\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
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.
- Show that the conormal bundle \(N^*H\subset T^*M\) is a Lagrangian submanifold.
- Let \(B_\epsilon(H)\subset X\) be a small neighborhood of \(H\subset M\subset T^*M\) . Find an embedded Lagrangian submanifold \(L\subset T^*M\) which, outside of a small neighborhood \(B_\epsilon(H)\subset X\), agrees with \(N^*H\cup M\),
\[ L\setminus (B_\epsilon(H))=(N^*H\cup M)\setminus H.\]
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?