Let $\gamma\_0$ and $\gamma\_1$ be two simple closed curves in a closed surface $S$. 



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> What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ such that $\partial \Sigma = \gamma\_0 \times \{0\} \cup \gamma\_1 \times \{1\}$?
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Of course, in order for such a surface $\Sigma$ to exist, the two curves $\gamma\_0$ and $\gamma\_1$ must represent the same class in $H\_1(S,\mathbb Z\_2)$. Note that $\Sigma$ may be non-orientable.


If $2n$ is the minimum geometric intersection number of $\gamma\_1$ and $\gamma\_2$, it is easy to construct a $\Sigma$ with $\chi(\Sigma)\geqslant \chi(S) - n$. Is there a converse estimate of this kind? Do we have $\chi(\Sigma) \leqslant -n$ when $S$ is a torus?



Maximal euler characteristic of surfaces bounding two fixed curves
Here is a graph $\mathcal{G}(S,x)$ associated to a surface and homology class (similar to the 1-skeleton of the curve complex): For a fixed class in $x\in H\_1(S;\mathbb{Z}\_2)$, consider isotopy classes of embedded multicurves representing the homology class $x$ (one may assume no components are parallel and there are no trivial components). Make this collection the vertices of the graph $\mathcal{G}(S,x)$. Connect two vertices $A, B$ to be adjacent in the graph $\mathcal{G}(S,x)$ if $A\cup B$ are disjointly embedded, and after removing all parallel curves of $A\cup B$, the remaining components bound a pair of pants or a twice-punctured projective plane (this second can only happen if $S$ is non-orientable). I'm not sure if such a complex has been defined before, but there is a somewhat analogous complex defined in the integral homology case by [Bestvina, Bux, and Margalit.](http://www.ams.org/mathscinet-getitem?mr=2552249) Also, this is related to a technique of [Hatcher-Thurston](http://www.ams.org/mathscinet-getitem?mr=778125) to undertand surfaces in two-bridge knot complements. 


I claim that the maximal Euler characteristic of a surface bounding $\gamma\_0\times 0 \cup \gamma\_1\times 1$ is the negative of the distance between $\gamma\_0$ and $\gamma\_1$ in $\mathcal{G}(S,x)$. Put the product metric on $S\times [0,1]$, and make $\Sigma$ into a minimal surface with respect to this metric (Theorem 6.12 of [Hass-Scott](http://www.ams.org/journals/tran/1988-310-01/S0002-9947-1988-0965747-6/S0002-9947-1988-0965747-6.pdf)). Then $S\times t, t\in [0,1]$ gives a foliation by totally geodesic surfaces, and by the maximum principle, they can be tangent to $\Sigma$ in only saddle tangencies (see an [argument of Hass](http://www.ams.org/mathscinet-getitem?mr=847517), we will assume things are perturbed to be generaic). Thus, for all but finitely many $t$, $S\times t$ meets $\Sigma$ in a finite collection of curves, giving rise to a vertex of $\mathcal{G}(S,x)$. As one passes through a tangency point $S\times t\_0$ (assuming things are generic), the intersection changes by a saddle move, giving a surface in $S$ of Euler characteristic $-1$ bounding the curves before and after the tangency. There can never be a closed trivial curve occurring, because this would give rise to a center tangency. Thus, each saddle tangency gives an edge between the adjacent vertices of $\mathcal{G}(S,x)$, and therefore $\Sigma$ gives a path between $\gamma\_0$ and $\gamma\_1$ in $\mathcal{G}(S,x)$. Conversely, any such path gives rise to a surface. 


Of course, there will be many geodesics connecting $\gamma\_0$ and $\gamma\_1$ in $\mathcal{G}(S,x)$, given by any Morse function on $(\Sigma, \gamma\_0,\gamma\_1)$ with only index 1 critical points, and I don't expect the distance function to be easy to compute (probably one should use normal surface theory to compute it). 


