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

Abstract

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space $\HH^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3, g)$ with pinched negative sectional curvature $-b^2 \leq K \leq -a^2 < 0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian $Δ_\Def = Δ_B + \Ric$ 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 $\HH^3$: (i) the semigroup factorisation $e^{tΔ_\Def} = e^{-2t}e^{tΔ_B}$ fails because $\Ric$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with $Δ_\Def$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V = \Ric + 2a^2 g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\PP, Δ_\Def]$ to the complementary projector $(I-\PP)$ 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 $1/2 - 3/(2p)$ is unchanged from the $\HH^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.