We show that certain functions whose nodal sets lie near a fixed nondegenerate minimal hypersurface satisfy a strong min-max principle for the Allen--Cahn energy which is analogous to the strong min-max principle for non-degenerate minimal hypersurfaces first proved by Brian White.
The wave equation for level sets is not a Huygens' equation
The co-points are cut points of level sets for Busemann functions
From constant mean curvature hypersurfaces to the gradient theory of phase transitions
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