Uniqueness of some cylindrical tangent cones to special Lagrangians

Abstract

Abstract We show that if an exact special Lagrangian

$$N\subset {\mathbb {C}}^n$$

N ⊂

C

n

has a multiplicity one, cylindrical tangent cone of the form

$${\mathbb {R}}^{k}\times {\textbf{C}}$$

R

k

× C

where

$${\textbf{C}}$$

C

is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided

$${\textbf{C}}$$

C

satisfies an integrability condition. This applies, for example, when

$${\textbf{C}}= {\textbf{C}}_{HL}^{m}$$

C =

C

HL

m

is the Harvey-Lawson

$$T^{m-1}$$

T

m - 1

cone for

$$m\ne 8,9$$

m ≠ 8 , 9

.