Quasigeodesic flows, partial hyperbolicity and geometry of 3-manifolds

• Grant number: 25/11384-1
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: February 01, 2026
• End date: January 31, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: André Salles de Carvalho
• Grantee: Anindya Chanda
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/07076-4 - Dynamics and geometry in low dimensions, AP.TEM

Abstract

Our research plan lies at the intersection of dynamical systems and geometry, with a focus on understanding the interplay between Anosov and pseudo-Anosov flows and partially hyperbolic maps and the large-scale geometry of 3-manifolds supporting them. A central question we investigate is when the flowlines or leaves of one foliations associated to these systems are quasigeodesic-that is, when they closely resemble geodesics in the universal cover-since this property often bridges dynamical behavior with underlying geometric structures.Our proposed work consists of four interconnected projects:Project A explores Anosov flows on CAT(0) manifolds, a setting that generalizes hyperbolic geometry. While the relationship between Anosov flows and the Gromov boundary is well-studied in hyperbolic manifolds, little is known in CAT(0) settings. We aim to investigate whether weak-stable and weak-unstable foliations of such flows can be extended continuously to the visual boundary-analogous to the celebrated Cannon-Thurston theorem.Project B addresses leafwise quasigeodesic behavior in partially hyperbolic diffeomorphisms, which generalize Anosov systems. Recent breakthroughs suggest that many partially hyperbolic maps arise as collapsed Anosov flows (CAF)-flows that collapse onto simpler Anosov structures. We are particularly interested in understanding whether the center foliations in these systems are leafwise quasigeodesic, a property essential to classifying and characterizing CAFs.Project C investigates "zippers"-newly defined geometric structures on the boundary of hyperbolic spaces that emerge from group actions and quasigeodesic flows. These structures connect group theory, dynamics, and foliation theory. My goal is to further develop zipper constructions using Anosov-like group actions on R2, generalizing the flow-based approach to broader contexts.Project D is motivated by a recent joint work where we identified the first quasigeodesic Anosov flows on non-hyperbolic, non-Seifert fibred manifolds. We plan to expand this study to discover more such examples and understand which constructions (e.g., those based on hyperbolic plugs) preserve quasigeodesic behavior.These projects address several well-recognized areas in geometry and dynamics, including the study of Anosov flows and partial hyperbolicity, the geometry of foliations, and the role of group actions in low-dimensional topology. We believe this research will contribute new tools and perspectives to our understanding of 3-manifolds, bridging geometry, topology, and dynamical systems. (AU)