Universal thin-shell limits for the viscous operator on Riemannian hypersurfaces

TL;DR

This paper establishes universal limits for the viscous operator on Riemannian hypersurfaces, showing how boundary conditions influence the intrinsic and extrinsic geometric operators in fluid dynamics.

Contribution

It generalizes sphere-specific results to arbitrary hypersurfaces, revealing universal deformation and Hodge Laplacian limits under specific boundary conditions.

Findings

Stress-free boundary conditions yield the deformation Laplacian universally.

Hodge boundary conditions yield the Hodge Laplacian universally.

Derived a family of boundary conditions interpolating between limits, coupling extrinsic geometry in intermediate regimes.

Abstract

We decompose the ambient Bochner Laplacian acting on tangential vector fields on a thin shell around an arbitrary smooth hypersurface $M^n \hookrightarrow \R^{n+1}$ into an intrinsic piece and a radial boundary-shear piece. The intrinsic piece is the deformation Laplacian $\Delta_B^{(n)} + \Ric^{(n)}$ on every hypersurface, regardless of extrinsic geometry. The boundary-shear piece is determined entirely by the normal profile of the velocity field. We prove that stress-free (Navier slip) boundary conditions yield the deformation Laplacian universally, and that Hodge (zero tangential vorticity) boundary conditions yield the Hodge Laplacian universally. Both results hold on any smooth hypersurface, not only on surfaces of constant curvature. This extends the sphere-specific results of Temam-Ziane and Miura to the general case and explains the extension-dependence found by Chan, Czubak, and Yoneda on the ellipsoid as a physical boundary-condition dependence. We also derive a continuous one-parameter family of boundary conditions interpolating between the two limits, producing an effective viscous operator $\Delta_\alpha = \Delta_\Def - 2\alpha\,\Ric - 4\alpha(1-\alpha)S^2$ that couples to the extrinsic geometry (through the shape operator squared) only in the intermediate partial-slip regime.