Let $\gamma_0$ and $\gamma_1$ be two simple closed curves in a closed surface $S$.

What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ such that $\partial \Sigma = \gamma_0 \times \{0\} \cup \gamma_1 \times \{1\}$?

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?

Let $\gamma_0$ and $\gamma_1$ be two simple closed curves in a closed surface $S$.

What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ such that $\partial \Sigma = \gamma_0 \times \{0\} \cup \gamma_1 \times \{1\}$?

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?