Min-max, phase transitions and minimal hypersurfaces

Min-max, phase transitions and minimal hypersurfaces

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Marco Guaraco, IMPA-Brazil
Fine Hall 314

Please note special time.  There is a strong correspondence between critical points of in the theory of phase transitions and critical points of the area functional in theory of minimal hypersurfaces. Historically, research has focused in studying the case of minima or stable critical points. We use ideas from Pitts to extend these results to the case of unstable critical points of any index. As an application we obtain a new min-max method for constructing embedded minimal hypersurfaces in an arbitrary closed manifold of any dimension. Our approach is variational, but it is substantially different from Almgren-Pitts theory. We also study the correspondence of critical points from a "global" variational point of view in the case of those obtained by min-max methods. In particular, we compare our construction of a minimal hypersurface with that of Almgren-Pitts and its refinement by Simon-Smith in dimension 3.