Lecture 11. Laplace-Beltrami operators on Rn

In this lecture we define Riemannian structures and corresponding LaplaceBeltrami operators. We first study Riemannian structures on Rn to avoid technicalities in the presentation of the main ideas and then, in a later lecture, will extend our results to the manifold case.

We start with the following basic definition:

Definition: A Riemannian structure on \mathbb{R}^n is a smooth map g from \mathbb{R}^n to the set of symmetric positive matrices.

In other words, a Riemannian structure induces at each point x \in \mathbb{R}^n an inner product g_x, and the dependence x \rightarrow g_x is asked to be smooth.

A natural way to define Riemannian structures, is to start from a family of smooth vector fields V_1,\cdots, V_n such that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n. It is then easily seen that there is a unique Riemannian structure on \mathbb{R}^n that makes for x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) an orthonormal basis.

Conversely, given a Riemannian structure on \mathbb{R}^n, it is possible to find smooth vector fields V_1,\cdots, V_n on \mathbb{R}^n such that for x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) an orthonormal basis for this Riemannian structure.

In this course, we shall mainly deal with such a point of view on Riemannian structures and use as much as possible the language of vector fields. This point of view is not restrictive and will allow more easy extensions to the sub-Riemannian case in a later part of the course.

Let us consider a family of smooth vector fields V_1,\cdots, V_n such that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n. Without loss of generality we may assume that \mathbf{det} ( V_1(x),\cdots, V_n(x)) > 0. Our goal is to associate to this Riemannian structure a canonical diffusion operator.

As a first step, we associate with the vector fields V_1,\cdots, V_n a natural Borel measure \mu which is the measure with density d\mu =\frac{1}{ \mathbf{det} ( V_1(x),\cdots, V_n(x))} dx with respect to the Lebesgue measure. This is the so-called Riemannian measure. The diffusion operator we want to consider shall be symmetric with respect to this measure.

Remark: Let (U_1,\cdots,U_n) be another system of smooth vector fields on \mathbb{R}^n such that for every x \in \mathbb{R}^n, (U_1(x),\cdots,U_n(x)) is an orthonormal basis with respect to the inner product g_x. The systems of vector fields (U_1,\cdots,U_n) and (V_1,\cdots,V_n) are related one to each other through an orthogonal mapping. This implies that |\mathbf{det} ( V_1(x),\cdots, V_n(x))|=| \mathbf{det} ( U_1(x),\cdots, U_n(x))|. In other words, the Riemannian measure \mu only depends on the Riemannian structure g.

Due to the fact that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n, we may find smooth functions \omega^{k}_{ij}‘s on $\mathbb{R}^n$ such that [V_i,V_j]=\sum_{k=1}^n \omega_{ij}^k V_k. Those functions are called the structure constants of the Riemannian structure. Every relevant geometric quantities may be expressed in terms of these functions. Of course, these functions depend on the choice of the vector fields $V_i$’s and thus are not Riemannian invariants, but several combinations of them, like curvature quantities, are Riemannian invariants.

The following proposition expresses the formal adjoint with respect to the Lebesgue measure of the vector field V_i which is seen as an operator acting on the spave of smooth and compactly supported functions.

Proposition: If f,g\in \mathcal{C}_c(\mathbb{R}^n, \mathbb{R}) are smooth and compactly supported functions, then we have \int_{\mathbb{R}^n} (V_i f ) g d\mu =\int_{\mathbb{R}^n} f (V_i^* g) d\mu, where V_i^*=-V_i+ \sum_{k=1}^n \omega_{ik}^k. 

Proof: Let us denote V_i=\sum_{j=1}^n v_i^j \frac{\partial}{\partial x_j}, and m(x)= \mathbf{det} ( V_1(x),\cdots, V_n(x)). We have \int_{\mathbb{R}^n} (V_i f ) g d\mu =\int_{\mathbb{R}^n}\sum_{j=1}^n v_i^j \frac{\partial f}{\partial x_j} g \frac{dx}{m} =-\sum_{j=1}^n\int_{\mathbb{R}^n} f \left( m \frac{\partial}{\partial x_j} \frac{1}{m} g v_i^j \right) d\mu. We now compute \sum_{j=1}^n m \frac{\partial}{\partial x_j} \frac{1}{m} v_i^j=-\sum_{j=1}^n v_i^j \frac{1}{m} \frac{\partial m}{\partial x_j} +\sum_{j=1}^n\frac{\partial v_i^j}{\partial x_j}. We then observe that \frac{\partial m}{\partial x_j} = \frac{\partial}{\partial x_j} \mathbf{det} ( V_1(x),\cdots, V_n(x)) =\sum_{k=1}^n \mathbf{det} \left( V_1(x),\cdots,\frac{\partial V_k}{\partial x_j}(x),\cdots, V_n(x)\right).
Thus, we obtain
-\sum_{j=1}^n v_i^j \frac{\partial m}{\partial x_j} +\sum_{j=1}^n m \frac{\partial v_i^j}{\partial x_j} = \sum_{k=1}^n \mathbf{det} \left( V_1(x),\cdots,-\sum_{j=1}^n v_i^j \frac{\partial V_k}{\partial x_j}(x),\cdots, V_n(x)\right)
+\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,\frac{\partial v_i^j}{\partial x_j} V_k(x) ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,[V_i,V_k](x) ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,\sum_{j=1}^n \omega_{ik}^j(x) V_j(x) ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n \omega_{ik}^k (x)\mathbf{det} \left( V_1(x),\cdots,V_k(x) ,\cdots, V_n(x)\right)=-m\sum_{k=1}^n \omega_{ik}^k. \square

With this integration by parts formula in hands we are led to the following natural definition
Definition: The diffusion operator L =-\sum_{i=1}^n V_i^* V_i =\sum_{i=1}^n V_i^2 -\sum_{i,k}^n \omega_{ik}^k V_i is called the Laplace-Beltrami operator associated with the Riemannian structure g

The following straightforward properties of L are let as an exercise to the reader:

  • L is an elliptic operator;
  • The Riemannian measure \mu is invariant for L;
  • The operator L is symmetric with respect to \mu.

Exercise: Let (U_1,\cdots,U_n) be another system of smooth vector fields on \mathbb{R}^n such that for every x \in \mathbb{R}^n, (U_1(x),\cdots,U_n(x)) is an orthonormal basis with respect to the inner product g_x. Show that \sum_{i=1}^n V_i^* V_i=\sum_{i=1}^n U_i^* U_i. In other words, the Laplace-Beltrami operator is a Riemannian invariant: It only depends on the Riemannian structure g.

In order to apply the diffusion semigroup theory developed in the first lectures and construct without ambiguity the semigroup associated to the Laplace-Beltrami operator L, we need to know if L is essentially self-adjoint. Interestingly, this property of essential self-adjointness is clodely related to a metric property of the underlying Riemannian structure: The completeness of the associated distance.

Given an absolutely continuous curve \gamma: [0,T] \rightarrow \mathbb{R}^n, we define its Riemannian length by L_g (\gamma)=\int_0^T \sqrt{g_{\gamma(s)} (\gamma'(s),\gamma'(s) )} ds. If x,y \in \mathbb{R}^n, let us denote by \mathcal{C}(x,y) the set of absolutely continuous curves \gamma: [0,1] \rightarrow \mathbb{R}^n such that \gamma(0)=x, \gamma(1)=y. The Riemannian distance between x and y is defined by d(x,y)=\inf_{\gamma \in \mathcal{C}(x,y)} L_g(\gamma). By using reparametrization, we may define the Riemannian distance in a equivalent way by using the notion of sub-unit curve. Let \gamma: [0,T] \to \mathbb{R}^n be an absolutely continuous curve. Since the vector fields V_1,\cdots,V_n‘s form a basis of \mathbb{R}^n at each point, we may find continuous functions \alpha_i:[0,T] \to \mathbb{R}^n such that \gamma'(t)=\sum_{i=1}^n \alpha_i(t) V_i (\gamma(t)). The curve \gamma is then said to be sub-unit if for almost every t \in [0,T], \sum_{i=1}^n \alpha_i(t)^2 \le 1. By using reparametrization, it is easily seen that for x,y \in \mathbb{R}^n, d(x,y)=\inf \left\{ T \text{ such that there exists a sub-unit curve } \gamma:[0,T] \to \mathbb{R}^n, \gamma(0)=x, \gamma(1)=y \right\}.

Exercise: Let \theta_1, \cdots , \theta_n \in \mathbb{R}. Show that for every x \in \mathbb{R}^n, d \left(x, e^{\sum_{i=1}^n \theta_i V_i } x \right) \le \sum_{i=1}^n \theta_i^2. 

An important fact is that d hence defined is indeed a distance that induces the usual topology of \mathbb{R}^n.

Definition: The function d defined above is a distance that induces the usual topology of \mathbb{R}^n.

Proof: Since any curve can be parametrized backwards and forwards, we have d(x,y)=d(y,x). The triangle inequality is easily proved by using juxtaposition of curves. Plainly d(x,x)=0, so it remains to prove that if x \neq y then d(x,y) > 0. Let x,y \in \mathbb{R}^n such that x \neq y. Let us denote R=\| x-y \|. The closed Euclidean ball \bar{B}_e (x,R) is compact, therefore there exist two constants \alpha, \beta > 0 such that for every z \in \bar{B}_e (x,R) and u \in \mathbb{R}^n, \alpha^2 \| u \|^2 \le g_z(u,u) \le \beta^2 \| u \|^2. Let now \gamma:[0,1]\to \mathbb{R}^n be an absolutely continuous curve such that \gamma(0)=x, \gamma(1)=y. Let \tau =\inf \{ t, \| \gamma(t) \|=R \}.
We have
L_g(\gamma)
=\int_0^1 \sqrt{g_{\gamma(s)} (\gamma'(s),\gamma'(s) )} ds
\ge \int_0^\tau \sqrt{g_{\gamma(s)} (\gamma'(s),\gamma'(s) )} ds
\ge \alpha \int_0^\tau \| \gamma'(s) \| ds
\ge \alpha \int_0^\tau \| \gamma'(s) \| ds
\ge \alpha \| \gamma(\tau) - \gamma (0) \|
\ge \alpha \| x -y \|.
As a consequence, we deduce that d(x,y) \ge \alpha^2 \| x -y \| > 0. Therefore d is indeed a distance. Moreover, it is shown as above that for every z \in \mathbb{R}^n, R> 0, there are constants C_1,C_2> 0 such that for every x,y \in \bar{B}_e (z,R), C_1 \| x-y \| \le d(x,y) \le C_2 \| x-y \|. This implies that d induces the usual topology of \mathbb{R}^n \square

As shown in the following proposition, the distance d is intrinsically associated to the Laplace-Beltrami operator.

Proposition: For x,y \in \mathbb{R}^n, we have d(x,y) =\sup \{ | f(x)-f(y) |, f \in \mathcal{C}_c^\infty(\mathbb{R}^n,\mathbb{R}), \| \Gamma(f,f) \|_\infty \le 1 \}. 

Proof: Let x,y \in \mathbb{R}^n. We denote \delta (x,y)=\sup \{ | f(x)-f(y) |, f \in \mathcal{C}_c^\infty(\mathbb{R}^n,\mathbb{R}), \| \Gamma(f,f) \|_\infty \le 1 \}. Let \gamma: [0,T] \to \mathbb{R} be a sub-unit curve such that \gamma(0)=x, \gamma(T)=x. We can find \alpha_1,\cdots,\alpha_n:[0,T] \to \mathbb{R}^n such that \gamma'(t)=\sum_{i=1}^n V_i (\gamma(t)) \alpha_i (t), and \sum_{i=1}^n \alpha_i^2 \le 1. Let now f\in \mathcal{C}^\infty(\mathbb{R}^n,\mathbb{R}), \| \Gamma(f,f) \|_\infty \le 1. From the change of variable formula we have, f(\gamma(T))= f(\gamma(0)) + \sum_{i=1}^n \int_0^T V_i f(\gamma(s)) \alpha_i(s) ds. Therefore, from Cauchy-Schwarz inequality, \left| f(y)-f(x) \right| \le T. As a consequence \delta (x,y) \le d(x,y).

We now prove the converse inequality which is trickier. The idea is to use the function f(y)=d(x,y) that satisfies | f(x) -f(y)|=d(x,y) and "\Gamma(f,f) =1". However, giving a precise meaning to \Gamma(f,f) =1 is not so easy, because it turns out that f is not differentiable at x. It suggests to use an approximation of the identity to regularize f and avoid the discussion of this differentiability issue. More precisely, fix x_o ,y_o\in \mathbb{R}^n, and for N \ge 1, consider the function \Psi_N(y)= \eta \left( \int_{\mathbb{R}^n} \rho_N (t) d(x_o,y-t) dt \right), where \rho \in \mathcal{C}_c (\mathbb{R}^n,\mathbb{R}), \rho \ge 0, \int_{\mathbb{R}^n} \rho=1, \rho_N(t)=N^n \rho(N t) and \eta \in \mathcal{C}_c (\mathbb{R}^n,\mathbb{R}), \eta \ge 0, has the property that \| \eta ' \|_\infty \le 1 and \eta (u)=u for u \in [0,1+d(x_o,y_o)]. Since for any \theta \in \mathbb{R}^{n}, \| \theta \|=1, t \ge 0 | d(x_o,y)-d\left(x_o, e^{t \sum_{i=1}^n \theta_i V_i} y\right)| \le d\left(y, e^{t \sum_{i=1}^n \theta_i V_i} y\right) \le t, it is easy to see that \Gamma(\Psi_N,\Psi_N) \le 1+\frac{C}{N}, for some constant > 0. Hence \delta(x_o,y_o) \ge \lim \inf_{N \to +\infty} | \Psi_N(y_o) -\Psi_N(x_o) |=d(x_o,y_o) \square

The following theorem is known as the HopfRinow theorem, it provides a necessary and sufficient condition for the completeness of the metric space (\mathbb{R}^n,d).

Proposition: The metric space (\mathbb{R}^n,d) is complete (i.e. Cauchy sequences are convergent) if and only the compact sets are the closed and bounded sets.

Proof: It is clear that if closed and bounded sets are compact then the metric space (\mathbb{R}^n,d) is complete; It comes from the fact that Cauchy sequence are convergent if and only if they have at least one cluster value. So, we need to prove that closed and bounded sets for the distance d are compact provided that (\mathbb{R}^n,d) is complete. To check this, it is enough to prove that closed balls are compact. Let x \in \mathbb{R}^n. Observe that if the closed ball \bar{B}(x, r) is compact for some r> 0, then \bar{B}(x, \rho) is closed for any \rho < r. Define R=\sup \{ r> 0, \bar{B}(x, r) \text{ is compact } \}. Since d induces the usual topology of \mathbb{R}^n, R>0. Let us assume that R < +\infty and let us show that it leads to a contradiction. We first show that \bar{B}(x, R) is compact. Since (\mathbb{R}^n,d) is assumed to be complete, it suffices to prove that \bar{B}(x, R) is totally bounded: That is, for every \varepsilon > 0 there is a finite set S_\varepsilon such that every point of \bar{B}(x, R) belongs to a \varepsilon-neighborhood of S_\varepsilon.

So, let \varepsilon > 0 small enough. By definition of R, the ball \bar{B}(x, R-\varepsilon / 4) is compact; It is therefore totally bounded. We can find a finite set S=\{ y_1,\cdots,y_N\} such that every point of \bar{B}(x, R-\varepsilon / 4) lies in a \varepsilon / 2-neighborhood of S. Let now y \in \bar{B}(x, R). We claim that there exists y' \in \bar{B}(x, R-\varepsilon / 4) such that d(y,y') \le \varepsilon /2 . If y \in \bar{B}(x, R-\varepsilon / 4), there is nothing to prove, we may therefore assume that y \notin \bar{B}(x, R-\varepsilon / 4). Consider then a sub-unit curve \gamma: [0,R+\varepsilon / 4] such that \gamma(0)=x, \gamma(R+\varepsilon/4)=y. Let \tau =\inf \{t, \gamma(t) \notin \bar{B}(x, R-\varepsilon / 4) \}. We have \tau \ge R-\varepsilon / 4. On the other hand, d(\gamma(\tau), \gamma(R+\varepsilon/4)) \le R+\varepsilon/4 -\tau. As a consequence, d(\gamma(\tau),y) \le \varepsilon /2. In every case, there exists therefore y' \in \bar{B}(x, R-\varepsilon / 4) such that d(y,y') \le \varepsilon /2 . We may then pick y_k in S such that d(y_k,y') \le \varepsilon / 2. From the triangle inequality, we have d(y,y_k) \le \varepsilon. So, at the end, it turns out that every point of \bar{B}(x, R) lies in a \varepsilon-neighborhood of S. This shows that \bar{B}(x, R) is totally bounded and therefore compact because (\mathbb{R}^n,d) is assumed to be complete. Actually, the previous argument shows more, it shows that if every point of \bar{B}(x, R) lies in a \varepsilon /2-neighborhood of a finite S, then every point of \bar{B}(x, R+\varepsilon/4) will lie \varepsilon-neighborhood of S, so that the ball \bar{B}(x, R+\varepsilon/4) is also compact. This contradicts the fact the definition of R. Therefore every closed ball is a compact set, due to the arbitrariness of x \square

Checking that the metric space (\mathbb{R}^n,d) is complete is not always easy in concrete situations. From the Hopf-Rinow theorem, it suffices to prove that the closed balls are compact. The following proposition is therefore useful.

Proposition: Suppose that the vector fields V_1,\cdots,V_n‘s have globally Lipschitz coefficients . Then the closed ball \bar B(x,R) is compact for every x \in \mathbb{R}^n and R > 0. As a consequence the metric space (\mathbb{R}^n,d) is complete. 
Proof: By the hypothesis on the V_j‘s there exists a constant M > 0 such
that \|V(x)\|=\left(\sum_{j=1}^n \|V_j(x)\|^2\right)^\frac{1}{2}\leq M(1+\|x\|) for any x\in\mathbb{R}^n. Fix x_o,y\in\mathbb{R}^n and let \gamma:[0,T]\to\mathbb{R}^n, be a sub-unit curve such that \gamma(0)=x_o, \gamma(T)=y. Letting y(t)=\|\gamma(t)\|^2 we obtain y'(t)=2\leq 2 \|\gamma(t)\|\|\gamma'(t)\| \leq 2\|\gamma(t)\|\|V(\gamma(t))\|. We infer that y'(t)\leq C\left(\sqrt{y(t)}+y(t) \right), for some C>0 depending only on M. Integrating the latter inequality one has \|\gamma(t)\| \leq (A+\| x_o \|) e^{\frac{C}{2}T},\quad t\in [0,T]. for some constant A >0. The previous estimate shows in particular, that B(x_o,R)\subset B_e(0, (A+\| x_o \|) e^{CR}). We conclude that \bar B(x_o,R) is Euclidean compact. Since the metric and the Euclidean topology coincide, it is also d-compact \square

Completeness of the metric space (\mathbb{R}^n,d) is related to the essential self-adjointness of the Laplace-Beltrami operator.

Theorem: If the metric space (\mathbb{R}^n,d) is complete, then the Laplace-Beltrami operator L is essentially self-adjoint.

Proof: We know that if there exists an increasing sequence h_n\in C_c(\mathbb{R}^n,\mathbb{R}) such that h_n\nearrow 1 on \mathbb{R}^n, and ||\Gamma(h_n,h_n)||_{\infty} \to 0, as n\to \infty, then the operator L is essentially self-adjoint. We are therefore reduced to prove the existence of such a sequence. Let us fix a base point x_0\in\mathbb{R}^n. We can find an exhaustion function \rho\in C^\infty(\mathbb{R}^n,\mathbb{R}) such that |\rho - d(x_0,\cdot)| \le L,\ \ \ \ \ \ |\Gamma(\rho,\rho)|\le L \ \ \text{on } \mathbb{R}^n. By the completeness of (\mathbb{R}^n,d) and the Hopf-Rinow theorem, the level sets \Omega_s = \{x\in \mathbb{M} \mid \rho(x) < s\} are relatively compact and, furthermore, \Omega_s \nearrow \mathbb{R}^n as s\to \infty. We now pick an increasing sequence of functions \phi_n\in C^\infty([0,\infty)) such that \phi_n\equiv 1 on [0,n], \phi_n \equiv 0 outside [0,2n], and |\phi_n'|\le \frac{2}{n}. If we set h_n(x) = \phi_n(\rho(x)), then we have h_n\in C_c(\mathbb{R}^n,\mathbb{R}), h_n\nearrow1 on \mathbb{R}^n as n\to \infty, and ||\Gamma(h_n,h_n)||_{\infty} \le \frac{2L}{n} \square

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