A Theory of Sub-Finsler Surface Area in the Heisenberg Group
Sánchez, Ayla Paloma.
2017
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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 an... read mored 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 - ID:
- p8419060s
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