# Convex caps with prescribed edges and curvature

Let $G$ be the edge graph of a convex subdivision of a convex polygon $P$ in the plane. I would like to construct a convex polyhedral cap $C$ (with zero boundary values) over $P$ whose edges project onto $G$ and whose interior vertices have prescribed curvature.

It is well known that for $C$ to satisfy the edge condition, $G$ must meet certain requirements in the context of Maxwell-Cremona correspondence; see the answer to this earlier question. So let us assume that $G$ meets these conditions.

It is also known that one can construct a convex cap over $P$ whose interior vertices project onto those of $G$ and have prescribed curvature values as long as these values are all positive and add up to less than $2\pi$. This is a lemma of Pogorelov; see p. 319 of Pak's Lecture Notes.

In short, one can prescribe the edges of $C$, and one can also prescribe the curvature of the interior vertices of $C$ independently, but to what extent can one do both at the same time:

Question: Suppose that we have a convex subdivision of a convex polygon which can be lifted to a convex cap. What are all the possible values for the curvatures at the interior vertices of this cap?

I know that theoretically one can compute these curvatures in a given situation via the Maxwell-Cremona machinery; however, I am wondering if there are some relatively nice integrability conditions which can be formulated in general, to somehow yield a more refined version of the Maxwell-Cremona correspondence.

Given Gaussian curvatures at the vertices, there is a unique lift that realizes these curvatures, as you can see from Igor's note.

Given a graph, the set of liftings that projects to this graph form a (very nice) polyhedral cone called secondary cone. See Section 5.2.1 of Triangulations by de Loera, Rambau and Santos.

So the question is actually about how to compute Gaussian curvatures from a lifting. I don't see a "nice" way of doing this. Some elementary but nasty trigonometry will be involved, and will destroy all the nice linear properties of lifting. I would say that Gaussian curvature is not the nice parameter to describe the set of liftings for a fixed combinatorics.

• Your answer seems too pessimistic. There are a number of things that one can say. For instance, one can make all the the curvatures arbitrarily small by an affine transformation or rescaling of the cap. Second, one can make the curvature of the highest point of the cap arbitrarily large, while making the rest of the curvatures arbitrarily small, again by rescaling. Third the set of possible values for the curvatures are open when the subdivision is simplicial. Is there not anything else that one can say? For instance, for which vertices can one make the curvature large? Commented Oct 12, 2017 at 12:16
• @MohammadGhomi The things you mention here are all much simpler without curvature, by noticing that the map from lifting to curvature is continuous. For example: when you say "rescaling" you mean rescaling the height; when you say open, you mean the interior of a secondary cone. These topological statements are all consequences of the secondary cone / fan, and are much nicer with lifting instead of Gaussian curvature. If you insist, I suggest looking at mean curvature (dihedral angle $\times$ edge length), which might behave nicer. Although still no more info than the secondary cone. Commented Oct 12, 2017 at 17:24
• Thanks for elaboration. But my aim is not to parametrize the space of liftings at all. I understand that such a parametrization already exist in terms of the cone which you mention. Rather the goal is to understand the curvature at the vertices, due to certain important applications where one needs to study the Gauss curvature at the vertices. The mean curvature would not be at all relevant in that context, and would not be helpful. Commented Oct 12, 2017 at 18:09