%0 PDF %T A Theory of Sub-Finsler Surface Area in the Heisenberg Group %A Sánchez, Andrew. %D 2017-06-29T09:04:37.223-04:00 %8 2017-07-07 %R http://localhost/files/p8419060s %X Abstract: This work is concerned with the metric properties of continuous Heisenberg group under Carnot-Caratheodory metrics. Such metrics are in general sub-Finsler, but a particular choice of CC-metric yields the sub-Riemannian case, which has long been studied by analysts and geometers alike. In the sub-Riemannian case, there is a clear definition of surface area that can be easily expressed and computed. This thesis is a comprehensive study of surface area in general for the previously unstudied sub-Finsler cases, and proposes four inequivalent definitions of surface area based on Minkowski content that agree in the sub-Riemannian setting. A still-open conjecture by Pansu in the sub-Riemannian case is concerned with maximizing the the isoperimetric ratio (Volume)^(3/4) / (Surface area). Included in this thesis is a study of this isoperimetric problem in sub-Finsler cases using the theory of surface area established within, with bounds on the isoperimetric constants for the four notions, computed examples yielding best known constants in the CC-metrics that arise from the L-1 and L-Infinity norms, and theorems concerning first variation of perimeter and mean curvature.; Thesis (Ph.D.)--Tufts University, 2017.; Submitted to the Dept. of Mathematics.; Advisor: Moon Duchin.; Committee: Kim Ruane, Fulton Gonzalez, and Luca Capogna.; Keyword: Mathematics. %[ 2022-10-11 %9 Text %~ Tufts Digital Library %W Institution