2023/03/14 by Marc Arnaudon, Arnaudon, Marc, Koléhè Coulibaly-Pasquier +3
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2303.07921
openalex publication_date 2023/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate renormalized curvature flow (RCF) and stochastic renormalized curvature flow (SRCF) for convex sets in the plane.RCF is the gradient descent flow for logarithm of σ/λ2 where σ is the perimeter and λ is the volume. SRCF is RCF perturbated by a Brownian noise and has the remarkable property that it can be intertwined with the Brownian motion, yielding a generalization of Pitman "2M-X" theorem. We prove that along RCF, entropy Et for curvature as well as ht:=σt/λt are non-increasing. We deduce infinite lifetime and convergence to a disk after normalization.For SRCF the situation is more complicated. The process (ht)t is always a supermartingale. For (Et)t to be a supermartingale, we need that the starting set is invariant by the isometry group Gn generated by the reflection with respect to the vertical line and the rotation of angle 2π/n with n≥ 3. But for proving infinite lifetime, we need invariance of the starting set by Gn with n≥ 7. We provide the first SRCF with infinite lifetime which cannot be reduced to a finite dimensional flow. Gage inequality plays a major role in our study of the regularity of flows, as well as a careful investigation of morphological skeletons. We characterize symmetric convex sets with star shaped skeletons in terms of properties of their Gauss map. Finally, we establish a new isoperimetric estimate for these sets, of order 1/n4 where n is the number of branches of the skeleton.