FRG: Collaborative Research: Geometric Structures on Higher Teichmuller Spaces

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A surface is a space which looks locally like the 2-dimensional plane, e.g. the surface of a basketball or a pretzel. Surfaces arise naturally in many scientific fields. A geometric structure is a way of measuring distances and angles on a surface or more complicated object. Studying spaces of geometric structures (or shapes) on a fixed object gives further information about their nature. The classical Teichmuller theory studies a space which parametrizes certain geometric structures (of constant curvature) on a fixed surface. Teichmuller theory has impacted diverse areas in mathematics, including algebraic geometry, complex analysis, low-dimensional topology, and dynamics, as well as theoretical physics through its connections with string theory. A metric on Teichmuller space is a way of measuring the distance, or difference, between two such geometric structures. The PIs plan to study metrics on a generalization of this theory called Higher Teichmuller Theory. Higher Teichmuller spaces may be viewed as deformation spaces of geometric structures on higher-dimensional spaces. It shares some of the nice properties of the classical theory and has become a very active field of research. The PIs will mentor graduate students who will be engaged in aspects of the project. They will also run a program which helps science and engineering students from low-resource high schools transition to college studies. Higher Teichmuller theory studies spaces of "geometric" representations of a hyperbolic group into a semi-simple Lie group. The main goal is to develop a theory which shares the richness, beauty and versatility of classical Teichmuller theory. The Higher theory has exploded in popularity because of the interactions it fosters between the subjects of geometric topology, real and complex differential geometry, Lie theory, algebraic geometry, string theory, and dynamics. Bridgeman, Canary, Labourie and Sambarino used thermodynamic formalism to construct a pressure metric on many higher Teichmuller spaces which is motivated by Thurston's definition of the Weil-Petersson metric on Teichmuller space (and its reformulations by Bonahon and McMullen). In the special case of the Hitchin component, the pressure metric is a mapping class group invariant, analytic Riemannian metric whose restriction to the Fuchsian locus is a multiple of the Weil-Petersson metric. Wolf developed an analogous approach to the Weil-Petersson metric, and has results on the isometry group and curvature of the Weil-Petersson metric, degeneration of hyperbolic structures, and on harmonic maps (Hitchin equations) approaches to Teichmuller theory. Wentworth has worked on the pressure metric, Weil-Petersson geometry, Higgs bundles and harmonic maps. The PIs together propose to study the isometry group, curvature and metric completion of both the pressure metric and variants on Hitchin components and quasifuchsian spaces, aiming to understand the pressure metric on general higher Teichmuller spaces.