# PDE on manifolds

I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE on manifolds. I am a topologist (homotopy theorist) by training so i prefer things to be coordinate free, but this may not be possible. For example something relating various notions of curvature to PDE, or something on viewing the PDE globally in terms of acting on sections would be great.

If this isn't the best forum my apologies.

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There are some reasonable answers below, but unless you're willing to do a lot of work, it might be best to choose a topic that builds on what you're learning in the course. Any chance you want to describe what will be done in the course? –  Deane Yang Feb 9 '10 at 1:18

There are lots of possible answers to your question, but maybe here are some ideas. They aren't papers, but good projects.

• Method of Characteristics in First Order Nonlinear PDE can be interpreted very cleanly using contact topology and symplectic forms. This frees one up from coordinates, but you can then use the geometry to write down the full-blown Hamilton-Jacobi equations. See Vladimir Arnold's "Lectures on Partial Differential Equations" Chapter 2. In general a lot of dynamical systems problems can be recast completely in differential form theoretic notation. For a physics perspective Jose and Saletan's "Classical Dynamics: A Contemporary Approach" has some of this.

• Depending on how much you've done, one can prove the Hodge Decomposition Theorem using basic Sobolev space theory, Lax-Milgram and Fredholm Alternative. This isn't coordinate-independent per se, but just uses general functional-analytic machinery. We did this in a PDE class recently, and I only have my notes as a reference, but Griffiths and Harris's "Principles of Algebraic Geometry" seems to do the proof starting on page 84.

• You could also look at Nash's original paper on his embedding theorem, it basically reduces to a fix-point problem. However, this is necessarily coordinate-driven.

Good luck.

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Gage and Hamilton's paper on curvature flow (curve straightening) for curves in $\mathbb R^2$ could be nice to present.

MR0840401 (87m:53003)

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I think curvature flow is a great example of a dynamical system (with good practical applications), but I'm not sure it really answers the question. Although the thing that evolves is (some representation of) a manifold, the space it lives on is just Euclidean space...right? –  Darsh Ranjan Feb 9 '10 at 4:33
Where you want to think this DE exists is to some extent a matter of choice. The curve is in Euclidean space but the PDE is for time-dependant maps $S^1 \to \mathbb R^2$ so you could say the manifold is $S^1$. –  Ryan Budney Feb 9 '10 at 6:51
Good point. I think I confused myself earlier. I retract my criticism. –  Darsh Ranjan Feb 9 '10 at 7:49

I agree with your views concerning coordinate-freeness. Furthermore I find the work of Richard Melrose very inspiring since he is - although clearly dealing with pde and index theory - at every step concerned with explicit coordinate-invariance of the statements. Thus his stuff can be clearly understood agreed upon by a differential topologist.
Here ist his homepage, look yourself:
http://www-math.mit.edu/~rbm/
Look for example at his blow-up explanations or the stuff on Fourier transformation and pseudodifferential operators from a differential topologist's perspective.

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Dr.Melrose has a wealth of extraordinary lecture notes indeed on various subjects in modern analysis at his website-any mathematics student serious about analysis should bookmark them! –  The Mathemagician Jan 15 '12 at 4:45

To make things coordinate-free, it is sufficient to reformulate differential equations in the form that makes use of exterior derivatives and exterior products of differential forms. Any set of differential equations can be cast into this form, the only subtlety being that it may require an infinite collection of differential forms to be introduced. As topologist you may be aware of Sullivan's work "Infinitesimal computations in topology" in which a sort of such equations were studied. Though to do real PDE it is necessary to work with zero-degree forms too, which he did not. Such equations have applications in physics, e.g. you may write the equations describing black-hole without referring to any coordinates.

The typical system has the form $d W^A=F^A(W)$, where $W^A$ is a set of some forms valued in some linear spaces, $W^A$ do not necessary have the same degree, $F^A(W)$ expands only in terms of exterior products of $W^A$ with constant coefficients.

As an example, take one-forms $\Omega^I$, $F^I=f^I_{JK}\Omega^I\Omega^K$, then $d\Omega^I=f^I_{JK}\Omega^I\Omega^K$, the integrability for this equations implies $f^I_{JK}$ be the structure constants for some Lie algebra. The covariant constancy equations can be formulated in the same form.

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The tour-de-force of elliptic pde on manifolds is the Yamabe problem. There the pde is a second-order, elliptic, and semilinear with a Sobolev critical exponent. The analysis can become incredibly difficult if you want to solve the whole problem, but if you focus on the early steps (the ones done by Yamabe, Trudinger) you could have a good problem. I mention this here because it was my exposure to this problem that prepared me to do research in geometric analysis (PDE on Riemannian manifolds) and I think this is the case for many people in conformal differential geometry. References would include Lee&Parker's "The Yamabe Problem" and lecture notes on Emmanuel Hebey's website.

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All previous replies have their own merit, however I am surprised not to find the first thing that popped in my head: That Frobenius' theorem concerning the integrability of a differential system on the tangent bundle. And maybe as a homotopy-oriented researcher (kudos to that) you could relate a simple result and use it as an intro to the Atiyah-Singer index theorem.

Or maybe not. Frobenius' theorem is pretty enough to hold the show on its own.

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That is a great idea, I think the background necessary to have it make sense for the other students would have been high. I should keep this in mind though. Funny story, we thought about having a course on index theory but it is unclear whether or not any PDE students would sign up. –  Sean Tilson Dec 13 '11 at 4:37