The restriction problem on the ellipsoid

TL;DR

This paper develops a geometric invariant formula for a viscosity operator on an ellipsoid in 3D space, using restriction arguments, and provides an asymptotic expansion based on the ellipsoid's eccentricity.

Contribution

It introduces a novel geometric invariant formula for viscosity operators on ellipsoids and analyzes its asymptotic behavior related to eccentricity.

Findings

Derived a geometric invariant viscosity operator formula for ellipsoids.

Provided an asymptotic expansion of the formula in terms of eccentricity.

Enhanced understanding of fluid flow on curved surfaces with geometric invariants.

Abstract

Following a restriction argument in the Euclidean space, we derive a geometric invariant formula for a possible viscosity operator for an incompressible fluid flow on an ellipsoid embedded in $\mathbb R^3$. We also give an asymptotic expansion of the formula in terms of the eccentricity associated with the ellipsoid.