Isometric Rigidity of the Wasserstein Space W1(G) Over Carnot Groups

This paper aims to study isometries of the 1-Wasserstein space W1(G) over Carnot groups endowed with horizontally strictly convex norms. Well-known examples of horizontally strictly convex norms on Carnot groups are the Heisenberg group Hn endowed with the Heisenberg-Korányi norm, or with the Naor-Lee norm; and H -type Iwasawa groups endowed with a Korányi-type norm. We prove that on a general Carnot group there always exists a horizontally strictly convex norm. The main result of the paper says that if (G, NG) is a Carnot group where NG is a horizontally strictly convex norm on G, then the Wasserstein space W1(G) is isometrically rigid. That is, for every isometry Φ : W1(G) → W1(G) there exists an isometry ψ : G → G such that Φ = ψ# .