Uniqueness of some cylindrical tangent cones to special Lagrangians

TL;DR

This paper proves the uniqueness of certain cylindrical tangent cones to special Lagrangian submanifolds under specific conditions, advancing understanding of their local geometric structure.

Contribution

It establishes the uniqueness of cylindrical tangent cones for exact special Lagrangians with certain integrability conditions, including notable examples like Harvey-Lawson cones.

Findings

Uniqueness of tangent cones under integrability conditions

Applicability to Harvey-Lawson cones in specific dimensions

Advancement in understanding local geometry of special Lagrangians

Abstract

We show that if an exact special Lagrangian $N\subset \mathbb{C}^n$ has a multiplicity one, cylindrical tangent cone of the form $\mathbb{R}^{k}\times \mathbf{C}$ where $\mathbf{C}$ is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $\mathbf{C}$ satisfies an integrability condition. This applies, for example, when $\mathbf{C}= \mathbf{C}_{HL}^{m}$ is the Harvey-Lawson $T^{m-1}$ cone for $m\ne 8,9$.