Configuration-Space Curvature and the Navier–Stokes Singular Set

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Abstract

At a singular point of a suitable weak solution of the three-dimensional Navier–Stokes equations, the partial-regularity theory of Caffarelli, Kohn and Nirenberg leaves two possibilities: either a configuration-space curvature concentrates, or the strain energy and the enstrophy come into balance. Working from the partial-regularity theory and the Liouville theory for divergence-free drifts, we control the first possibility coercively and constrain the second. The contribution is a geometric identity: the curvature of the $L^2$ metric on the group $\operatorname{SDiff}(\mathbb{T}^3)$ of volume-preserving diffeomorphisms is the pressure Hessian, the Gauss curvature of $\operatorname{SDiff}(\mathbb{T}^3)$ inside the flat group $\operatorname{Diff}(\mathbb{T}^3)$ whose second fundamental form is the pressure gradient; its $L^2$ size is the strain–enstrophy imbalance. The imbalance carries a transport law in which inertial production, curvature work, and viscosity each redistribute it with zero net budget. Under a scale-critical $L^3$ bound, the local form of the Escauriaza–Seregin–Šverák condition, the balanced possibility forces a bounded ancient pressureless flow that a Liouville theorem for divergence-free drifts excludes, so every such singular point, the Type-I points among them, is curvature-concentrating. Beyond the critical class the imbalance is the structure of the nonlinearity the energy identity does not see, the structure an energy-preserving averaged equation discards, so the curvature is where regularity is to be decided.