Compactified Morse trajectory spaces

table of contents

Consider a smooth manifold $L$ equipped with a Morse function $f:L\to \mathbb{R}$ and a metric so that the gradient vector field $\nabla f$ satisfies the Morse-Smale conditions. Then the Morse trajectory spaces

${\begin{alignedat}{4}{\mathcal {M}}(L,L)&=\{\gamma :&[0,a]\;\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),a\geq 0\},\\{\mathcal {M}}(p^{-},L)&=\{\gamma :&(-\infty ,0]\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to -\infty }}\gamma (s)=p^{-}\},\\{\mathcal {M}}(L,p^{+})&=\{\gamma :&[0,\infty )\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to +\infty }}\gamma (s)=p^{+}\},\\{\mathcal {M}}(p^{-},p^{+})&=\{\gamma :&\mathbb{R} \;\;\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to \pm \infty }}\gamma (s)=p^{\pm }\}/\mathbb{R} .\end{alignedat}}$

can - under an additional technical assumption specified in [1] - be compactified to smooth manifolds with boundary and corners ${\mathcal {M}}(\cdot ,\cdot )$. These compactifications are constructed such that the codimension-1 strata of the boundary are given by single breaking at a critical point (except in the first case we have to add one copy of $L$ to represent trajectories of length 0),

$\textstyle \partial ^{1}\overline {\mathcal {M}}(\cdot ,\cdot )\;=\;{\bigl (}\;L\cup \;{\bigr )}\;\bigsqcup _{{q\in {\text{Crit}}(f)}}\overline {\mathcal {M}}(\cdot ,q)\times \overline {\mathcal {M}}(q,\cdot ).$

TODO:

• Introduce smooth evaluation maps ${{\rm {ev}}}^{\pm }:\overline {\mathcal {M}}(\cdot ,\cdot )\to L$,
• Define the renormalized length $\ell :\overline {\mathcal {M}}(L,L)\to [0,1]$ by $\ell (\gamma )={\tfrac {1}{1+a}}$ for $\gamma :[0,a]\to L$ and $\ell (\underline {\gamma })=1$ for all generalized (broken) Morse trajectories $\underline {\gamma }$
• Define the metric $d_{{\overline {\mathcal {M}}}}(\underline {\gamma },\underline {\gamma }')=d_{{{\rm {Hausdorff}}}}{\bigl (}\overline {{{\rm {im}}}(\underline {\gamma })},\overline {{{\rm {im}}}(\underline {\gamma })}{\bigr )}+{\bigl |}\ell (\underline {\gamma })-\ell (\underline {\gamma }'){\bigr |}$ as sum of Hausdorff distance between images and difference of renormalized lengths.
• discuss boundary&corner stratification, in particular note that $L\subset \partial ^{1}\overline {\mathcal {M}}(L,L)$ (the set of trajectories with $\ell =0$) is isolated from all other boundary strata (made up of generalized trajectories with $\ell =\infty$)