Exponential stability for the three-dimensional Navier-Stokes equations on negatively curved manifolds

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space ${\text{HH}}^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3,g)$ with pinched negative sectional curvature $-b^2\le K\le -a^2<0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian ${\mathrm{\Delta }}_{\text{Def}}={\mathrm{\Delta }}_B+\text{Ric}$ remains the viscous operator, selected by Lagrangian kinematics. We prove that the {exact} system admits a unique global mild solution for small $L^3$ data, with exponential decay at a rate determined by the spectral gap of the Stokes operator. The extension overcomes three obstacles absent on ${\text{HH}}^3$: (i) the semigroup factorisation $e^{t{\mathrm{\Delta }}_{\text{Def}}}=e^{-2t}{e}^{t{\mathrm{\Delta }}_B}$ fails because $\text{Ric}$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with ${\mathrm{\Delta }}_{\text{Def}}$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V=\text{Ric}+2a^{2}g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\text{PP},{\mathrm{\Delta }}_{\text{Def}}]$ to the complementary projector $(I-\text{PP})$ applied to the shifted Ricci endomorphism, giving a clean zeroth-order bound proportional to the curvature variation $b^2-a^2$. This is the sole source of a curvature pinching constraint. We resolve (iii) via McKean’s theorem, the diamagnetic inequality, and the Weitzenböck identity. The Fujita-Kato temporal singularity exponent $1\mathrm{/}2-3\mathrm{/}(2p)$ is unchanged from the ${\text{HH}}^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.