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Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. We prove that there exists a closed hypersurface M whose mean curvature attains the values prescribed by g (joint work with Neshan Wickramasekera, Cambridge). Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C^2 immersed and two-sided (it admits a global unit normal); the scalar mean curvature at x is g(x) with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when g is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

The construction of M is carried out largely by means of PDE principles: (i) a minmax for an Allen--Cahn (or Modica-Mortola) energy, involving a parameter that, when sent to 0, leads to an interface from which the desired PMC hypersurface is extracted; (ii) quasi-linear elliptic PDE and geometric-measure-theory arguments, to obtain regularity conclusions for said interface; (iii) parabolic semi-linear PDE (together with specific features of the Allen-Cahn framework), to tackle cancellation phenomena that can happen when sending to 0 the Allen-Cahn parameter.