Moving Manifolds and the Poincare Conjecture
David V. Svintradze
TL;DR
The paper develops the calculus of moving surfaces (CMS) as an extrinsic, energy-based framework for evolving embedded hypersurfaces in $\mathbb{R}^{n+1}$. By coupling normal velocity to extrinsic curvature and enforcing topology conservation, CMS identifies constant-mean-curvature (CMC) configurations as stationary states, and, via Alexandrov’s theorem, demonstrates that simply connected compact initial data relax to round spheres in any dimension. This provides a deterministic geometric route to sphericalization that complements Ricci flow by preserving topology and avoiding surgeries. The approach unifies dynamics, curvature diffusion, and topological invariants within a single variational scheme, offering a Poincaré-type conclusion in a broad geometric setting.
Abstract
We present a differential geometric formulation of the Poincare problem using the calculus of moving surfaces (CMS). In this framework, an n dimensional compact hypersurface evolves under a velocity field that couples motion to the extrinsic curvature tensor while preserving topology through smooth diffeomorphic flow. A variational energy principle identifies constant mean curvature (CMC) manifolds as the unique stationary equilibria of CMS dynamics. Consequently, the evolution of any compact simply connected hypersurface relaxes to a CMC equilibrium and, in the isotropic case, to the round sphere. Unlike Ricci flow approaches, which are dimension restricted and require topological surgery, the CMS formulation holds for all dimensions and preserves manifold topology for all time. This provides a deterministic geometric mechanical route to the Poincare conclusion, unifying dynamics, topology, and equilibrium geometry within a single analytic framework.