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We consider the finite-size scaling of equilibrium droplet shapes for fluidadsorption (at bulk two-phase co-existence) on heterogeneous substrates andalso in wedge geometries in which only a finite domain $\Lambda_{A}$ of thesubstrate is completely wet. For three-dimensional systems with short-rangedforces we use renormalization group ideas to establish that both the shape ofthe droplet height and the height-height correlations can be understood fromthe conformal invariance of an appropriate operator. This allows us to predictthe explicit scaling form of the droplet height for a number of differentdomain shapes. For systems with long-ranged forces, conformal invariance is notobeyed but the droplet shape is still shown to exhibit strong scalingbehaviour. We argue that droplet formation in heterogeneous wedge geometriesalso shows a number of different scaling regimes depending on the range of theforces. The conformal invariance of the wedge droplet shape for short-rangedforces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.

Droplet shapes on structured substrates and conformal invariance