Inflation of a planar domain
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Take a soap film or other capillary surface that spans a fixed boundary at one end of a container and bounds the volume inside. If you increase the pressure in the bounded volume the soap film will expand outwards into the unbounded volume. In the absence of gravitational effects, if the boundary of the soap film remains stationary then the film will be a constant mean curvature surface at each point in time during the expansion. We model this process mathematically as an inflation, a one parameter family of constant mean curvature surfaces with the same boundary and with increasing bounded volume. Such families have been shown to exist as graphs over planar domains. However, this places an artificial restriction on an inflation as not all constant mean curvature surfaces can be represented as graphs over a plane. We avoid these restrictions by using an alternative representation of the surfaces. Specifically, we consider surfaces as graphs, not over a planar domain, but over a known nearby constant mean curvature surface. In so doing we prove the existence of new constant mean curvature surfaces beyond the limits of previous approaches.