Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

Abstract

We consider graphs $${{\Sigma^n \subset \mathbb{R}^m}}$$ with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate $$ {\sup_{\Sigma \cap B_R} |A|^2 \leq \frac{C}{R^2}} $$ up to dimension n ≤ 5, where C is a constant depending on natural geometric data of Σ only. This generalizes previous results of Smoczyk et al. (Calc Var Partial Differ Equs 2006) and Wang (Preprint, 2004) for minimal graphs with flat normal bundle.