[Paper Review] The deformation of symplectic critical surfaces in a Kähler surface-II---Compactness

Xiaoli Han, Jiayu Li+1 more|arXiv (Cornell University)|Jul 6, 2016

Geometry and complex manifolds11 references3 citations

TL;DR

This paper establishes the compactness of sequences of $β$-symplectic critical surfaces in a compact Kähler surface under uniform bounds on the $L^q$-norm of $1/\cos^q\alpha$ for $q>4$ and on genus. Using Moser iteration and small energy regularity, it proves subsequential $C^l$ convergence on compact subsets away from a finite singular set, with each singularity resolved by a holomorphic bubble in $\mathbb{C}^2$, and the limit surface extends smoothly across the singularities.

ABSTRACT

In this paper we consider the compactness of $β$-symplectic critical surfaces in a Kähler surface. Let $M$ be a compact Kähler surface and $Σ_i\subset M$ be a sequence of closed $β_i$-symplectic critical surfaces with $β_i oβ_0\in (0,\infty)$. Suppose the quantity $\int_{Σ_i}\frac{1}{\cos^qα_i}dμ_i$ (for some $q>4$) and the genus of $Σ_{i}$ are bounded, then there exists a finite set of points ${\mathcal S}\subset M$ and a subsequence $Σ_{i'}$ that converges uniformly in the $C^l$ topology (for any $l

Motivation & Objective

• To establish compactness for sequences of $\beta$-symplectic critical surfaces in a compact Kähler surface as $\beta_i \to \beta_0 \in (0,\infty)$, addressing the closedness of the set of stable $\beta$-critical surfaces.
• To control geometric quantities such as $\cos\alpha$ and total curvature uniformly across the sequence, enabling convergence analysis.
• To show that singularities in the limit surface are isolated and resolved by holomorphic bubbles in $\mathbb{C}^2$, with the tangent cone at each singularity being a union of planes intersecting at a point.
• To prove that the limit surface is a $\beta_0$-symplectic critical surface that extends smoothly across the singular set, generalizing minimal surface compactness results.

Proposed method

• Establish a small energy regularity theorem for $\beta$-symplectic critical surfaces, ensuring regularity when energy is sufficiently small.
• Apply Moser's iteration to the elliptic equation satisfied by $\cos\alpha$, using the $L^q$-bound on $1/\cos^q\alpha$ for $q>4$ as a starting condition.
• Derive uniform lower bounds for $\cos\alpha$ and upper bounds for total curvature via the iteration and curvature estimates.
• Use the monotonicity formula (proved in the appendix) to control area growth and energy concentration, crucial for the small energy regularity result.
• Construct a subsequence converging in $C^l$ topology on compact subsets of $M \setminus \mathcal{S}$, where $\mathcal{S}$ is a finite singular set.
• Analyze the bubble tree structure at each singular point $p_\gamma \in \mathcal{S}$, showing each bubble is a smooth holomorphic curve in $\mathbb{C}^2$ with a flat cone tangent cone.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of $\beta_i$-symplectic critical surfaces in a compact Kähler surface converge to a $\beta_0$-symplectic critical surface?
• RQ2 Can uniform bounds on $\int_\Sigma \frac{1}{\cos^q\alpha} d\mu$ for $q>4$ and on genus prevent energy concentration and ensure regularity?
• RQ3 What is the structure of the limit surface at points where convergence fails, and how do the singularities arise?
• RQ4 How do the bubbles that form at singularities relate to holomorphic curves in $\mathbb{C}^2$, and what is their tangent cone structure?
• RQ5 Can the limit surface be extended smoothly across the singular set $\mathcal{S}$, and if so, under what conditions?

Key findings

• A subsequence of $\beta_i$-symplectic critical surfaces converges uniformly in $C^l$ topology (for any $l<\infty$) on compact subsets of $M \setminus \mathcal{S}$ to a $\beta_0$-symplectic critical surface $\Sigma$, where $\mathcal{S}$ is a finite singular set.
• The limit surface $\Sigma$ extends smoothly across each point in $\mathcal{S} \cap \overline{\Omega}$ for each connected component $\Omega$ of $\Sigma$.
• At each singular point $p_\gamma \in \mathcal{S}$, a bubble forms that is a smooth holomorphic curve in $\mathbb{C}^2$, and the tangent cone at $p_\gamma$ is a flat cone consisting of planes intersecting only at the origin.
• The convergence also holds in extrinsic Hausdorff distance, ensuring geometric stability of the limit surface.
• The uniform $L^q$-bound on $1/\cos^q\alpha$ for $q>4$ is essential to initiate Moser iteration, which yields uniform lower bounds on $\cos\alpha$ and upper bounds on total curvature.
• The small energy regularity theorem ensures that points without energy concentration are regular, and the monotonicity formula is key to controlling area and curvature growth.