Inserted: 23 apr 2022
Last Updated: 24 may 2022
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 solve a conjecture of Allard [Invent. Math.,1983] in dimension $3$.