Let ${\mathbin{\mathbb{C}}}^{m}$ have coordinates $(z_{1},\dots,z_{m})$ and complex structure $J$, and define a Kähler metric $g$, Kähler form $\omega$ and $(m,0)$-form $\Omega$ on ${\mathbin{\mathbb{C}}}^{m}$ by

Then $({\mathbin{\mathbb{C}}}^{m},J,g,\Omega)$ is the simplest example of a Calabi–Yau $m$-fold.

Define a real 1-form $\lambda$ on ${\mathbin{\mathbb{C}}}^{m}$ called the Liouville form by

Then ${\rm d}\lambda=\omega$. Thus, if $L$ is a Lagrangian in ${\mathbin{\mathbb{C}}}^{m}$ then ${\rm d}(\lambda|_{L})=0$. We call $L$ an exact Lagrangian if $\lambda|_{L}={\rm d}f$ for some smooth $f:L\rightarrow{\mathbin{\mathbb{R}}}$.

A (singular) Lagrangian $C$ in ${\mathbin{\mathbb{C}}}^{m}$ is called a cone if $C=tC$ for all $t>0$, where $tC=\{t\,{\bf z}:{\bf z}\in C\}$. Let $C$ be a closed Lagrangian cone in ${\mathbin{\mathbb{C}}}^{m}$ with an isolated singularity at 0. Then $\Sigma=C\cap{\cal S}^{2m-1}$ is a compact, nonsingular Legendrian $(m\!-\!1)$-submanifold of ${\cal S}^{2m-1}$, not necessarily connected. Let $g_{\smash{\scriptscriptstyle\Sigma}}$ be the metric on $\Sigma$ induced by the metric $g$ on ${\mathbin{\mathbb{C}}}^{m}$ in (2.2), and $r$ the radius function on ${\mathbin{\mathbb{C}}}^{m}$. Define $\iota:\Sigma\times(0,\infty)\rightarrow{\mathbin{\mathbb{C}}}^{m}$ by $\iota(\sigma,r)=r\sigma$. Then the image of $\iota$ is $C\setminus\{0\}$, and $\iota^{*}(g)=r^{2}g_{\smash{\scriptscriptstyle\Sigma}}+{\rm d}r^{2}$ is the cone metric on $C\setminus\{0\}$.

Let $L$ be a closed, nonsingular Lagrangian $m$-fold in ${\mathbin{\mathbb{C}}}^{m}$, e.g. $L$ could be special Lagrangian, or a Lagrangian LMCF expander. We call $L$ asymptotically conical (AC) with rate $\rho<2$ and cone $C$ if there exists a compact subset $K\subset L$ and a diffeomorphism $\varphi:\Sigma\times(T,\infty)\rightarrow L\setminus K$ for some $T>0$, such that

Here $\nabla,|\,.\,|$ are computed using the cone metric $\iota^{*}(g)$. Note that if $\rho<\sigma<2$ and $L$ is AC with rate $\rho$, then $L$ is also AC with rate $\sigma$.