Abstract:

The Allen-Cahn equation is a reaction-diffusion equation that originally comes from mathematical-physics, but in recent decades has found a new home in the field of geometric analysis. The equation describes the separation of a space, in our case a manifold, into two states, with a boundary in between. Imagine oil and water separating out in a bottle. This has become a very successful method of producing hypersurfaces with properties that mathematicians are interested in. In this talk I plan to give a heuristic idea of how the Allen-Cahn equation can prove existence of minimal hypersurfaces on closed Riemannian Manifolds with particular focus on Min-Max techniques from PDE theory. The proof of existence of minimal hypersurfaces by the Allen-Cahn equation is due to Guaraco. The appropriate work can be found at: https://arxiv.org/abs/1505.06698, and will be the main reference for this talk. Attached picture is found in the cited work by Guaraco.