Spectral rigidity and topological completeness of the scalar conformal flow

Determine the spectral rigidity and topological completeness of the scalar conformal flow C(v, τ), defined via the conformal metric evolution g_ij(τ) = C(v, τ) g_ij^0 and proposed for geometric smoothing and classification of three-manifolds; specifically, ascertain whether this flow exhibits spectral rigidity and whether its induced topological classification of compact 3-manifolds is complete under the framework introduced.

Background

The paper introduces a velocity-dependent scalar deformation function C(v) and extends it to a flow C(v, τ) that drives a conformal evolution of the metric g_ij(τ) = C(v, τ) g_ij^0. This scalar flow is presented as a singularity-free analogue of Ricci flow aimed at smoothing geometry and supporting the classification of 3-manifolds.

While the authors outline convergence properties and discuss invariant limits suggestive of S^3 classification under certain assumptions, they explicitly state that questions concerning spectral rigidity and the completeness of the resulting topological classification are not resolved within this work and remain open.