Minimal Surfaces in Hyperbolic 3-Manifolds

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Minimal surfaces are essential objects studied in differential geometry and considered as the mathematical model of, for example, soap films. Being locally area minimizing, they are quite special and have applications in various domains, e.g., chemistry, materials science, biology, low dimensional topology and mathematical physics. In general relativity, minimal surfaces appear as models for the apparent horizons of black holes. In biology and material science, minimal surfaces are used in the design of materials with key applications. Recently, the PI used well-known notions about minimal surfaces to explain some fundamental questions in topological data analysis which have powerful applications in several fields in data science. In addition to this research, the PI aims to focus on teaching and training of undergraduate and graduate students as well as advancing the field by organizing seminars, conferences and writing expository materials. In particular, the project will study the existence and significant properties of minimal surfaces in hyperbolic 3-manifolds. Hyperbolic 3-manifolds are one of the most important families of manifolds in the low dimensional topology. Unfortunately, as the topological understanding of these manifolds is quite challenging, the study of minimal surfaces in this setting have not been considered by those in geometric analysis for many years. Even though there are several breakthrough results in geometric analysis in the past decade, minimal surfaces in hyperbolic 3-manifolds are still an uncharted territory, and many fundamental questions are still open. In this project, the PI aims to completely resolve the existence question of minimal surfaces in infinite volume hyperbolic 3-manifolds, and study one of Thurston’s famous conjectures about the minimal foliations in closed hyperbolic 3-manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.