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Interior gradients estimates and existence theorems for constant mean curvature graphs in $$M^n\times\mathbb R$$. (English) Zbl 1145.53048
Introduction: We establish a priori interior gradient estimates and existence theorems for $$n$$-dimensional graphs $$S= \{(x,u(x)): x\in\Omega\}$$ of constant mean curvature $$H>0$$ in an $$n+1$$-dimensional Riemannian manifolds of the form $$M^n\times\mathbb R$$, where $$M^n$$ is simply connected and complete and $$\Omega$$ is a bounded domain in $$M$$. Our aim is to illustrate the use of intrinsic methods that hold in great generality to obtain apriori estimates. In particular, we solve the Dirichlet problem for constant mean curvature graphs analogous to the results of J. Serrin [Philos. Trans. R. Soc. Lond., Ser. A 264, 413–496 (1969; Zbl 0181.38003), Proc. Lond. Math. Soc. (3) 21, 361–384 (1970; Zbl 0199.16604)].

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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