arXiv subjects

Balarka Sen

Publications and source records attributed to Balarka Sen.

8 recordsLinked to original sources

arXiv2026

Urysohn width and macroscopic scalar curvature

We show that the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above. This is based on (1) a novel estimate on the codimension two Urysohn width of circle bundles over manifolds with large hypersphericity radius, and (2) a notion of ruling for Riemannian manifolds that yields circle bundles with total spaces admitting metrics of positive macroscopic scalar curvature. Along the way, we also show that Urysohn width is not continuous under Cheeger-Gromov collapsing limits. This article is a continuation of our study of metric invariants and scalar curvature for circle bundles over large Riemannian manifolds initiated in [KS25].

math.DG↗

arXiv2025

Contact domination

In this note, we prove that every closed connected oriented odd-dimensional manifold admits a map of non-zero degree (i.e., a domination) from a tight contact manifold of the same dimension. This provides an odd-dimensional counterpart of a symplectic domination result due to Joel Fine and Dmitri Panov. We prove that the dominating contact manifold can be ensured to be Liouville-fillable, but not Weinstein-fillable in general. We discuss an application for contact divisors arising as zero sets of asymptotically contact-holomorphic sections.

math.SG↗

arXiv2025

Positive scalar curvature and exotic structures on simply connected four manifolds

We address Gromov's band width inequality and Rosenberg's $S^1$-stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a related result for non-simply connected smooth four manifolds.

math.DG↗

arXiv2024

Circle bundles with PSC over large manifolds

We construct infinitely many examples of macroscopically large manifolds of dimension $m \geq 4$ equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $\lceil m/2 \rceil + 1$. In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.

math.DG↗

arXiv2024

Tight contact structures on hyperbolic homology 3-spheres

We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class of oriented plane distributions. As a corollary, we give a recipe to construct hyperbolic L-spaces admitting arbitrarily many distinct tight contact structures. We also introduce a notion of geometric limits of contact structures compatible with geometric limits of hyperbolic manifolds and study the behavior of the tight contact structures we construct under geometric limits.

math.GT↗

arXiv2023

h-principle for loose Legendrian embeddings

This article provides an exposition of Emmy Murphy's work on loose Legendrian embeddings. After a brief review of the rudiments of contact topology, we state and discuss some foundational results from the theory of h-principles, providing many relevant examples from contact topology on the way. We then proceed to prove Murphy's h-principle for loose Legendrian embeddings. We also provide an accessible exposition of some background material from microlocal sheaf theory. As applications, we demonstrate the existence of non-loose Legendrian embeddings, and prove a version of Gromov's nonsqueezing theorem for loose charts.

math.SG↗

arXiv2022

h-Principle for Stratified Spaces

We extend Gromov and Eliashberg-Mishachev's h-principle on manifolds to stratified spaces. This is done in both the sheaf-theoretic framework of Gromov and the smooth jets framework of Eliashberg-Mishachev. The generalization involves developing 1) the notion of stratified continuous sheaves to extend Gromov's theory, 2) the notion of smooth stratified bundles to extend Eliashberg-Mishachev's theory. A new feature is the role played by homotopy fiber sheaves. We show, in particular, that stratumwise flexibility of stratified continuous sheaves along with flexibility of homotopy fiber sheaves furnishes the parametric h-principle. We extend the Eliashberg-Mishachev holonomic approximation theorem to stratified spaces. We also prove a stratified analog of the Smale-Hirsch immersion theorem.

math.GT↗

arXiv2022

The Gromov-Tischler theorem for stratified spaces

We define a notion of a symplectic structure on stratified spaces, and demonstrate that given a symplectic structure on a stratified space $X$ with integral cohomology class, $X$ can be symplectically embedded in some complex projective space equipped with the standard K\"ahler form. This extends a theorem, due to Gromov and Tischler for manifolds, to stratified spaces.

math.SG↗