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The Area and the Volume of a Constant Mean Curvature Surface

In this chapter we address the study of the area and the volume of a compact cmc surface with boundary. We shall see how the control of the area or the volume provides information concerning the geometry of the surface. In order to motivate the type of problems that we consider here, let Ω ⊂ R2 be a bounded domain. The domain Ω is itself a minimal surface with boundary Γ = ∂Ω and its volume is V = 0. For small values of H close to 0, there exist H -graphs on Ω spanning Γ and whose volume is also small. We may expect that given Ω and a sufficiently small volume V , the only compact cmc surface with boundary ∂Ω and volume V is a graph. This evidence is supported when we blow air across a planar wire. Once a plane closed wire is introduced into a soapy solution, the first surface that appears is precisely the planar domain bounded by the wire. If we blow a small amount of air against the surface, the bubble formed looks like a graph attached to the wire. We shall derive a formula giving a lower bound of the area in terms of the height of the surface with respect to the plane containing the boundary. We remark that the estimates of the height appearing in Theorem 2.3.5 for cmc graphs and Corollary 4.2.5 for embedded surfaces similarly relate the height to the value of the mean curvature. We will show that the solution of the isoperimetric problem for a convex planar domain is a graph if the volume is sufficiently small. In the case where the volume is large, we expect that the surface looks like a large spherical cap. We will prove this fact in Sect. 6.4 for the particular case where the surface is embedded spanning a convex curve and the surface lies on one side of the boundary plane. Some of the results in this chapter and their proofs are taken from [LM96, RR96].

6.1 A Monotonicity Formula for the Area Consider a plane P ⊂ R3 given by P = {p ∈ R3 : p − p0 , a = 0} with |a| = 1. If S ⊂ R3 , define the part of S above (resp. below) the plane P as the set S + = {p ∈ S : p − p0 , a > 0} (resp. S − = {p ∈ S : p − p0 , a < 0}). Analogously, the height R. López, Constant Mean Curvature Surfaces with Boundary, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-39626-7_6, © Springer-Verlag Berlin Heidelberg 2013

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6 The Area and the Volume of a Constant Mean Curvature Surface

h+ (resp. h− ) of S above (resp. below) P is h+ = sup{p − p0 , a : p ∈ S + }, (resp. h− = − inf{p − p0 , a : p ∈ S − }). Theorem 6.1.1 Let x : M → R3 be an immersion of a compact surface M with constant mean curvature H and whose boundary is included in a plane P . Then h+ ≤

A+ |H | , 2π

(6.1)

where A+ is the area of x(M)+ . Equality holds if and only if x(M)+ is a planar domain of P or a spherical cap. Proof If H = 0, the result is trivial by Theorem 3.1.12. Assume now H = 0 and let us orient M with the unit normal vector field N so that H > 0. After a change of coordinates, assume that P = {p ∈ R3 : p, a = 0}, where |a| = 1. Suppose that x(M)+ = ∅, on the