Geometry & Topology

Let $\gamma(t) = (x(t), y(t))$ be a smooth curve in the plane. Its signed area is defined by $$S(\gamma) = \frac{1}{2}\int_\gamma (x dy - y dx).$$ When $\gamma$ is a simple closed curve, this agrees, up to sign, with the area it encloses. Using 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 this result leads to a coarea formula for Lipschitz maps from $\mathbb H^1$ to $\mathbb R^2$, where $\mathbb H^1$ denotes the first subRiemannian Heisenberg group. This is a vector-valued coarea formula in the subRiemannian setting under only Lipschitz regularity, in a regime that was beyond the reach of previously available methods in geometric measure theory. Joint work with Robert Young (NYU).