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The Geometric Heat Equation-WTF?

  1. Dec 1, 2005 #1
    The Geometric Heat Equation--WTF??

    I need some help getting from point A to B. Let's say we have the plain ol' heat equation
    [tex]u_t=\Delta u[/tex]
    where the [tex]u=u\left(x,t\right)[/tex], and that's all good. Then, we also have the so-called geometric heat equation
    [tex]\dfrac{\partial F}{\partial t}=kN[/tex]
    where [tex]F:\mathcal{M}\rightarrow\mathcal{M}'[/tex] is a smooth map between btween Riemannian manifolds, [tex]k[/tex] the curvature, and [tex]N[/tex] the unit normal vector. Intuitivly, I can see how these are the same, but I cannot seem to work out how to go from [tex]\Delta[/tex] to [tex]kN[/tex].
    Of course, I can explain my woes further if anyone has any ideas, so let me know what you all think. Thanks!
  2. jcsd
  3. Dec 6, 2005 #2
    Your kN is called The Laplace-Beltrami equation u can see this on google
  4. Dec 6, 2005 #3
    Well, I don't think so...

    As I understand it, the Laplace-Beltrami equation is merely a generalization of the Laplacian as it applies to taking the laplacian of a k-form on a manifold.

    What I am after is how to perfom the specific acrobatics to get from the "regular" heat equation to the geometric one, but in a general sort of way. That is, the geometric one is typically applied to evolving closed curves in the plane, where it becomes obvious how to get from one to the other by simply parameterizing the curve by it's arc length. But let's assume that we have the more general situation of a closed (Riemannain) submanifold evolving on/within another (Riemannian) manifold. Then what would the so-called geometric heat equation look like?
  5. Dec 22, 2005 #4

    OK, so here's where I'm at. We can, with some trickery, view the action of the laplacian on a function on a Riemannian manifold as

    [tex]\Delta f = \dfrac{1}{\sqrt{\det g}}\dfrac{\partial}{\partial x^j}\left(g^{ij}\sqrt{\det g}\dfrac{\partial}{\partial x^i}f\right)[/tex]

    and we also can calculate the Gaussian curvature via


    [tex]=-\dfrac{\left<X\otimes Y\otimes X\otimes Y,R\right>}{g_{ij}X^iX^jg_{kl}Y^kY^l-\left(g_{ij}X^iY^j\right)^2}[/tex]

    which, for the "common" case of a two dimensional manifold embedded in [tex]\mathbb{R}^3[/tex] reduces to

    [tex]K=-\dfrac{R_{1212}}{\det g}[/tex]

    which would allow one to compare this to [tex]\Delta f =Kn[/tex], in theory. However, I am still at a loss as to what I am getting here. Specifically, the question of what [tex]n[/tex] is even supposed to be in a general setting is a bit of a mystery to me... I have come across the relation

    [tex]h_{ij}=-\left<\dfrac{\partial^2}{\partial x^i\partial x^j},n\right>[/tex]

    a few times in various literature, where [tex]h_{ij}[/tex] is the second fundamental form, but as well as the fact that I cannot figure out what this pairing is, I (still) cannot see how to state [tex]n[/tex] in general. Would I need to specify a Darboux frame, and embed my manifold in [tex]\mathbb{R}^{n+1}[/tex] so that I may view the orginal manifold as a hypersurface or something? I'm rather lost at the moment, so any help would be greatly appreciated...
    Last edited: Dec 23, 2005
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