\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

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There exists a standard model of two Lagrangian submanifolds intersecting transversely at a point. Therefore, it suffices to construct a Lagrangian surgery neck for the standard intersection neighborhood \(X=\CC^n\), \(L_1=\RR^n\) and \(L_2=\jmath\RR^n\). We start by picking a surgery profile curve, \begin{align*} \gamma: [-R, R] \to& \CC\\ t\mapsto& (a(t)+\jmath b(t)) \end{align*} with the property that \(a(t), b(t)\) are non-decreasing, and there exists a value \(t_0\) so that We denote the are bounded between the real axis, imaginary axis, and curve \(\gamma\) by \(\lambda\). An example is drawn in figure 0.0.1.
figure 0.0.1:Surgery Profile for Polterovich surgery
This data provides a construction for the Lagrangian surgery neck: \[ L_1\#_\gamma L_2:=\left\{(\gamma(t)\cdot x_1,\ldots, \gamma(t)\cdot x_n) \text{ such that } x_i \in \RR^n,t\in \RR, \sum_{i} x_i^2=1\right\}. \] Note that when \(t < t_0\) this parameterizes \((\RR\setminus B_r(0))\subset \CC^n\), and when \(t > t_0\) the chart parameterizes \((\jmath \RR \setminus B_r(0))\subset \CC^n\). Therefore, this construction satisfies the condition that the surgery Lagrangian agrees with the surgery components outside of a small neighborhood of the surgery point. This Lagrangian has the topology of \(S^{n-1}\times \RR\), which is the local model for the connect sum \(\RR^n\#_0\RR^n\). Then (Polterovich surgery of Lagrangian submanifolds) follows by taking \(L_1\#_\gamma L_2\) for any suitable choice of \(\gamma\).