Nonsmooth methods in geometric function theory and geometric measure theory on the Heisenberg group

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Abstract Tyson The proposed research centers on a suite of problems at the interface between differential geometry, geometric measure theory and geometric function theory in the Heisenberg group and more general sub-Riemannian (Carnot-Caratheodory) spaces. The unifying theme is the development and application in this context of effective tools from nonsmooth metric geometry. One series of problems focuses on the geometry of submanifolds with possible application to the celebrated Heisenberg isoperimetry conjecture of Pierre Pansu. Sub-Riemannian analogs of the classical machinery of differential geometry have recently been introduced by Garofalo et al, Pauls, and Franchi et al, among others. In joint work with Capogna and Pauls, the PI will further develop this machinery in order to gain a more intrinsic understanding of the geometry of Carnot-Caratheodory submanifolds. This investigation is currently limited to surfaces given either intrinsically or extrinsically, as level sets of or parameterized by highly regular functions. Dimension jump phenomena and the size of the characteristic set for graphs of weakly regular functions will be investigated by the PI's graduate student John Maki in his thesis. A second line of research focuses on sub-Riemannian geometric function theory and fractal geometry. This includes the search for effective symmetrization procedures, existence, extension and regularity problems for quasiconformal maps, metric regularity of rough domains (John and uniform domains, domains satisfying a quasihyperbolic growth condition), and the structure of self-affine tilings. Finally (joint with Z. M. Balogh), the nonsmooth first-order calculus of Cheeger-Keith will be investigated in connection with exotic metrics on the Heisenberg group with an eye towards constructing new examples of spaces on which such calculus can be developed. Sub-Riemannian geometry is the "geometry of constrained motion"; it provides a mathematical model for any physical situation in which allowable motion is subject to a priori geometric constraints. Historically, its roots lie in Carnot's work on thermodynamics, but the subject has progressed significantly beyond these motivating questions to a central position in modern nonsmooth geometric analysis, and has recently seen remarkable applications in numerous areas, including robotic path planning, remote control of satellites, digital image reconstruction and computer vision, neurobiology, and the mathematics of finance. There are direct links between one aspect of the proposed research (sub-Riemannian differential geometry of submanifolds and the isoperimetric problem) and emerging models for the function and structure of the mammalian visual cortex. Nonsmooth techniques and methods are essential in geometric analysis in view of the incompleteness of classical spaces of smooth functions and sets; solutions to differential equations and variational problems cannot be guaranteed unless the domain of definition is widened to a suitably large family of (nonsmooth) candidates (although in hindsight, smoothness for such solutions can often be established a posteriori). The proposal includes an outreach component, joint with Capogna and Pauls, involving cross-training of graduate students and postdocs, a series of conferences, workshops and summer schools, expository articles and monographs, and an online forum for researchers in sub-Riemannian geometry aimed at developing a North American presence in this exciting field on par with the established centers of study in Europe and Australia.