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Some remarks on surfaces of prescribed mean curvature. (English) Zbl 0773.53002
Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 123-148 (1991).
Throughout this paper, we study properties of prescribed mean curvature surfaces in $$R^3$$; particularly, surfaces of constant mean curvature and surfaces presented in non parametric form.
In section 3, we study surfaces given as graphs over a domain $$\Omega$$ in the plane. We prove that if the mean curvature of the graph is bounded in a neighborhood $$D^*$$ of a puncture of $$\Omega$$, then the graph is also bounded over $$D^*$$. Our proof uses a one parameter family of Delaunay surfaces as barriers (R. Finn introduced this technique [J. Anal. Math. 14, 139–160 (1965; Zbl 0163.34604)]). When the mean curvature is zero, this boundedness result implies the theorem of Bers: the minimal surface equation has removable isolated singularities. R. Finn generalized this to prove the constant mean curvature equation has removable isolated singularities [Commun. Pure Appl. Math. 9, 415–423 (1956; Zbl 0070.32201)], (for more general results of R. Finn, see [Trans. Am. Math. Soc. 75, 385–404 (1953; Zbl 0053.39205)] and [Lect. Notes Math. 1357, 156–197 (1988; Zbl 0692.35006)]). More generally, our result implies that isolated singularities of the prescribed mean curvature equation are removable, provided the mean curvature function is $$C^1$$ at the puncture. Consider an embedded Delaunay surface $${\mathcal D}$$ in $$R^3$$ of mean curvature $$1/2a$$.
In section 4 we prove a maximum principle inside $${\mathcal D}$$, which allows us to derive the following corollary: Let $$M$$ be a connected complete (unbounded) real analytic properly immersed surface in $$R^3$$, with compact boundary. Let $$H$$ be the mean curvature vector of $$M$$. If $$\vert H\vert \leq 1/2a$$ and if $$M$$ lies inside $${\mathcal D}$$, then $$M={\mathcal D}$$.
[For the entire collection see Zbl 0718.00010.]

##### MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35J60 Nonlinear elliptic equations 35B50 Maximum principles in context of PDEs