Estimates for stable hypersurfaces of prescribed \(F\)-mean curvature (Q811867)

Curvature estimates for stable minimal hypersurfaces \(x:M\to N^{n+1}\) have been proved by \textit{R. Schoen, L. Simon, S.-T. Yau} [Acta Math. 134, 275--288 (1975; Zbl 0323.53039)]. It turns out that the method of Schoen-Simon-Yau is in fact general enough to be applicable to other important situations. In the paper under review the author considers immersed hypersurfaces \(X:M\to \mathbb R^{n+1}\) with prescribed anisotropic mean curvature \(H_F={\mathcal H}_F(X)\). Such hypersurfaces can be characterized as critical points of parametric functions of the type \[ {\mathcal F}^0(X)=\int_MF(N)\,d\mu+\int_M<Q(X),N>d\mu, \] with an elliptic Lagrangian \(F\) depending on normal directions and a smooth vector field \(Q\) satisfying \(\text{ div}_{\mathbb R^{n+1}}Q={\mathcal H}_F\). If \(F\) is \(C^3\)-close to the area integrand, the author establishes curvature estimates for stable hypersurfaces of dimension \(n\leq 5\). The case \(Q\equiv 0\) was studied by the author in two previous papers see [Calc. Var. Partial Differ. Equ. 23, 391--414 (2005; Zbl 1075.53053); Ann. Inst. H. Ponicaré, Anal. Non Linéaire 22, 543--555 (2005; Zbl 1088.53042)].