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.]

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.]

Reviewer: Michele Emmer (Roma)

##### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

35J60 | Nonlinear elliptic equations |

35B50 | Maximum principles in context of PDEs |