The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

Antonio De Rosa; Guido De Philippis

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The anisotropic Min-Max theory: Existence of anisotropic minimal and CMC surfaces

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arxiv 2205.12931 v1 pith:SQPT6VHT submitted 2022-05-23 math.DG math.AP

Guido De Philippis , Antonio De Rosa This is my paper

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keywords anisotropicsurfacesclosedconstantcurvatureexistencemeanmin-max

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abstract

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth $3$-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.

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