Explore a 50-minute mathematical lecture on mean curvature flow (MCF) in the Heisenberg group setting, delivered at the Centre International de Rencontres Mathématiques. Delve into the evolution of hypersurfaces where velocity at each point is determined by the mean curvature vector, with particular focus on mean convex MCF adaptation. Learn about the relationship between this flow and the Heisenberg isoperimetric problem, examining solution existence and uniqueness while discovering why the Pansu sphere doesn't evolve homothetically under MCF. Based on collaborative research with Gaia Bombardieri and Mattia Fogagnolo, this presentation from the "Frontiers in Sub-Riemannian Geometry" thematic meeting offers deep mathematical insights enhanced with chapter markers, keywords, abstracts, and bibliographies through CIRM's Audiovisual Mathematics Library.