Lecture 18 (revised): The Laplace-Beltrami Operator

In this lecture we take a close look at the Laplacian, and its generalization to curved spaces via the Laplace-Beltrami operator. The Laplacian is one of the most fundamental objects in geometry and physics, and which plays a major role in algorithms. Here we consider several perspectives to build up some basic intuition about what the Laplacian is, and what it means.

3 thoughts on “Lecture 18 (revised): The Laplace-Beltrami Operator”

  1. By the way, there’s a great talk by Peyman Milanfar (Director of Computational Imaging at Google) that touches on many of the same themes, but with applications to imaging rather than geometry—including some examples of how the Laplacian is used in Google’s camera hardware. Check it out here.

  2. Why is $\Delta u$ the Hessian of Dirichlet Energy? Why isn’t it the gradient of Dirichlet Energy? Do we require the domain of a harmonic function to be compact and connected to apply maxima’s principle?

    1. A really simple analogy would be the ordinary function

      \[ f: \mathbb{R} \to \mathbb{R}; x \mapsto \frac{1}{2} ax^2, \]

      where \(a \in \mathbb{R}\) is a constant. The gradient of this function is \(ax\), and the second derivative or Hessian is just \(a\). Likewise, consider the function

      \[ f: \mathbb{R}^2 \to \mathbb{R}; x \mapsto \frac{1}{2} x^T A x, \]

      where \(A \in \mathbb{R}^{n \times n}\) is now a symmetric matrix. Then we have a gradient

      \[
      \nabla f(x) = Ax,
      \]

      and a Hessian

      \[
      \nabla^2 f(x) = A,
      \]

      i.e., the Hessian is again just the constant part. The Dirichlet energy just continues this story: think of the Dirichlet energy \(E_D\) as a map from functions \(u: M \to \mathbb{R}\) to a real number \(\mathbb{R}\). We can express Dirichlet energy as a quadratic form

      \[
      E_D(u) = \frac{1}{2}\langle\!\langle \Delta u, u \rangle\!\rangle,
      \]

      where the double brackets denote the standard \(L^2\) inner product on functions, i.e.,

      \[
      \langle\!\langle u, v \rangle\!\rangle = \int_M u(x) v(x)\ dx
      \]

      for any two functions \(u,v: M \to \mathbb{R}\). Then you can write the gradient of Dirichlet energy as

      \[
      \nabla E_D(u) = \Delta u
      \]

      and the Hessian of Dirichlet energy as

      \[
      \nabla^2 E_D(u) = \Delta,
      \]

      i.e., just as in the first two examples, the Hessian captures the constant part of the quadratic form, which in this case is just the Laplace operator.

      I am intentionally sweeping a lot under the rug here by not being more careful with my discussion of what kinds of functions are allowed, i.e., what the actual domain of the Dirichlet energy is. But let’s consider the original definition of Dirichlet energy (as seen in the slides),

      \[
      E_D(u) := \int_M |\nabla u|^2\ dA.
      \]

      In order for this definition to be meaningful, we know that:

      • the function \(u\) must have a well-defined gradient \(\nabla u\), and
      • the integral of the squared norm of this gradient must also be well-defined.

      Because these same conditions come up frequently, there is a special name for the set of all such functions: the Sobolev space \(W^{1,2}(M;\mathbb{R})\). The first number in the superscript indicates that the function has at least one derivative, and the second number indicates that the square of that derivative (and all lower-order derivatives, including the function itself) is integrable. The \(M\) inside the parentheses indicates that \(M\) is the domain of these functions; likewise \(\mathbb{R}\) indicates that they take values in \(\mathbb{R}\). This so-called “function space” comes up often enough (and has some important special properties) that it is also denoted by just \(H^1(M; \mathbb{R}\), and often the arguments \(M\) and \(\mathbb{R}\) are dropped if the meaning is clear from context. You can get a more detailed overview on the Wikipedia page. Using this notation, we can write the Dirichlet energy as a map

      \[ E_D: H^1(M;\mathbb{R}) \to \mathbb{R}, \]

      i.e., as a function whose input is a real-valued function on \(M\) with square-integrable first derivative (gradient), and which produces a real value as output. Notice that nothing needs to be assumed here about compactness of \(M\): for instance, even though a function supported on all of the plane \(M = \mathbb{R}^2\) may not have finite Dirichlet energy, we have already excluded such functions from our set.

      You also ask whether the domain of a harmonic function needs to be compact and connected in order for the maximum principle to apply. I guess formally the answer is “yes” because the standard statement about the maximum principle has something to do with the maximum (and minimum) values being achieved on the boundary. But you can of course have functions satisfying the Laplace equation \(\Delta u = 0\) on a noncompact set, for instance, the function \(f(x,y) = x^2 – y^2\) over the whole Euclidean plane \(\mathbb{R}^2\), and this function will have no local extrema. If you want to dig deeper there are some subtleties not covered in lecture; the Wikipedia article is again a decent starting place.

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