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--- monotonicity [2023/06/07 12:45] – spencer
+++ monotonicity [2023/06/07 13:12] (current) – [Bellettini-Tian almost-monotonicity] spencer
@@ Line 29: / Line 29: @@
 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.
-==== Walpuski monotonicity ====
+==== Bellettini-Tian almost-monotonicity ====
-==== Almgren 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.