Topological Rigidity and the Mathematical H-Theorem for the Ricci Flow

We propose a thermodynamic characterization of 3-manifold topology via the global dynamicalproperties of the Ricci flow. While Perelman’s W-entropy is known to be monotonicalong smooth solutions, its global behavior across surgeries and infinite-time limits in topologicallycomplex manifolds is typically singular. In this work, we reverse the perspectiveand introduce the Mathematical H-Theorem (MHT).We investigate the consequences of assuming the existence of a globally admissible Ricciflow governed by a strict Lyapunov functional H := −W, which is monotonic, bounded frombelow, and well-defined across surgeries. We prove that the existence of such a functional isrigidly linked to the topological simplicity of the underlying manifold. In particular, we showthat manifolds containing hyperbolic components or non-trivial JSJ decompositions exhibita dynamical “phase separation” phenomenon, leading to incompatible geometric asymptoticsthat obstruct the existence of a single globally bounded energy functional.Consequently, we establish that any 3-manifold admitting a Mathematical H-Theoremmust be geometrically simple (spherical, flat, or S2 × S1-type). As a corollary, we obtaina dynamical reformulation of the Poincar´e conjecture, characterizing the 3-sphere as theunique simply connected 3-manifold admitting a globally stable, entropy-dissipating Ricciflow evolution.Note: This work proposes a conceptual framework integrating geometric analysis with thermodynamic stability constraints. It establishes a topological selection principle based on the global Lyapunov properties of the Ricci flow.