Analysis Seminar

Speaker: Gioachinno Antonelli

Title: Area of Hölder planar curves and the coarea formula for Lipschitz maps on the Heisenberg group

Abstract: Let \gamma=(x,y) be a smooth curve in the plane. Its signed area is defined by S(\gamma) := 1/2 \int_\gamma (x dy − y dx). When \gamma is a simple closed curve, this agrees, up to sign, with the area it encloses. By Young integration, S(\gamma) is well defined for every \alpha-Hölder curve with \alpha>1/2. In this talk, I will discuss a new sharp existence result for S(\gamma) at the critical exponent \alpha = 1/2, under an additional square-summability assumption. I will explain how ideas behind this result lead to a proof of the coarea formula for Lipschitz maps from H^1 to R^2, where H^1 denotes the first sub-Riemannian Heisenberg group. This gives a vector-valued coarea formula in the sub-Riemannian setting for merely Lipschitz maps, in a regime that was beyond the reach of previously available methods in geometric measure theory. This is joint work with Robert Young (NYU).