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

exercises on Lagrangian submanifolds

exercise 0.0.1

Prove (sections in the cotangent bundle) without using coordinates. Try showing that \(\omega|_L=\pm d\alpha\).

exercise 0.0.2

Show that there are no Lagrangian spheres inside of \(\CP^2\).

exercise 0.0.3

Prove the converse of (Hamiltonian associated to exact homotopy), showing that every Hamiltonian Lagrangian isotopy is generated by the flow of a function \(H_t: L\times \RR\to \RR\).

exercise 0.0.4

Let \(M\) be an oriented manifold, and \(H\subset M\) be an oriented hypersurface. Let \(X=T^*M\) be the cotangent bundle equipped with the canonical symplectic form.

exercise 0.0.5

Let \(\Sigma\) be a symplectic manifold with \(\dim_\RR(\Sigma)=2\). Let \(L_0, L_1\subset \Sigma\) be two Lagrangian submanifolds which intersect transversely at points \(p_i\). Show that there exists a Lagrangian submanifold \(L'\subset X\) which agrees with \(L_0, L_1\) outside a small neighborhood of the \(p_i\): \[L'\setminus \left(\bigcup_{i} B_\epsilon(p_i)\right)=(L_0\cup L_1) \setminus \left(\bigcup_{i} B_\epsilon(p_i)\right).\]

exercise 0.0.6

Suppose that \(\alpha\) is a closed one form. Find a Hamiltonian isotopy of \(T^*N\) for which sends the zero section to \(\alpha\).

exercise 0.0.7

Let \(H_t: L\times [0, c]\to \RR\) be a time-dependent Hamiltonian. Suppose for each \(t_0\) the function \(H_{t_0}: L\to \RR\) is Morse. Let \(\li_t: L\to X\) be the induced isotopy. Show that for each critical point \(x\in \Crit(H_0)\), there is a sequence of points and times \(x_i\in L\) , \(t_i\in [0, c]\) so that \(\lim_{i\to\infty} t_i=0\), \(\li_{t_i}(x_i)\subset\li_{t_i}(L)\cap \li_{0}(L)\) and \(\lim_{i\to\infty}\li_{t_i}(x_i)=x\).

exercise 0.0.8

Which circles on \(S^2\) can be displaced by Hamiltonian isotopy?