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Register a new userFilling of closed Surfaces
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Authors
Bidyut Sanki
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Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of $F_g$ are in the same $text{Mod}(F_g)$-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of $F_2$ whose complement is a single disc (i.e., a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of $text{Mod}(F_2)$. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of $F_2$ is two. Finally, given positive integers $g$ and $k$ with $(g, k) eq (2, 1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.
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Let $F_g$ be a closed orientable surface of genus $g$. A set $Omega = { gamma_1, dots, gamma_s}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a emph{filling system} or simply a emph{filling} of $F_g$, if $F_gsetminus Omega$ is a union of $b$ topological discs for some $bgeq 1$. A filling system is called emph{minimal}, if $b=1$. The emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $ggeq 2, bgeq 1text{ with }(g,b) eq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2leq sleq 2g+b-1$ (resp. $3leq sleq 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $ggeq 2$, we show that for a minimal filling $Omega$ of size $s$, the emph{geometric intersection numbers} satisfy $max leftlbrace i(gamma_i, gamma_j)| i eq jrightrbraceleq 2g-s+1$, and for each such $s$ there exists a minimal filling $Omega=leftlbrace gamma_1, dots, gamma_s rightrbrace$ such that $maxleftlbrace i(gamma_i, gamma_j) | i eq jrightrbrace = 2g-s+1$.
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The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces. This paper proves that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $geq1$, when the approximating triangle meshes are reasonably good. To the best of the authors knowledge, this is the first convergence result on computing uniformizations for surfaces of genus $>1$.
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