arXiv subjects

Bhola Nath Saha

Publications and source records attributed to Bhola Nath Saha.

6 recordsLinked to original sources

arXiv2026

Length of Filling Pairs on Punctured Surface

A pair $(α, β)$ of simple closed curves on a surface $S_{g,n}$ of genus $g$ and with $n$ punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks and once punctured disks. In this article, we study the length of filling pairs on once-punctured hyperbolic surfaces. In particular, we find a lower bound of the length of filling pairs which depends only on the topology of the surface.

math.GT↗

arXiv2026

Complexity in Bolza surface

A surface in the Teichmüller space, where the systole function attains its maximum, is called a maximal surface. For genus two there exists a unique maximal surface which is called the Bolza surface. In this article, we study the complexity of the set of systolic geodesics on the Bolza surface. We show that any non-systolic geodesic intersects the systolic geodesics in $2n$ points, where $n\geq 5$. Furthermore, we show that there are $12$ second systolic geodesics on the Bolza surface and they form a triangulation of the surface.

math.DG↗

arXiv2026

Bolza-like surfaces in the Thurston set

A surface in the Teichmüller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a triangulation of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into $(p, q, r)$-triangles for some integers $p,q,r$. In this article, we will provide a construction of Bolza-like surfaces for infinitely many genera $g\geq 9$. Next, we see an intriguing application of Bolza-like surfaces. In particular, we construct global maximal surfaces using these Bolza-like surfaces. Furthermore, we study a symmetric property satisfied by the systolic geodesics of our Bolza-like surfaces. We show that any simple closed geodesic intersects the systolic geodesics at an even number of points.

math.GT↗

arXiv2026

The diameter function is a topological Morse function

Schmutz Schaller developed techniques for studying Teichmüller space using the systole function. These were presented in \cite{SchmutzMorse}, \cite{SchmutzVoronoi} as a hyperbolic analogue of Voronoi's theory of quadratic forms in the theory of Euclidean lattice packings and coverings \cite{VoronoiPDQF}. It is known that the packing density function on Euclidean space is a topological Morse function, \cite{TMFAsh}, and the same is true of the systole function on Teichmüller space, \cite{Akrout}, \cite{SchmutzMorse}. The study of hyperbolic packing and covering problems is technical, for example, the density depends on the scale, and very little is known about optimisers of the density \cite{tóth2022ballpackingshyperbolicspace}. At least in the Euclidean setting, there are also fewer techniques available for studying sphere covering as opposed to sphere packing problems, as the covering problems seem to have less discernible structure. One approach to studying efficient circle coverings in the hyperbolic plane is to study the critical points of the diameter function on Teichmüller space. This paper shows that the diameter function on Teichmüller space is a topological Morse function. As a mapping class group-equivariant topological Morse function, critical points of the diameter function are related to the homology of moduli space. It would seem that for small genus, the systole function and diameter function have a larger proportion of common critical points than at higher genus.

math.GT↗

arXiv2026

Length minimization of filling pairs on hyperbolic surfaces

A filling pair $(α, β)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $α\cupβ$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair.

math.GT↗

arXiv2024

Filling with separating curves

A pair $(α, β)$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $α$ is separating, then we call it as separating filling pair. In this article, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\M$ on the set of such filling pairs. Furthermore, we construct a Morse function $\mathcal{F}_g$ on the moduli space $\mathcal{M}_g$ which, for a given hyperbolic surface $X$, outputs the length of shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function $\mathcal{F}_g$ is the same as the number of $\M$-orbits of such filling pairs.

math.GT↗