Zusammenfassung für den Vortrag am 24.04.2018 (15:15 Uhr)Oberseminar ANALYSIS - PROBABILITY
Felix Schulze (University College London)
Optimal isoperimetric inequalities for 2-dimensional surfaces in Hadamard-Cartan manifolds in any codimension
Let (Mn,g) be simply connected, complete, with non-positive sectional curvatures, and Σ a 2-dimensional closed integral current (or ﬂat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. ﬂat chain mod 2) such that ∂S = Σ. We use a weak mean curvature ﬂow, obtained via elliptic regularisation, starting from Σ, to show that S satisﬁes the optimal Euclidean isoperimetric inequality: 6M[S] ≤ (M[Σ])3∕2 . We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by −κ < 0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric diﬀerence along the approximating ﬂows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with ﬁrst variation summable in L2.