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--- monotonicity [2023/06/07 12:24] – created spencer
+++ monotonicity [2023/06/07 13:12] (current) – [Bellettini-Tian almost-monotonicity] spencer
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 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.