A Theory of Sub-Finsler Surface Area in the Heisenberg Group
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 ... read moreand 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.read less