AHHH so heres that Yamabe Problem thread. I'm very far from being an expert, so take what I say with some big blocks of salt. If anyone would like to add in with comments or concerns, please do! In any case I hope it’s enjoyable :)

The tagline is that the affirmative resolution of the Yamabe Problem is a landmark result in geometry and PDEs which has a remarkably simple statement, but which took a quarter of a century, and multiple brilliant minds from around the world to finally settle.

The proof uses differential geometry, functional analysis and PDEs, geometric measure theory, general relativity, and maybe even some sacrifices to appease the math gods. Let's start with a motivating result: the Uniformization Theorem, which says:

for a closed Riem. 2-mfd. (M,g), we can find a conformal metric e^{2f}g which has constant curvature. Basically, we are warping space without changing the angles at which curves intersect, and finding that this new world curves the same way in every direction at every point

On a 2-mfd, all curvature data is determined by the metric and a single function, the scalar curvature (which in dim>2 is an average of all 2-dim. “sectional” curvatures in different pairwise orthogonal directions).

Fundamentally, the Uniformization Theorem is therefore about finding one function (the conformal factor e^{2f}) which satisfies one condition (that the single scalar curvature function be constant).

Yamabe sought to solve the Poincare Conjecture, and wanted to start by finding simple metrics via conformal changes. Constant (sectional) curvature metrics are the simplest, but in dim>2 it’s hopeless to find a conformal change to make a general metric have constant curvature!

Indeed the curvature tensor then depends on more than a single function, and we would need to control many sectional curvatures via a single conformal change. The scalar curvature is still just a single function, so Yamabe thought this was the right function to make constant.

So that’s the Yamabe Problem: Given a closed Riem. mfd (M,g) of dim>2, find a conformal metric h to g which has constant scalar curvature. In the early 60’s Yamabe presented a proof of this fact, and all was seemingly well until Neil Trudinger discovered a very subtle oversight..

Let’s take a glimpse at Yamabe’s approach. I’ll write some formulas, but don’t worry too much about what they explicitly say if you don’t want (as theyre a tad clunky). The important part is what they represent and how they relate.

One can write out how the scalar curvature S_g of g evolves under a conformal change. If our conformal change is written h:=u^{p-2}g, where u is a smooth positive function and p=2*=2n/(n-2) is the Sobolev conjugate of 2, then the equation is

S_h={-cΔu+S_g u}u^{1-p}

where c is another dimensional constant. We want to find u so that S_h is constant, say λ, so this equation can be written as a sort of non-linear eigenvalue problem called Yamabe’s Equation:

Lu:=-cΔu+S_g u=λu^{p-1}.

This equation ends up being the Euler-Lagrange equation for a functional we call Q_g:

Q_g(u)=[\\int_M c|∇u|^2+Su^2 dvol_g]/(||u||_p)^2.

The point is this: To solve the Yamabe Problem we need to solve Yamabe’s Equation, and this is equivalent to minimizing Q_g over all u (say in W^{1,2}(M)) and doing some good ol elliptic regularity. Yamabe thus converted his original problem to one in the calculus of variations.

An analyst would at this point try something called the “direct method,” which means they would study an infimizing sequence {u_k} so that Q_g(u_k) converges to inf(Q_g(-))=λ(M), which we call the Yamabe constant of M.

Then, they would try to use compactness methods in functional analysis to find a converging subsequence of the u_k with a limit function u. Hopefully, u is a positive, smooth minimizer of Q_g, thus completely solving the problem.

But it’s bad news buckaroos, because that pesky exponent p is exactly the critical exponent where the compactness statement of the Sobolev Embedding Theorem fails! This torpedoes our hope of finding a quick proof using the direct method, because we can’t just

extract a good converging subsequence and limit function using the classical methods with the Embedding Theorem. Yamabe’s idea on how to avoid this is simple and beautiful. Consider the sub-critical problems Lu=λu^t-1, where t<p.

For these lower values of t, the problem is solvable using the direct method, and any student of PDEs might even solve something like it as homework. Yamabe took a limit of these solutions as t increased to p, hoping that they would converge to a solution of the critical problem.

This is where his error arose. He claimed to have uniform Holder estimates on the solutions as t increased to p, but Trudinger noticed that counterexamples can be found on the round sphere! Something else would need to be done to take a limit of these sub-critical solutions.

The rest of the story happens in three parts: (1) We tackle and fully solve the problem on the round sphere. (2) We show that the Yamabe Problem can be solved if λ(M)<λ(S^n). (3) By some clever case work we show all closed Riem. mfds of dim>2 have λ(M)<λ(S^n).

PART 1: Using stereographic projection, the minimization problem for Q on S^n converts to an equivalent problem on R^n. In fact, if you write out the quotient for Q_g_std on R^n (using that the scalar curvature vanishes), this equivalent problem is

exactly equivalent to determining the sharp constant in the Sobolev Inequality on R^n! Using geometric measure theory, Talenti and Aubin were able to explicitly give the sharp constant and characterized the extremal functions which give equality in the inequality,

thus solving the Yamabe Problem on the round sphere. These extremal functions will prove to be extremely useful in the next parts.

I'll add the next parts in a sec

PART 2: One possible way to proceed here is to use the sharp Sobolev Inequality to show that Yamabe’s uniform estimates actually do hold if λ(M)<λ(S^n), so that his limiting argument works as intended. Another really cool approach is due to Lions, who showed that if the required

bounds were to fail, then the functions u_k would need to concentrate at some point, and this “bubbling” would add λ(S^n) to the energy which cant occur if λ(M)<λ(S^n). The point is, strict inequality gives us room to “absorb the errors” of not having compactness.

PART 3: We need one more definition: A manifold is locally conformally flat at p iff its Weyl tensor W (the traceless part of the curvature tensor, and a conformal invariant) vanishes at p, iff up to a conformal change the manifold has zero curvature near p.

The two theorems we need to finish up are the following, which show that λ(M)<λ(S^n) by producing specific test functions that make λ(M)<=Q<λ(S^n):

The first theorem is due to Aubin: If dim>5, and if there is some p where M is not loc. conf. flat, then λ(M)<λ(S^n).
The second theorem is due to Schoen: If dim=3,4, 5, or if M is loc. conf. flat at some p, then λ(M)<λ(S^n).

AUBIN’S THEOREM: By utilizing really nice normal coordinates at the point p, we can make powerful estimates on the metric which give us control over the volume form and scalar curvature, and which relate the laplacian of the scalar curvature at p to the norm of |W(p)|>0.

In the proof, the dimension assumption is used to ensure we have enough decay in the integrands we estimate.
Our candidate test function on M is a Sobolev extremal function (pulled back to M via the chart) which is scaled so that it becomes really pointy and concentrated at p.

We multiply it by a cutoff function to keep it inside our normal coordinate ball where we can do our estimates. Sure enough, if we write out the Q energy of this test function, we get an expansion whose first term is λ(S^n), whose second term is negative since -|W(p)|<0,

and with all other terms negligible for a pointy enough test function. Thus, by taking our scaling factor small enough, we push the Q energy below λ(S^n).

SCHOEN’S THEOREM: Again we work in our cleverly chosen coordinates. With these hypotheses, the local data of M doesn’t provide enough information to argue locally like Aubin, so we are forced to make a “global test function.”

Schoen’s remarkable idea is the following: we can reduce to the case where L has positive Green’s function (a function G which is smooth on M\\{p} and has LG=δ_p on M in the sense of distributions) and so we can use G as a conformal factor to get a new metric h=G^{p-2}g.

Crucially (and just as in the sphere problem), this “generalized stereographic projection” from (M\\{p}, h) to (M, g) has S_h=0.

We can then estimate G in the conformal normal coordinates, and this leads us to an estimate for the new metric h on M\\{p}. In fact, if we poke a hole in our normal ball at p and invert the coordinates to throw the hole out to infinity, the expansion for h in these new

coordinates tells us that M\\{p} is asymptotically flat at infinity. The point is that we have garnered a good understanding of this heretofore mysterious area around p, and are in a good position to start making some estimates.

Our candidate test function is again based on a Sobolev extremal function, but this time we’re going to dilate it so that it “melts” all over the manifold instead of concentrating like before. We expand the Q energy on this test function in terms of the dilation, and in the

process utilize a volume comparison for spheres in the h metric and in euclidean space. The volume ratio involves a quantity called the “distortion factor,” and the remarkable coincidence is that in our setting, it agrees (up to a positive factor) with the mass of h as defined in

general relativity. If we can show that it is positive, then the Taylor expansion of Q will have first term λ(S^n), second term with strictly negative coefficient due to the distortion factor, and with other terms negligible for large dilation factor.

Schoen and Yau’s Positive Mass Theorem (roughly) tells us that if M is asymptotically flat like in our case and has non-negative scalar curvature, then this mass is non-negative, vanishing iff M is isometric to flat space. If (M\\{p},h) is isometric to flat space, it’s conformal

to the round sphere, and we would be done. Otherwise, since S_h=0, the PMT tells us that the mass, and thus the distortion factor, is positive. Thus, we can take the dilation scale large enough to force Q below λ(S^n), and this completes the resolution of the Yamabe Problem!

FINALLY, I should add that some fantastic references for this are (1) Lee and Parker's incredible survey, (2) the nonlinear problems in geometry book by T. Aubin, and (3) some excellent notes by R. Neumayer, which you can find online