Lecture Notes of Seminario Interdisciplinare di Matematica
We formulate the isoperimetric problem for the class of C2 smooth cylindrically symmetric surfaces in the Heisenberg group in terms of Legendrian foliations. The known results for the sub-Riemannian isoperimetric problem yield a new isoperimetric inequality in the plane: For any strictly convex, C2 loop γ ∈ R2 , bounding a planar region ω, we have I(ω) 3 4 ≤ √ π 3 1 4 (8π) 3 4 L3, where I(ω) = R ω |z| 2dz is the moment of inertia and L3 is the length of the curve γ 3 . Moreover if equality is achieved then γ is a circle.
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Capogna, L. (2007). Isoperimetric inequalities in the Heisenberg group and in the plane. Lecture Notes of Seminario Interdisciplinare di Matematica, 6, 93-106.