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Today I’d like to discuss (in the Tricks Wiki format) a fundamental trick in “soft” analysis, sometimes known as the “limiting argument” or “epsilon regularisation argument”.

Title: Give yourself an epsilon of room.

Quick description: You want to prove some statement S_0 about some object x_0 (which could be a number, a point, a function, a set, etc.).  To do so, pick a small \varepsilon > 0, and first prove a weaker statement S_\varepsilon (which allows for “losses” which go to zero as \varepsilon \to 0) about some perturbed object x_\varepsilon.  Then, take limits \varepsilon \to 0.  Provided that the dependency and continuity of the weaker conclusion S_\varepsilon on \varepsilon are sufficiently controlled, and x_\varepsilon is converging to x_0 in an appropriately strong sense, you will recover the original statement.

One can of course play a similar game when proving a statement S_\infty about some object X_\infty, by first proving a weaker statement S_N on some approximation X_N to X_\infty for some large parameter N, and then send N \to \infty at the end.

General discussion: Here are some typical examples of a target statement S_0, and the approximating statements S_\varepsilon that would converge to S:

S_0 S_\varepsilon
f(x_0) = g(x_0) f(x_\varepsilon) = g(x_\varepsilon) + o(1)
f(x_0) \leq g(x_0) f(x_\varepsilon) \leq g(x_\varepsilon) + o(1)
f(x_0) > 0 f(x_\varepsilon) \geq c - o(1) for some c>0 independent of \varepsilon
f(x_0) is finite f(x_\varepsilon) is bounded uniformly in \varepsilon
f(x_0) \geq f(x) for all x \in X (i.e. x_0 maximises f) f(x_\varepsilon) \geq f(x)-o(1) for all x \in X (i.e. x_\varepsilon nearly maximises f)
f_n(x_0) converges as n \to \infty f_n(x_\varepsilon) fluctuates by at most o(1) for sufficiently large n
f_0 is a measurable function f_\varepsilon is a measurable function converging pointwise to f_0
f_0 is a continuous function f_\varepsilon is an equicontinuous family of functions converging pointwise to f_0 OR f_\varepsilon is continuous and converges (locally) uniformly to f_0
The event E_0 holds almost surely The event E_\varepsilon holds with probability 1-o(1)
The statement P_0(x) holds for almost every x The statement P_\varepsilon(x) holds for x outside of a set of measure o(1)

Of course, to justify the convergence of S_\varepsilon to S_0, it is necessary that x_\varepsilon converge to x_0 (or f_\varepsilon converge to f_0, etc.) in a suitably strong sense. (But for the purposes of proving just upper bounds, such as f(x_0) \leq M, one can often get by with quite weak forms of convergence, thanks to tools such as Fatou’s lemma or the weak closure of the unit ball.)  Similarly, we need some continuity (or at least semi-continuity) hypotheses on the functions f, g appearing above.

It is also necessary in many cases that the control S_\varepsilon on the approximating object x_\varepsilon is somehow “uniform in \varepsilon“, although for “\sigma-closed” conclusions, such as measurability, this is not required. [It is important to note that it is only the final conclusion S_\varepsilon on x_\varepsilon that needs to have this uniformity in \varepsilon; one is permitted to have some intermediate stages in the derivation of S_\varepsilon that depend on \varepsilon in a non-uniform manner, so long as these non-uniformities cancel out or otherwise disappear at the end of the argument.]

By giving oneself an epsilon of room, one can evade a lot of familiar issues in soft analysis.  For instance, by replacing “rough”, “infinite-complexity”, “continuous”,  “global”, or otherwise “infinitary” objects x_0 with “smooth”, “finite-complexity”, “discrete”, “local”, or otherwise “finitary” approximants x_\varepsilon, one can finesse most issues regarding the justification of various formal operations (e.g. exchanging limits, sums, derivatives, and integrals).  [It is important to be aware, though, that any quantitative measure on how smooth, discrete, finite, etc. x_\varepsilon should be expected to degrade in the limit \varepsilon \to 0, and so one should take extreme caution in using such quantitative measures to derive estimates that are uniform in \varepsilon.]  Similarly, issues such as whether the supremum M := \sup \{ f(x): x \in X \} of a function on a set is actually attained by some maximiser x_0 become moot if one is willing to settle instead for an almost-maximiser x_\varepsilon, e.g. one which comes within an epsilon of that supremum M (or which is larger than 1/\varepsilon, if M turns out to be infinite).  Last, but not least, one can use the epsilon room to avoid degenerate solutions, for instance by perturbing a non-negative function to be strictly positive, perturbing a non-strictly monotone function to be strictly monotone, and so forth.

To summarise: one can view the epsilon regularisation argument as a “loan” in which one borrows an epsilon here and there in order to be able to ignore soft analysis difficulties, and can temporarily be able to utilise estimates which are non-uniform in epsilon, but at the end of the day one needs to “pay back” the loan by establishing a final “hard analysis” estimate which is uniform in epsilon (or whose error terms decay to zero as epsilon goes to zero).

A variant: It may seem that the epsilon regularisation trick is useless if one is already in “hard analysis” situations when all objects are already “finitary”, and all formal computations easily justified.  However, there is an important variant of this trick which applies in this case: namely, instead of sending the epsilon parameter to zero, choose epsilon to be a sufficiently small (but not infinitesimally small) quantity, depending on other parameters in the problem, so that one can eventually neglect various error terms and to obtain a useful bound at the end of the day.  (For instance, any result proven using the Szemerédi regularity lemma is likely to be of this type.)  Since one is not sending epsilon to zero, not every term in the final bound needs to be uniform in epsilon, though for quantitative applications one still would like the dependencies on such parameters to be as favourable as possible.

Prerequisites: Graduate real analysis.  (Actually, this isn’t so much a prerequisite as it is a corequisite: the limiting argument plays a central role in many fundamental results in real analysis.)  Some examples also require some exposure to PDE.

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We now begin the study of (smooth) solutions t \mapsto (M(t),g(t)) to the Ricci flow equation

\frac{d}{dt} g_{\alpha \beta} = - 2 \hbox{Ric}_{\alpha \beta}, (1)

particularly for compact manifolds in three dimensions. Our first basic tool will be the maximum principle for parabolic equations, which we will use to bound (sub-)solutions to nonlinear parabolic PDE by (super-)solutions, and vice versa. Because the various curvatures \hbox{Riem}_{\alpha \beta \gamma}^\delta, \hbox{Ric}_{\alpha \beta}, R of a manifold undergoing Ricci flow do indeed obey nonlinear parabolic PDE (see equations (31) from Lecture 1), we will be able to obtain some important lower bounds on curvature, and in particular establishes that the curvature is either bounded, or else that the positive components of the curvature dominate the negative components. This latter phenomenon, known as the Hamilton-Ivey pinching phenomenon, is particularly important when studying singularities of Ricci flow, as it means that the geometry of such singularities is almost completely dominated by regions of non-negative (and often quite high) curvature.

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