Lecture 6. Transverse Weitzenbock formula and heat equation on one-forms

Let \mathbb M be a smooth, connected manifold with dimension n+m. We assume that \mathbb M is equipped with a Riemannian foliation \mathcal{F} with bundle like metric g and totally geodesic m-dimensional leaves.

We define the canonical variation of g as the one-parameter family of Riemannian metrics:

g_{\varepsilon}=g_\mathcal{H} \oplus \frac{1}{\varepsilon }g_{\mathcal{V}}, \quad \varepsilon >0.
We now introduce some tensors and definitions that will play an important role in the sequel.

 

For Z \in \Gamma^\infty(T\mathbb M), there is a unique skew-symmetric endomorphism J_Z:\mathcal{H}_x \to \mathcal{H}_x such that for all horizontal vector fields X and Y,

g_\mathcal{H} (J_Z (X),Y)= g_\mathcal{V} (Z,T(X,Y)).
where T is the torsion tensor of \nabla. We then extend J_{Z} to be 0 on \mathcal{V}_x. If Z_1,\cdots,Z_m is a local vertical frame, the operator \sum_{l=1}^m J_{Z_l}J_{Z_l} does not depend on the choice of the frame and shall concisely be denoted by \mathbf{J}^2. For instance, if \mathbb M is a K-contact manifold equipped with the Reeb foliation, then \mathbf{J} is an almost complex structure, \mathbf{J}^2=-\mathbf{Id}_{\mathcal{H}}.
The horizontal divergence of the torsion T is the (1,1) tensor which is defined in a local horizontal frame X_1,\cdots,X_n by

\delta_\mathcal{H} T (X)=- \sum_{j=1}^n(\nabla_{X_j} T) (X_j,X), \quad X \in \Gamma^\infty(\mathbb M).
The g-adjoint of \delta_\mathcal{H}T will be denoted \delta_\mathcal{H} T^*.

In the sequel, we shall need to perform computations on one-forms. For that purpose we introduce some definitions and notations on the cotangent bundle.

We say that a one-form to be horizontal (resp. vertical) if it vanishes on the vertical bundle \mathcal{V} (resp. on the horizontal bundle \mathcal{H}). We thus have a splitting of the cotangent space

T^*_x \mathbb M= \mathcal{H}^*(x) \oplus \mathcal{V}^*(x)

The metric g_\varepsilon induces then a metric on the cotangent bundle which we still denote g_\varepsilon. By using similar notations and conventions as before we have for every \eta in T^*_x \mathbb M,

\| \eta \|^2_{\varepsilon} =\| \eta \|_\mathcal{H}^2+\varepsilon \| \eta \|_\mathcal{V}^2.

 

By using the duality given by the metric g, (1,1) tensors can also be seen as linear maps on the cotangent bundle T^* \mathbb M. More precisely, if A is a (1,1) tensor, we will still denote by A the fiberwise linear map on the cotangent bundle which is defined as the g-adjoint of the dual map of A. The same convention will be made for any (r,s) tensor.

 

We define then the horizontal Ricci curvature \mathfrak{Ric}_{\mathcal{H}} as the fiberwise symmetric linear map on one-forms such that for every smooth functions f,g,

\langle \mathfrak{Ric}_{\mathcal{H}} (df), dg \rangle=\mathbf{Ricci} (\nabla_\mathcal{H} f ,\nabla_\mathcal{H} g),
where \mathbf{Ricci} is the Ricci curvature of the connection \nabla.
If V is a horizontal vector field and \varepsilon >0, we consider the fiberwise linear map from the space of one-forms into itself which is given for \eta \in \Gamma^\infty(T^* \mathbb M) and Y \in \Gamma^\infty(T \mathbb M) by

\mathfrak{T}^\varepsilon_V \eta (Y) = \begin{cases} \frac{1}{\varepsilon} \eta (J_Y V), \quad Y \in \Gamma^\infty(\mathcal{V}) \\ -\eta (T(V,Y)), Y \in \Gamma^\infty(\mathcal{H}) \end{cases}
We observe that \mathfrak{T}^\varepsilon_V is skew-symmetric for the metric g_\varepsilon so that \nabla -\mathfrak{T}^\varepsilon is a g_\varepsilon-metric connection.

If \eta is a one-form, we define the horizontal gradient of \eta in a local frame as the (0,2) tensor

\nabla_\mathcal{H} \eta =\sum_{i=1}^n \nabla_{X_i} \eta \otimes \theta_i.

 

Similarly, we will use the notation

\mathfrak{T}^\varepsilon_\mathcal{H} \eta =\sum_{i=1}^n \mathfrak{T}^\varepsilon_{X_i} \eta \otimes \theta_i.

 

Finally, we will still denote by \Delta_\mathcal{H} the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus

\Delta_{\mathcal{H}}=-\nabla_{\mathcal{H}}^* \nabla_{\mathcal{H}}=\sum_{i=1}^n \nabla_{X_i}\nabla_{X_i} -\nabla_{\nabla_{X_i} X_i}.
For \varepsilon >0, we consider the following operator which is defined on one-forms by

\square_\varepsilon=-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)-\frac{1}{ \varepsilon}\mathbf{J}^2+\frac{1}{\varepsilon} \delta_\mathcal{H} T- \mathfrak{Ric}_{\mathcal{H}},
where the adjoint is understood with respect to the metric g_{\varepsilon}. It is easily seen that, in a local horizontal frame,
-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) =\sum_{i=1}^n (\nabla_{X_i} -\mathfrak{T}^\varepsilon_{X_i})^2 - ( \nabla_{\nabla_{X_i} X_i}- \mathfrak{T}^\varepsilon_{\nabla_{X_i} X_i}),

We can also consider the operator which is defined on one-forms by

\square_\infty:=\sum_{i=1}^n (\nabla_{X_i} -\mathfrak{T}^\infty_{X_i})^2 - ( \nabla_{\nabla_{X_i} X_i}- \mathfrak{T}^\infty_{\nabla_{X_i} X_i}) - \mathfrak{Ric}_{\mathcal{H}}

It is clear that for every smooth one-form \alpha on \mathbb M and every x \in \mathbb M the following holds

\lim_{\varepsilon \to \infty} \square_\varepsilon \alpha (x)=\square_\infty \alpha (x).

 

The following theorem that was proved in this paper is the main result of the lecture:

Theorem: Let 0 < \varepsilon \le +\infty. For every f \in C^\infty(\mathbb M), we have
d \Delta_{\mathcal{H}} f=\square_\varepsilon df.

Proof:
We only sketch the proof and refer to the original paper for the details. If Z_1,\cdots,Z_m is a local vertical frame of the leaves, we denote

\mathfrak J(\eta)=-\sum_{l=1}^mJ_{Z_l}(\iota_{Z_l}d\eta_{\mathcal V}),
where \eta_{\mathcal V} is the the projection of \eta onto the vertical cotangent bundle. It does not depend on the choice of the frame and therefore defines a globally defined tensor.
Also, let us consider the map \mathcal{T} \colon \Gamma^\infty(\wedge^2 T^*\mathbb M)\to \Gamma^\infty( T^*\mathbb M) which is given in a local coframe \theta_i \in \Gamma^\infty(\mathcal{H}^*), \nu_k \in \Gamma^\infty(\mathcal{V}^*)

\mathcal{T}(\theta_i\wedge\theta_j)=-\gamma_{ij}^l \nu_l,\quad \mathcal{T}(\theta_i\wedge\nu_k)=\mathcal{T}(\nu_k\wedge\nu_l)=0.
A direct computation shows then that

-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon) =
\Delta_{\mathcal H} +2\mathfrak J-\frac{2}{\varepsilon}\mathcal{T}\circ d+\delta_\mathcal{H} T^*-\frac{1}{\varepsilon}\delta_\mathcal{H} T+\frac{1}{\varepsilon}\mathbf{J}^2.
Thus, we just need to prove that if \square_\infty is the operator defined on one-forms by

\square_\infty=\Delta_{\mathcal H}+2\mathfrak J-\mathfrak{Ric}_{\mathcal{H}}+\delta_\mathcal{H} T^* ,
then for any f\in C^\infty(\mathbb M),

d\Delta_{\mathcal H} f=\square_\infty df.
A computation in local frame shows that

d\Delta_{\mathcal H} f- d\Delta_{\mathcal H} f = 2\mathfrak J(df) -\mathfrak{Ric}_{\mathcal{H}}(df) +\delta_\mathcal{H} T^* (df),
which completes the proof \square

 

We also have the following Bochner’s type identity.

Theorem: For any \eta \in \Gamma^\infty(T^* \mathbb M),
\frac{1}{2} \Delta_\mathcal{H} \| \eta \|_{\varepsilon}^2 -\langle \square_\varepsilon \eta , \eta \rangle_{\varepsilon}= \| \nabla_{\mathcal{H}} \eta -\mathfrak{T}^\varepsilon_{\mathcal{H}} \eta \|_{\varepsilon}^2 + \left\langle \mathfrak{Ric}_{\mathcal{H}} (\eta), \eta \right\rangle_\mathcal{H} -\left \langle \delta_\mathcal{H} T (\eta) , \eta \right\rangle_\mathcal{V} +\frac{1}{\varepsilon} \langle \mathbf{J}^2 (\eta) , \eta \rangle_\mathcal{H}.

 

We now turn to probabilistic applications.

 

We denote by (X_t)_{t\geq 0} the horizontal Brownian motion on \mathbb M. The lifetime of the process is denoted by \mathbf{e}. We assume that the metric g is complete and \mathcal{H} is bracket generating. As a consequence, one can define (P_t)_{t \ge 0} the heat semigroup associated to (X_t)_{t\geq 0} as being the semigroup generated by the self-adjoint extension of \frac{1}{2} \Delta_\mathcal{H}.

 

 

We define a process \tau^\varepsilon_t:T_{X_t}^*\mathbb{M}\rightarrow T^*_{X_0}\mathbb{M} by the formula
\tau^{\varepsilon}_t=\mathcal{M}_{t}^{\varepsilon}\Theta_{t}^{\varepsilon}, \quad t < \mathbf{e}
where the process \Theta_t^{\varepsilon}: T_{X_t}^{*}\mathbb{M}\rightarrow T_{X_0}^{*}\mathbb{M} is the stochastic parallel transport with respect to the connection \nabla -\mathfrak{T}^\varepsilon along the paths of (X_t)_{t\geq 0}. The multiplicative functional (\mathcal{M}_t^{\varepsilon})_{t\geq 0} is defined as the solution of the following ordinary differential equation

\frac{d\mathcal{M}_t^{\varepsilon}}{dt}=-\frac{1}{2}\mathcal{M}_t^{\varepsilon}\Theta_t^{\varepsilon}\left(\frac{1}{\varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+\mathfrak{Ric}_{\mathcal{H}} \right)(\Theta_t^{\varepsilon})^{-1}, ~~\mathcal{M}_0^{\varepsilon}=\mathbf{Id}.

Observe that the process \tau^\varepsilon_t:T_{X_t}^*\mathbb{M}\rightarrow T^*_{X_0}\mathbb{M} is a solution of the following covariant Stratonovitch stochastic differential equation:

d[\tau^\varepsilon_t \alpha(X_t)]=\tau^\varepsilon_t\left( \nabla_{\circ dX_t}-\mathfrak{T}_{\circ dX_t}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right)dt\right) \alpha(X_t),~~\tau_0=\mathbf{Id},
where \alpha is any smooth one-form.

 

From Gronwall’s lemma and the fact that \Theta_t^{\varepsilon} is an isometry, we easily deduce that

Lemma: Let \varepsilon >0. Assume that there exists a constant C_\varepsilon \ge 0 such that for every \alpha \in \Gamma^\infty (T^*\mathbb{M}),

\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon
Then, there exists a constant \tilde{C}_\varepsilon \ge 0 , such that for every t \ge 0,

\| \tau^\varepsilon_t \alpha(X_t) \|_\varepsilon \le e^{ \tilde{C}_\varepsilon t }\| \alpha(X_t) \|_\varepsilon

 

For x \in \mathbb M, as usual we will denote

\mathbb{P}_x=\mathbb{P} \left( \cdot \mid X_0=x \right).

Theorem: Assume that there exists a constant C_\varepsilon \ge 0 such that for every \alpha \in \Gamma^\infty (T^*\mathbb{M}),

\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon
Let \eta be a one-form on \mathbb{M} which is smooth and compactly supported. The unique solution in L^2 of the Cauchy problem:

\begin{cases} \phi (0,x)=\eta (x) \\ \frac{\partial \phi}{\partial t} =\frac{1}{2} \square_\varepsilon \phi \end{cases}
is given by

\phi(t,x)=\mathbb{E}_x \left( \tau^\varepsilon_t \eta (X_t) 1_{t <\mathbf{e}}\right) .

Proof:

This is Feynman-Kac formula.
Sketch of the proof.

It is proved in this course, that the operator

-(\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)^* (\nabla_\mathcal{H} -\mathfrak{T}_\mathcal{H}^\varepsilon)

is essentially self-adjoint on the space of smooth and compactly supported one-forms. Thus, from the assumption, \frac{1}{2} \square_\varepsilon is the generator of a bounded semigroup Q_t^\varepsilon in L^2 that uniquely solves the above Cauchy problem.

 

From the Bochner’s identity, one has

\frac{1}{2} \Delta_\mathcal{H} \| \eta \|_{\varepsilon}^2 -\langle \square_\varepsilon \eta , \eta \rangle_{\varepsilon} \ge -C_\varepsilon \| \eta \|^2_\varepsilon.

From Shigekawa (L^p contraction for vector valued semigroups), this implies the a priori pointwise bound

\| Q_t^\varepsilon \eta \|^2_\varepsilon \le e^{2C_\varepsilon t} P_t (\| \eta\|^2_\varepsilon).

We now claim that the process

N_s=\tau^\varepsilon_s(Q_{T-s}^\varepsilon \eta) (X_s)1_{T <\mathbf{e}}, \quad 0 \le s \le T,

is a local martingale. Indeed, from Ito’s formula and the definition of \tau^\varepsilon, we have

dN_s=\tau^\varepsilon_s \left( \nabla_{\circ dX_s}-\mathfrak{T}_{\circ dX_s}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+\mathfrak{Ric}_{\mathcal{H}}\right)ds\right) (Q_{T-s}^\varepsilon \eta) (X_s)
+\tau^\varepsilon_s \frac{d}{ds} (Q_{T-s}^\varepsilon \eta) (X_s) ds.
We now  conclude from the fact that the bounded variation part of

\tau^\varepsilon_s \left( \nabla_{\circ dX_s}-\mathfrak{T}_{\circ dX_s}^{\varepsilon}-\frac{1}{2} \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right)ds\right)(Q_{T-s}^\varepsilon \eta) (X_s)

 

is given by \frac{1}{2} \tau^\varepsilon_s \square_\varepsilon (Q_{T-s}^\varepsilon \eta) (X_s) (X_s) ds.

Form the previous estimates, we conclude that N is a martingale \square

 

Corollary:  Let \varepsilon >0. Assume that there exists a constant C_\varepsilon \ge 0 such that for every \alpha \in \Gamma^\infty (T^*\mathbb{M}),

\left \langle \left( \frac{1}{ \varepsilon}\mathbf{J}^2-\frac{1}{\varepsilon} \delta_\mathcal{H} T+ \mathfrak{Ric}_{\mathcal{H}}\right) \alpha , \alpha \right\rangle_\varepsilon \ge -C_\varepsilon \| \alpha \|^2_\varepsilon

Then, for f \in C^\infty_0(\mathbb M), and t \ge 0

dP_tf (x)=\mathbb{E}_x ( \tau^\varepsilon_t df (X_t) 1_{t <\mathbf{e}})
As a consequence, \mathbb{P}_x(\mathbf{e}=+\infty)=1.

Proof: Let \phi(t,x)=dP_t f(x).
We have

\frac{\partial \phi}{\partial t} =d\Delta_{\mathcal{H}} P_t f= \frac{1}{2} \square_\varepsilon \phi.
Thus, from the previous theorem

dP_tf (x)=\mathbb{E}_x ( \tau^\varepsilon_t df (X_t) 1_{t <\mathbf{e}})
This representation implies the bound

\| dP_t f (x) \|_\varepsilon \le e^{\frac{C_\varepsilon}{2} t} (P_t \| df \|_\varepsilon) (x).

It is well-known that this type of gradient bound implies the stochastic completeness of P_t. More precisely, we can adapt an argument of Bakry. Let f,g \in C^\infty_0(\mathbb M), we have

\int_{\mathbb M} (P_t f -f) g d\mu  = \int_0^t \int_{\mathbb M}\left( \frac{\partial}{\partial s} P_s f \right) g d\mu ds
 =\frac{1}{2} \int_0^t \int_{\mathbb M}\left(\Delta_{\mathcal{H}} P_s f \right) g d\mu ds
- \int_0^t \int_{\mathbb M} \langle \nabla P_s f , \nabla g\rangle_\mathcal{H} d\mu ds.

By means of Cauchy-Schwarz inequality we
find
\left| \int_{\mathbb M} (P_t f -f) g d\mu \right| \le 2 \left(\int_0^t e^{\frac{C_\varepsilon}{2} s} ds\right) \| df \|_{\varepsilon,\infty} \int_{\mathbb M}\| \nabla_\mathcal{H} g \|^{\frac{1}{2}}d\mu.

We now apply the previous inequality with f = h_n, where h_n is an increasing sequence in  C_0^\infty(\mathbb M), 0 \le h_n \le 1, such that h_n\nearrow 1 on \mathbb{M}, and ||\Gamma (h_n)||_{\infty} \to 0, as n\to \infty.

By monotone convergence theorem we have P_t h_k(x)\nearrow P_t 1(x) for every x\in \mathbb M. We conclude that the
left-hand side converges to \int_{\mathbb M} (P_t 1 -1) g d\mu. Since the right-hand side converges to zero, we reach the conclusion

\int_{\mathbb M} (P_t 1 -1) g d\mu=0,\ \ \ g\in C^\infty_0(\mathbb M).
Since it is true for every g\in C^\infty_0(\mathbb M), it follows that P_t 1 =1.

\square

 

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