Min-max minimal spheres in infinitely many Berger 3-spheres are never connected by eternal mean curvature flow to lower-area minimal surfaces, due to a genus obstruction.
Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol
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When can minimal hypersurfaces be connected by mean curvature flow?
Min-max minimal spheres in infinitely many Berger 3-spheres are never connected by eternal mean curvature flow to lower-area minimal surfaces, due to a genus obstruction.