Proof of the Monotonicity Lemma

The monotonicity lemma is a local regularity result for $J$ -holomorphic curves, providing a lower bound on the area of a holomorphic curve passing through a point. It is the key analytical ingredient in proving the non-squeezing theorem and establishing compactness.

Statement

Theorem7.3Monotonicity Lemma

Let $(M, \omega, J, g)$ be an almost Kähler manifold and $u: \Sigma \to M$ a $J$ -holomorphic curve with $u(z_0) = p$ . There exist constants $\varepsilon > 0$ and $C > 0$ (depending only on $M, \omega, J$ ) such that if $u(\Sigma) \cap B(p, r) \neq \emptyset$ is proper in $B(p, r)$ for $r < \varepsilon$ , then: $\mathrm{Area}(u(\Sigma) \cap B(p, r)) \geq C \cdot r^2.$ In particular, if $u(z_0) = p$ , then $\int_{\Sigma \cap u^{-1}(B(p,r))} u^*\omega \geq \pi r^2 - O(r^3)$ .

Proof Sketch

Proof

The proof adapts the classical monotonicity formula for minimal surfaces.

Step 1: Since $u$ is $J$ -holomorphic and $g(v, w) = \omega(v, Jw)$ , the energy density equals the area density: $|du|^2 = 2u^*\omega$ . Thus the area of $u$ restricted to $u^{-1}(B(p, r))$ equals $\int_{u^{-1}(B(p,r))} u^*\omega$ .

Step 2: Define $A(r) = \int_{u^{-1}(B(p,r))} u^*\omega$ . Using the coarea formula and the isoperimetric inequality for holomorphic curves: $A'(r) \geq \frac{2A(r)}{r} - C_1 A(r)$ for some constant $C_1$ depending on the curvature of $(M, g)$ .

Step 3: The differential inequality $\frac{d}{dr}\left(\frac{A(r)}{r^2}\right) \geq -C_1 \frac{A(r)}{r^2}$ integrates to $A(r)/r^2 \geq A(\varepsilon)/\varepsilon^2 \cdot e^{-C_1 r}$ , giving the lower bound. As $r \to 0$ , $A(r)/\pi r^2 \to 1$ (the curve is approximately linear near $p$ ), establishing the sharp constant $C = \pi$ for small $r$ . $\square$

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RemarkApplications

The monotonicity lemma has three crucial applications: (1) non-squeezing: a holomorphic sphere through a point in the image of an embedded ball has area at least $\pi r^2$ ; (2) compactness: bubbling can only occur at finitely many points since each bubble carries at least $\hbar = \min\{\omega(A) : A \in H_2(M), \omega(A) > 0\}$ energy; (3) removal of singularities: a finite-energy holomorphic curve with an isolated singularity extends smoothly across it.