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

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