Nearby Special Lagrangians

TL;DR

This paper investigates the local structure of special Lagrangian submanifolds near a given one in a Calabi--Yau manifold, establishing conditions under which nearby special Lagrangians are close or cover the original.

Contribution

It provides new results on the local uniqueness and covering properties of special Lagrangians based on fundamental group conditions.

Findings

If $\pi_1Q$ is abelian, nearby special Lagrangians are $C^1$ close.

If $\pi_1Q$ is virtually solvable, nearby special Lagrangians of bounded degree are unbranched coverings.

Stronger results hold when $\pi_1Q$ is finite; weaker when it has no non-abelian free subgroups.

Abstract

Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $\pi_1Q$ is abelian then there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian with unobstructed Floer cohomology is $C^1$ close to $Q.$ We prove also that if $\pi_1Q$ is virtually solvable then for every positive integer $R$ there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian of degree $\le R$ and with unobstructed Floer cohomology is unbranched; that is, the projection $L\to Q$ is a covering map. We prove a stronger statement when $\pi_1Q$ is finite and a weaker statement when $\pi_1Q$ has no non-abelian free subgroups. The $\pi_1Q$ conditions, the Floer cohomology condition and the special Lagrangian condition are all essential as we show by counterexamples.