Question

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.

Answer 1

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

MR0840401 (87m:53003)

Answer 2

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.

Answer 3

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.

Answer 4

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.