Abstract We present a simple discrete formula for the elastic energy of a bilayer. The formula is convenient for rapidly computing equilibrium configurations of actuated bilayers of general initial shapes. We use maps of principal curvatures and minimum-curvature direction fields to analyze configurations. We find good agreement between the computations and an approximate analytical solution fo...

We show that the space of asymptotically conical self-expanders mean curvature flow is a smooth Banach manifold. An immediate consequence non-degenerate self-expanders—that is, those admit no non-trivial normal Jacobi fields fix asymptotic cone—are generic in certain sense.

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius R by a distance η. The mean curvature was reported to vanish at the rim where the d-cone is supported. We investigate the ratio of the two principal curvatures versus sheet thickness h over a wider dynamic range than was used previously...

We develop a min–max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable are asymptotic to the same cone and bound domain, there exists new self-expander trapped between two.

Meshes with planar quadrilateral faces are desirable discrete surface representations for architecture. The present paper introduces new classes of planar quad meshes, which discretize principal curvature lines of surfaces in so-called isotropic 3-space. Like their Euclidean counterparts, these isotropic principal meshes meshes are visually expressing fundamental shape characteristics and they ...

Richard Serra is considered one of the leading sculptors of our time. What is not usually pointed out is that much of his work is really mainstream mathematical sculpture. This includes the torqued ellipses, torqued spirals, and spaces between surfaces of zero, positive, and negative curvature. That is, spaces between cylindrical surfaces, conical surfaces, spherical surfaces, and saddle surfac...

A Ricci surface is a Riemannian 2-manifold (M, g) whose Gaussian curvature K satisfies K∆K+g(dK, dK)+4K = 0. Every minimal surface isometrically embedded in R is a Ricci surface of non-positive curvature. At the end of the 19 century Ricci-Curbastro has proved that conversely, every point x of a Ricci surface has a neighborhood which embeds isometrically in R as a minimal surface, provided K(x)...

We study conformal metrics with prescribed Gaussian curvature on surfaces conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive general existence result for at least two components. This seems to be the first this setting. Moreover, allow have both positive negative orders, that is cone angles less grater than $2\pi$.