when we use the blowup technique and bubbling analysis? what are the essence and pespective? could you give some examples to explain them or some reference？ thank you
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2$\begingroup$ -1 While in some sense I would actually like to see a really good answer to this, I feel I must discourage this question on the grounds that (from FAQ): "MathOverflow is not the appropriate place to ask somebody to write an expository article for you" and that this is a little bit along the lines of (from how to ask): "what's the deal with algebraic geometry?". So to continue quoting from these advice pages: "The great answer you're hoping for doesn't exist because there isn't a precise question". But you could have a look at arxiv.org/abs/math/0304396 $\endgroup$– SpencerCommented Oct 14, 2011 at 10:29
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2$\begingroup$ This question is indeed way too open ended. Also, looking at the asker's blog, the asker appears capable of providing more perspective and details and asking a more focused question. $\endgroup$– Deane YangCommented Oct 14, 2011 at 13:53
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In geometric analysis, I guess the first well-known blow-up argument was given in R. Schoen's solution to Yamabe problem. He used a Green function to blow up the metric near a point $x$ so that the punctured ball $B(x,r)\setminus \{x\}$ becomes an asymptotically flat end of a complete non-compact manifold. (The idea of Schoen comes from his study of "mass".)
In this case, because the blow-up factor is a positive function, the new metric differs from the original one only by a conformal change. In other cases, we usually use constants to blow-up the metric because we would like to understand the metric itself, rather than its conformal class.
For example, when studying geometric flows such as the Ricci flow or the mean curvature flow, "blow-up" means to dilate the metric near singularities. The motivation is to "look into the detail" and "kill the lower order terms". In Perelman's work, he can prove that after a suitable dilation, every finite-time singularity of 3-dimensional Ricci flow on a compact manifold must be a cylinder, ball or capped cylinder.(It's called Canonical Neighborhood Theorem.) On the other hand, there is also a blow-down argument which could be used to study the asymptotic behavior.
This is an (partial) answer from geometric viewpoint.
