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The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,

$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$

where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,

$YM(A) = \int d^4x Tr F_{\mu\nu} F_{\mu\nu} > 0$

The setting is quite general, either $M$ is a general 4-manifold or Kahler manifold and so all theorems, existence, uniqueness, etc, are quite general.

I'm wondering if there are further results somewhere for the specific case of $M = T^4$ the 4-torus. For example, is it known how the long time asymptotics look like in this case? Theorems about possible blow-ups? I'd think asymptotically the gradient flow drives the connection towards a critical point but is it known how it is approached, exponentially or polynomially in $t$?

Actually for $M=T^4$ I suspect $t^2 YM(A(t))$ goes to a constant for $t\to\infty$ as long as the initial condition for the flow is in a sufficiently small neighbourhood of the absolute minimum of $YM(A)$ but I can't prove it. What is certainly true is that $YM(A(t))$ is a decreasing function of $t$.

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2 Answers 2

up vote 3 down vote accepted

Donaldson and Kronheimer wrote their book by 1990. There were some further developments about long time behaviour of Yang-Mills flow on four manifolds by, among others, Struwe and collaborators. You may try starting with Schlatter's dissertation.

Crawling through MathSciNet reference links may get you "somewhere", but it is my impression that higher dimensional Yang-Mills heat flow is still somewhat of an open problem. Is there any reason why you expect $\mathbb{T}^4$ would be better behaved than, say, the unit ball?

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Thanks, the Schlatter reference was really useful! The reason I'm asking about $T^4$ is that this is the case I can study numerically. When I numerically solve the gradient flow it seems to have the property that $t^2 YM(A(t))$ goes to a constant, but I'm not really sure. –  Daniel Nov 24 '11 at 23:22
    
How about starting first in $\mathbb{T}^3$ or $\mathbb{T}^2$ (just to test out your numerical algorithm). Rade showed reference-global.com/doi/abs/10.1515/crll.1992.431.123 polynomial rates of convergence for global existence. –  Willie Wong Nov 25 '11 at 9:31
    
Sounds like a good idea, thanks for the reference, unfortunately our university doesn't appear to have digital subscription so can't access the article. Does anyone have a digital copy? –  Daniel Nov 25 '11 at 11:46
    
@Daniel: strangely enough, no-one I know has access to the digital copy. (And in a moment of lack of wisdom, I tossed out my paper copy after seeing that the article is "available on-line", only to find out it was, in fact, not the case for me.) This is one of those cases you'll just have to buckle down and go photocopy it from a paper copy of the journal. –  Willie Wong Nov 28 '11 at 7:47
    
@Daniel: ah, I just found out the "proper" way of accessing the digital copy. Try this link: digizeitschriften.de/dms/toc/?PPN=PPN243919689 If your institution has a subscription, you should be able to find the paper under "Band 431". Else you can send me an e-mail. –  Willie Wong Nov 28 '11 at 8:07

I think you should just take a look at the following paper:

http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.0845v1.pdf

(In particular it should exponential convergence.) Should there still be question, you might post them!

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That paper is only dealing with the flow on Riemann surfaces. –  Daniel Nov 18 '11 at 22:22

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