The Allen-Cahn energy is by now a well-understood way to approximate the area functional for hypersurfaces. Critical points of it converge to minimal hypersurfaces as we send the scaling parameter to zero, and the same holds for the gradient flow. Inspired by this parallel, De Giorgi proposed a conjecture which is analogous to the Bernstein problem for minimal graphs: given an entire critical point in dimension n < 9, monotone in one direction, is it necessarily a function of just one coordinate?
Savin solved this conjecture assuming local minimality, which can be seen to be implied by a mild additional assumption. We present an analogue in codimension two, for the abelian Yang-Mills-Higgs energy, which is known to approximate area in codimension two. The result is based on an improvement of flatness in the style of Allard and is partly inspired by an alternative proof of Savin's theorem by Wang. It also uses recent stability results in dimension two by Halavati. We also discuss some open questions.
This is joint work with Guido De Philippis (NYU Courant) and Aria Halavati (NYU Courant).
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