Surfaces of Finite III-type

In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces with respect to the third fundamental form of the surface. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.

Patterns for gridshells with negligible geometrical torsion at nodes

Curved envelope structural building envelopes have been quite popular in architecture in the past decades, and pose many challenges in their design, manufacturing and planning. In gridshells, a popular structural morphology for curved structure, designers will often strive to orient beams such that their top face is parallel to the envelope surface. However, this tends to induce geometrical torsion along the beam centerline, which complexifies significantly the manufacturing of the connection nodes or of the beams themselves. It is well known that such issue can be avoided by aligning beams with principal curvature directions of the envelope surface, thus yielding a quadrangular paneling. In this article, we study how other types of patterns (non-quadrangular) can be used to design torsion-free grid-shells. Based on asymptotic considerations, we derive a set of geometrical rules which, if fulfilled by a pattern, insure that a surface can be covered by this pattern with negligible torsion and limited deviation of beams from surface normals. A wide variety of patterns fulfill these rules, offering interesting possibilities for the design of curved architectural envelopes (Figure 1) is shown. As these rules are based on first order asymptotic analysis, we perform global validation on case studies. One main application is for structures in which face planarity is not necessary, for example ones cladded with ETFE cushions.

Deforming a Convex Hypersurface by Anisotropic Curvature Flows

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem.

Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature (“cylindrical estimates”) for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of the shrinking sphere amongst ancient solutions of these flows.