Differences

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--- monotonicity [2023/06/07 12:54] – [Almgren monotonicity] 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 ====
+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 ====