arXiv subjects

Atreyee Bhattacharya

Publications and source records attributed to Atreyee Bhattacharya.

12 recordsLinked to original sources

arXiv2026

Teichm\"{u}ller isometries induced by certain irreducible periodic mapping classes

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Teich}(S_g)$ be the Teichm\"{u}ller space of $S_g$. In this paper, we provide an algorithm for determining the isometries induced in $\mathrm{Teich}(S_g)$ by certain irreducible periodic mapping classes in Fenchel-Nielsen coordinates. As a demonstration of this method, we provide a description of the isometry induced by periodic mapping class of order $4g+2$ and also derive some of its geometric properties.

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arXiv2026

Ricci solitons as critical points of quadratic curvature functionals

Rigidity, stability and local minimizing properties of Einstein metrics as critical points of quadratic Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature have been extensively studied. However, there are non-Einstein critical points of these functionals that are not so well understood. In this paper, we study Ricci solitons, a generalization of Einstein metrics, that are critical points of a special quadratic curvature functional and analyze their rigidity.

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arXiv2025

Fenchel-Nielsen coordinates of the branch loci of cyclic actions

Let $S_g$ be a closed, connected, and oriented smooth surface of genus $g\geq 2$. Let the mapping class group of $S_g$ be denoted by $\mathrm{Mod}(S_g)$ and the Teichmüller space of $S_g$ by $\mathrm{Teich}(S_g)$. It is known that $\mathrm{Mod}(S_g)$ acts by isometries on $\mathrm{Teich}(S_g)$ with respect to the Weil-Petersson metric. In this paper, we develop algorithms to describe the Fenchel-Nielsen coordinates of fixed points of the actions of certain finite cyclic subgroups of $\mathrm{Mod}(S_g)$ on $\mathrm{Teich}(S_g)$. As applications of these algorithms, we compute the Fenchel-Nielsen coordinates of the fixed points of three cyclic subgroups of orders $10$, $8$, and $4$, in $\mathrm{Mod}(S_2)$.

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arXiv2025

Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.

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arXiv2025

Estimating the distances between hyperbolic structures in the moduli space

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of $H$ on the Teichmüller space $\mathrm{Teich}(S_g)$ of $S_g$. This paper provides a method to estimate the distance between the unique fixed points of certain irreducible cyclic actions on $S_g$. We begin by deriving an explicit description of a pants decomposition of $S_g$, the length of whose curves are bounded above by the Bers' constant. To obtain the estimate, our method then uses the quasi-isometry between $\mathrm{Teich}(S_g)$ and the pants graph $\mathcal{P}(S_g)$.

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arXiv2024

Rigidity of conformal submersions and quasi-Einstein manifolds

In this paper, we study two notions of rigidity, one of conformal submersions and the other of quasi Einstein manifolds, with an attempt to relate the two notions. Note that a smooth submersion between Riemannian manifolds is called conformal if it restricts to a conformal isometry on the horizontal distribution. A conformal submersion is said to be rigid if it reduces to a Riemannian submersion up to homothety. On the other hand, quasiEinstein manifolds are generalizations of Einstein manifolds that are of interest both in Riemannian geometry and theoretical physics. A Riemannian manifold $(M, g)$ is called quasi-Einstein if its Ricci tensor satisfies the identity: $R i c_g+ H e s s(f)-\frac{1}{m} d f \otimes d f=\lambda g$ for some $f \in C^{\infty}(M)$ and constants $\lambda \in \mathbb{R}$ and $0 0$. In particular, we study curvature conditions that force conformal submersions to be rigid, also leading to the rigidity of a related class of quasi-Einstein manifolds.

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arXiv2023

On Certain Rigidity Results of Compact Regular $(κ, μ) $-Manifolds

In this article, we investigate the Riemannian and semi-Riemannian metrics on the base space of the Boothby-Wang fibration of a closed regular non-Sasakian $(κ, μ)$-manifold. To this end, we study a natural class of deviations of the projection map from being (semi-)Riemannian submersions. We consider deviations that preserve the canonical bi-Legendrian structure on the given $(κ, μ)$-manifold. We present rigidity results for Riemannian and semi-Riemannian metrics on the base space which orthogonalize the natural bi-Lagrangian structure induced by the $(κ, μ)$-structure. This approach gives a unified framework to analyze rigidity results in both categories. More precisely, in the Riemannian category, we obtain uniqueness of Sasakian structure on the given $(κ, μ)$-manifold which orthogonalizes the canonical bi-Legendrian structure. In the semi-Riemannian category, we obtain an explicit description of the finitely many para-contact structures which orthogonalize the canonical bi-Legendrian structure.

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arXiv2021

Geometric realizations of cyclic actions on surfaces -- II

Let $\mathrm{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichmüller space $\mathrm{Teich}(S_g)$. When $H$ is cyclic with $|H| \geq 3$, we show that $\mathrm{Fix}(H)$ admits a decomposition as a product of two-dimensional strips at least one of which is of bounded width. For an arbitrary $H$ with at least one generator of order $\geq 3$, we derive a computable optimal upper bound for the restriction $\mathrm{sys} : \mathrm{Fix}(H) \to \mathbb{R}^+$ of the systole function. Furthermore, we show that in such a case, $\mathrm{Fix}(H)$ is not symplectomorphic to the Euclidean space of the same dimension. Finally, we apply our theory to recover three well-known results, namely: (a) Harvey's result giving the dimension of $\mathrm{Fix}(H)$, (b) Gilman's result that $H$ is irreducible if and only if the corresponding orbifold is a sphere with three cone points, and (c) the Nielsen realization theorem for cyclic groups.

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arXiv2019

Stability of Quadratic curvature Functionals at product Einstein manifolds

In this paper, we study Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.

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arXiv2013

On the Curvature ODE associated to the Ricci flow

In the vector space of algebraic curvature operators we study the reaction ODE $$\frac{dR}{dt} = R^2+R^{#}= Q(R)$$ which is associated to the evolution equation of the Riemann curvature oper- ator along the Ricci flow. More precisely, we analyze the stability of a special class of zeros of this ODE up to suitable normalization. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces $M \times \mathbb{R}^k, \ k \geq 0$ where $M$ is an Einstein (locally) symmetric space of compact type and not a spherical space form when $k = 0.$

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arXiv2012

Some Unstable Critical Metrics for $L^{\frac{n}{2}}$-norm of the Curvature Tensor

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by $\mathcal{R}_{\frac{n}{2}}(g):= \int_M |R(g)|^{\frac{n}{2}}dv_g$ where $R(g)$, $dv_g$ denote the Riemannian curvature and volume form corresponding to $g$. We show that there are locally symmetric spaces which are unstable critical points for this functional.

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arXiv2012

A gap theorem for Ricci-flat 4-manifolds

Let $(M,g)$ be a compact Ricci-flat 4-manifold. For $p \in M$ let $K_{max}(p)$ (respectively $K_{min}(p)$) denote the maximum (respectively the minimum) of sectional curvatures at $p$. We prove that if $$K_{max} (p) \le \ -c K_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac{2+\sqrt 6}{4}$, then $(M,g)$ is flat. We prove a similar result for compact Ricci-flat Kähler surfaces. Let $(M,g)$ be such a surface and for $p \in M$ let $H_{max}(p)$ (respectively $H_{min}(p)$) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at $p$. If $$H_{max} (p) \le -c H_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac {1+\sqrt 3}{2}$, then $(M,g)$ is flat.

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