For spherical curves in a rotationally symmetric 3-manifold with density, the weighted mean curvature flow splits into radial drift and a shape-changing flow on a sphere, with long-time limits governed by a parabolic/hyperbolic dichotomy.
Gaussian mean curvature flow
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The curve shortening flow with density of a spherical curve in codimension two
For spherical curves in a rotationally symmetric 3-manifold with density, the weighted mean curvature flow splits into radial drift and a shape-changing flow on a sphere, with long-time limits governed by a parabolic/hyperbolic dichotomy.