The Geometric Heat Equation-WTF?

In summary, the conversation discusses the relationship between the "regular" heat equation and the geometric heat equation, which is applied to evolving closed curves on a manifold. The geometric heat equation involves the Laplace-Beltrami equation, Gaussian curvature, and the second fundamental form, but the specific method of converting from the "regular" heat equation to the geometric one in a general setting is still unclear. The conversation ends with a request for further clarification and help.
  • #1
the.bone
9
0
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!
 
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  • #2
Your kN is called The Laplace-Beltrami equation u can see this on google
 
  • #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?
 
  • #4
***UPDATE***

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]K=-\dfrac{R\left(X,Y,X,Y\right)}{G\left(X,Y,X,Y\right)}[/tex]

[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:

1. What is the Geometric Heat Equation?

The Geometric Heat Equation is a partial differential equation that describes the distribution of heat in a geometric object over time. It is used in mathematics and physics to model heat transfer and diffusion processes.

2. How is the Geometric Heat Equation different from other heat equations?

The Geometric Heat Equation takes into account the geometry of the object being studied, whereas other heat equations do not. This allows for more accurate modeling of heat transfer in complex shapes and structures.

3. What is the significance of the Geometric Heat Equation?

The Geometric Heat Equation has applications in various fields such as engineering, physics, and mathematics. It is used to study heat transfer in objects with complex geometries, and has also been used in image processing and computer graphics.

4. How is the Geometric Heat Equation solved?

The Geometric Heat Equation can be solved using various mathematical techniques such as separation of variables, Fourier series, and numerical methods. The method used depends on the specific problem being studied and the available tools and resources.

5. What are some practical applications of the Geometric Heat Equation?

The Geometric Heat Equation has practical applications in fields such as heat transfer analysis in engineering, thermal imaging in medicine, and image processing in computer graphics. It is also used in the study of diffusion processes and can help predict the behavior of heat in complex systems.

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