Notes

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====== Monotonicity ======


There are all sorts of kinds of monotonicity for various PDEs.
Generally, one uses monotonicity together with an $\epsilon$-regularity result to get strong restriction on the structure of the singularity set. 
The most basic monotonicity results take the form $E(f, S_r) \le CE(f, S_R)$ for $E$ some (renormalized) energy of a solution $f$ to some PDE on spheres of radius $r < R$ and $C$ not dependent on $r, R$.
A monotonicity of this sort exists for harmonic maps ($d^* d f = 0$) with $E(f,\Omega) = r^{2-n} \|df\|_{L^2(\Omega)}^2$  and Yang-Mills connections $(d_A^* F_A = 0$) with $E(A, \Omega) = r^{4-n} \|F_A\|^2_{L^2(\Omega)}$.

==== Bochner monotonicity ====

Mark showed me this one. We start off with integrating a Bochner-type formula to get
$$\int_{B_R} \Delta \frac{|h|^2}{2} \mathrm{vol}_{B_R} = \int_{B_R} (|\nabla h|^2 + \langle \mathcal{R}h, h\rangle) \mathrm{vol}_{B_R},$$
with $\mathcal{R}$ a curvature term. By the divergence theorem, the left-hand side is
$$\int_{B_R} \Delta \frac{|h|^2}{2} = \int_{S_R} \partial_r \frac{|h|^2}{2} \mathrm{vol}_{S_R} \approx R^{n-1} \frac{\partial}{\partial r} \left( R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R} \right).$$
The last approximation is exact in Euclidean space, but holds to first order by working in geodesic coordinates. Thus we have
$$\frac{\partial}{\partial r} \left(R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R} \right) \approx R^{1-n} \int_{B_R} (|\nabla h|^2 + \langle \mathcal{R}h, h\rangle) \mathrm{vol}_{B_R},$$
which establishes monotonicity for the energy $R^{1-n} \int_{S_R} \frac{|h|^2}{2} \mathrm{vol}_{S_R}$ if we know that the approximation is good enough and the curvature is positive enough to make the integrand of the right hand side positive.

==== Price monotonicity ====

This monotonicity generally works for 'harmonic-type objects'.
The essence of proofs of such things is to write a sort of Lie derivative in two ways: once as an anticommutator $\{d, e^*(dr)\}$ and once in terms of the radial covariant derivative and second fundamental form, $L_{\partial_r} = \nabla_{\partial_r} + Q$ where $Q$ is the second fundamental form of a geodesic sphere in a geodesic ball.

By harmonicity, the anticommutator $\{d, e^*(dr)\}$ applied to an object $\alpha$ in question is simply $de^*(dr)\alpha$; integrating by parts in $(de^*(dr)\alpha, \alpha)$, coclosedness of $\alpha$ means only a boundary term survives and so $\int_{B_R} \langle de^*(dr)\alpha, \alpha) \rangle = \int_{S_R} |e^*(dr) \alpha|^2$; one compares this to the geometric term obtained by the other point of view.

See Di Cerbo-Stern for details.
The punchline is that in this situation it is actually //negative// curvature that makes monotonicity possible, while in the Bochner case it was positive curvature that helped.

Price monotonicity is what gives monotonicity for harmonic maps using the exterior derivative and Yang-Mills connections using the exterior covariant derivative, with respect to which the curvature is a harmonic (endomorphism-valued) 2-form.
 
==== Bellettini-Tian almost-monotonicity ====

This almost-monotonicity formula is for triholomorphic maps on hyperkahler manifolds; that is, maps $u \colon (M^m, I_1, I_2, I_3) \to (N^n, J_1, J_2, J_3)$ of hyperkahler manifolds that satisfy the triholomorphic map equation
$$du = \sum_{i=1}^3 J_i du I_i.$$

Triholomorphic maps enjoy monotonicity of the form $r^{2-m} \int_{B_r} |\nabla u|^2 = f(r) + O(r f(r))$ where $f$ is a non-decreasing function of $r$.
==== Walpuski almost-monotonicity ====

==== Almgren monotonicity ====

Almgren monotonicity is supposedly the result of comparing Price monotonicity to Bochner monoticity; I don't know why this is the case. In the Almgren setting we consider the following 'frequency function':
$$N_w(r) = \frac{r\int_{B_r} |\nabla w|^2}{\int_{S_r} |w|^2}.$$
Almgren monotonicity states that if $w$ is harmonic, then $N_w(r)$ is non-decreasing as a function of $r$.
Moreover, if $w$ is homogeneous of degree $k$, then integrating by parts and using harmonicity we have
$$r \left( \int_{S_r} \langle w, \nabla w\rangle - \int_{B_r} \langle w, \Delta w\rangle\right) = r\int_{S_r} \langle w, \nabla w\rangle = k \int_{S_r} |w|^2$$
by Euler's theorem so that $N_w(r) = k$.
This is where the 'frequency' part comes from.