Difference between revisions of "Compactified Morse trajectory spaces"
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Consider a smooth manifold <math>L</math> equipped with a Morse function <math>f:L\to\R</math> and a metric so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions. Then the Morse trajectory spaces | Consider a smooth manifold <math>L</math> equipped with a Morse function <math>f:L\to\R</math> and a metric so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions. Then the Morse trajectory spaces | ||
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| + | '''TODO:''' | ||
| + | * Introduce smooth evaluation maps <math>{\rm ev}^\pm: \overline\mathcal{M}(\cdot,\cdot) \to L</math>, | ||
| + | * Define the renormalized length <math>\ell: \overline \mathcal{M}(L,L) \to [0,1]</math> by <math>\ell(\gamma)= \tfrac{1}{1+a}</math> for <math>\gamma: [0,a] \to L</math> and <math>\ell(\underline{\gamma})=1</math> for all generalized (''broken'') Morse trajectories <math>\underline{\gamma}</math> | ||
| + | * Define the metric <math>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|</math> as sum of Hausdorff distance between images and difference of renormalized lengths. | ||
| + | * discuss boundary&corner stratification, in particular note that <math>L\subset \partial^1\overline\mathcal{M}(L,L)</math> (the set of trajectories with <math>\ell=0</math>) is isolated from all other boundary strata (made up of generalized trajectories with <math>\ell=\infty</math>) | ||
Latest revision as of 13:50, 9 June 2017
Consider a smooth manifold 
equipped with a Morse function 
and a metric so that the gradient vector field 
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}}](/images/math/a/c/3/ac36e80abe76a15538933ab31effdc2f.png)
can - under an additional technical assumption specified in [1] - be compactified to smooth manifolds with boundary and corners 
.
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 to represent trajectories of length 0),
TODO:
- Introduce smooth evaluation maps
, - Define the renormalized length
![\ell :\overline {\mathcal {M}}(L,L)\to [0,1]](/images/math/e/9/2/e9205489c81e4263977308f1f83fc81a.png)
by 
for ![\gamma :[0,a]\to L](/images/math/e/2/7/e27331e5ae4baf6049fb5fd6dfbd06d7.png)
and 
for all generalized (broken) Morse trajectories 
- Define the metric
as sum of Hausdorff distance between images and difference of renormalized lengths. - discuss boundary&corner stratification, in particular note that
(the set of trajectories with 
) is isolated from all other boundary strata (made up of generalized trajectories with 
)
