(\rho_1,\frac{d}{4},1,d)$ with $d = 2n$. Moreover, the hypothesis $\Gamma(f, \Gamma^Z(f))=\Gamma^Z( f, \Gamma(f))$ is satisfied.

With this examples in mind let us now come back to the general framework and give a good answer to Question 1 in many purely subelliptic examples. In the following theorem that I proved with N. Garofalo and M. Bonnefont, the hypothesis $\Gamma(f, \Gamma^Z(f))=\Gamma^Z( f, \Gamma(f))$ is supposed to be satisfied.

Theorem: Let $L$ be a subelliptic second order operator that satisfies CD$(\rho_1,\rho_2,\kappa,n)$, with $\rho_1 \ge 0$, $\rho_2 > 0$, $\kappa \ge 0$ and $n > 0$. Then, there exist constants $C_d, C_p > 0$, depending only on $\rho_1$ and $n$, for which one has for every $x\in \mathbb{M}$ and every $r > 0$:
$\mu(B(x,2r)) \le C_d\ \mu(B(x,r));$
and
$\int_{B(x,r)} |f - f_B|^2 d\mu \le C_p r^2 \int_{B(x,r)} \Gamma(f) d\mu,$
for every $f\in C^1( B(x,r))$.

Besides this result, the generalized curvature dimension inequality was also used to prove a Bonnet-Myers type theorem, the first of its kind in sub-Riemannian geometry.

The study of this generalized curvature dimension condition and of some of its extensions has generated quite a lot of research with my colleagues Michel Bonnefont, Nicola Garofalo, Isidro Munive and my students Bumsik Kim and Jing Wang.

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