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The interaction of surface geometry with morphogens.

Expressions are given for the Gauss and Mean curvatures of a surface of thickness h. The two curvatures, (K and H), which are given at each point of the middle surface, are adequate to describe the surface. The sheet thickness varies with position in the middle surface bisecting the apical and basal surfaces. The definitions of K and H are in terms of radii of curvature, but such radii are not appropriate variables for determining how morphogens in the surface may couple to the geometry. More suitable expressions are developed here. Two important geometrical constraints must be satisfied, namely the famous Gauss-Bonnet theorem, and an inequality stemming from the definition of the two curvatures. It is argued that these constraints are of great usefulness in determining the form of the coupling of morphogens to the geometry. In particular, when two key morphogens suffice to determine surface geometry, explicit expressions are suggested to determine both Gauss (K) and Mean (H) curvatures as functions of invariant morphogen densities.