arXiv subjects

Soma Maity

Publications and source records attributed to Soma Maity.

12 recordsLinked to original sources

arXiv2026

A Cheng-type Eigenvalue-Comparison Theorem for the Hodge Laplacian

We consider the class of closed Riemannian $n$-manifolds with Ricci curvature and injectivity radius bounded below by uniform constants, and an upper bound on the diameter. We establish a uniform upper bound for the eigenvalues of the Hodge Laplacian acting on differential forms on Riemannian manifolds in this class, similar to the classical eigenvalue comparison theorem proved by Cheng for the Laplace-Beltrami operator acting on smooth functions. This extends earlier work of Dodziuk and Lott, which required sectional curvature bounds in addition to bounds on other geometric quantities. As an application, we obtain uniform eigenvalue estimates for the connection Laplacian acting on $1$-forms.

math.DG↗

arXiv2026

Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

We establish uniform lower and upper bounds for the eigenvalues of the Hodge Laplacian acting on differential forms on closed Riemannian manifolds with a lower Ricci curvature bound, a positive lower bound on the injectivity radius, and an upper bound on the diameter. Our results extend earlier work of Dodziuk, Lott, and Mantuano, which required bounded sectional curvature, to the broader setting of lower Ricci curvature bounds. As applications, we obtain uniform eigenvalue bounds for the connection Laplacian acting on $1$-forms and establish a global Poincar\'e inequality for differential forms under the same geometric assumptions.

math.DG↗

arXiv2025

Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an $(\epsilon,\rho)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $\epsilon, \rho$ and the ratio $\frac{\epsilon}{\rho}$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$.

math.SP↗

arXiv2024

Volume growth functions of complete Riemannian manifolds with positive scalar curvature

Let $M$ be an open manifold of dimension at least $3$, which admits a complete metric of positive scalar curvature. For a function $v$ with bounded growth of derivative, whether $M$ admits a metric of positive scalar curvature with volume growth of the same growth type as $v$ is unknown. We answer this question positively in the case of manifolds, which are infinite connected sums of closed manifolds that admit metrics of positive scalar curvature. To define a metric of positive scalar curvature with a certain volume growth type on $M$, we use the Gromov-Lawson construction of metrics with positive scalar curvature on connected sums and Grimaldi-Pansu's construction of metrics of bounded geometry of certain volume growth type on open manifolds. We generalize this result to manifolds, which are infinite connected sums of similar closed manifolds along lower-dimensional spheres.

math.DG↗

arXiv2024

Volume growth on manifolds with more than one end

For an open manifold $M$ and a function $v$ with bounded growth of derivative, there exists a Riemannian metric of bounded geometry on $M$ such that the volume growth function lies in the same growth class as $v$. This was proved by R. Grimaldi and P. Pansu with the proof focusing on the case of manifolds with a single end. We prove this in the case of manifolds with multiple ends and call the constructed metrics Grimaldi-Pansu metrics. We give uniform bounds for the volume growth function of these metrics in terms of the given bgd-function in the case of a certain class of manifolds which can be written as connected sums of a finite collection of closed and compact manifolds. We study the volume doubling condition and the Relatively Connected Annulus (R.C.A.) property of the Grimaldi-Pansu metrics, which play an important role in studying geometric analysis on manifolds with finitely many ends.

math.DG↗

arXiv2022

Uniform Poincaré inequalities on measured metric spaces

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $μ.$ We establish a family of uniform Poincaré inequalities on $(X,d,μ)$ if it satisfies a local Poincaré inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $μ$ is doubling and supports $(P_{loc})$ then it satisfies a $(σ,β,σ)$-Poincaré inequality. If $(X,d,μ)$ is a $δ$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space then it supports a uniform Poincaré inequality and the Poincaré constant depends on the growth of the fundamental group.

math.MG↗

arXiv2021

Stability of the L^p-Norm of the Curvature Tensor at Kahler Space Forms

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by $R_p(g) :=\int_M|R(g)|^pdvg$ where $R(g)$, $dv_g$ denote the corresponding Riemannian curvature, volume form and p is a real number greater than or equal to 2. We prove that $R_p$ restricted to the space of Kahler metrics attains its local minima at a metric with constant holomorphic sectional curvature.

math.DG↗

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.

math.DG↗

arXiv2018

On the stability of L^p-norms of Curvature Tensor at Rank one symmetrics spaces

We study stability and local minimizing properties of $L^p$- norms of Riemannian curvature tensor denoted by $\mathcal{R}_p$ by variational methods. We compute the Hessian of $\mathcal{R}_p$ at compact rank 1 symmetric spaces and prove that they are stable for $\mathcal{R}_p$ for certain values of p > 2. A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of L^{n\2}- norm of Weyl curvature at these metrics.

math.DG↗

arXiv2012

On The Stability of The L^p Norm of The Curvature Tensor

We investigate stability and local minimizing properties of the Riemannian functional defined by the L^p norm of the curvature tensor on the space of Riemannian metrics on a closed manifold. Riemannian metrics with constant curvature and products of such metrics are critical points of this functional. We prove that these points are strictly stable for this functional and if (M; g) is a manifold of this type, g has a neighborhood U such that g is the strict minima on it.

math.DG↗

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.

math.DG↗

arXiv2011

On Wilking's criterion for the Ricci flow

B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$, which are nonnegative in a suitable sense, to every $Ad_{SO(n,\C)}$ invariant subset $S \subset {\bf so}(n,\C)$. For curvature operators of a Kähler manifold of complex dimension $n$, one considers $Ad_{GL(n,\C)}$ invariant subsets $S \subset {\bf gl}(n,\C)$. In this article we show: (i) If $S$ is an $Ad_{SO(n,\C)}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature and if $S$ is an $Ad_{GL(n,\C)}$ subset, then $C(S)$ is contained in the cone of Kähler curvature operators with nonnegative orthogonal bisectional curvature. (ii) If $S \subset {\bf so}(n,\C)$ is a closed $Ad_{SO(n,\C)}$ invariant subset and $C_+(S) \subset C(S)$ denotes the cone of curvature operators which are {\it positive} in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to either to a metric of constant positive sectional curvature or constant positive holomorphic sectional curvature or $M$ is a rank-1 symmetric space.

math.DG↗