Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

In this paper, we investigate closed strictly convex hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊 + n + 1 {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1.

Complete CSC Hypersurfaces Satisfying an Okumura-Type Inequality in Ricci Symmetric Manifolds

We investigate the spacelike hypersurface with constant scalar curvature (SCS) immersed in a Ricci symmetric manifold obeying standard curvature constraints. By supposing these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez, which is a weaker hypothesis than to assume that they have two distinct principal curvatures, we obtain a series of umbilicity and pinching results. In particular, when the Ricci symmetric manifold is an Einstein manifold, then we further obtain some rigidity classifications of such hypersufaces.

Theoretical Modeling of Conformal Criterion for Flexible Electronics Attached onto Complex Surface

Transferring completed electronic devices onto curvilinear surfaces is popular for fabricating three-dimensional curvilinear electronics with high performance, while the problem of conformality between the unstretchable devices and the surfaces needs to be considered. Prior conformability design based on conformal mechanics model is a feasible way to reduce the non-conformal contact. Former studies mainly focused on stretchable film electronics conforming onto soft bio-tissue with a sinusoidal form microscopic morphology or unstretchable film conforming onto rigid sphere substrate, which limits its applicability in the aspect of shape and modulus of the substrate. Here, a conformal mechanics model with general geometric shape and material is introduced by choosing a bicurvature surface as the target surface, and the conformal contact behavior of film electronics is analyzed. All eight fundamental local surface features is obtained by adjusting two principal curvatures of the bicurvature surface, and the conformal performance is simulated. A dimensionless conformal criterion is given by minimizing the total energy as a function of seven dimensionless parameters, including four in geometric and three in material. The model and analysis results are verified by the finite element analysis, and it can provide a guidance for prior conformability design of the curvilinear electronic devices during the planar manufacturing process.

Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Tubular and membranous shapes display a wide range of morphologies that are difficult to analyze within a common framework. By generalizing the classical Helfrich energy of biomembranes, we model them as solutions to a curvature optimization problem in which the principal curvatures may play asymmetric roles. We then give a novel phase-field formulation to approximate this geometric problem, and study its Gamma-limsup convergence. This results in an efficient GPU algorithm that we validate on well-known minimizers of the Willmore energy; the software for the implementation of our algorithm is freely available online. Exploring the space of parameters reveals that this comprehensive framework leads to a wide continuum of shape textures. This first step towards a unifying theory will have several implications, in biology for quantifying tubular shapes or designing bio-mimetic scaffolds, but also in computer graphics, materials science, or architecture.

Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

Footvector Representation of Curves and Surfaces

This paper proposes a foot mapping-based representation of curves and surfaces which is a geometric generalization of signed distance functions. We present a first-order characterization of the footvector mapping in terms of the differential geometric invariants of the represented shape and quantify the dependence of the spatial partial derivatives of the footvector mapping with respect to the principal curvatures at the footpoint. The practical applicability of foot mapping representations is highlighted by several fast iterative methods to compute the exact footvector mapping of the offset surface of CSG trees. The set operations for footpoint mappings are higher-order functions that map a tuple of functions to a single function, which poses a challenge for GPU implementations. We propose a code generation framework to overcome this that transforms CSG trees to the GLSL shader code.

Surface model of the human red blood cell simulating changes in membrane curvature under strain

We present mathematical simulations of shapes of red blood cells (RBCs) and their cytoskeleton when they are subjected to linear strain. The cell surface is described by a previously reported quartic equation in three dimensional (3D) Cartesian space. Using recently available functions in Mathematica to triangularize the surfaces we computed four types of curvature of the membrane. We also mapped changes in mesh-triangle area and curvatures as the RBCs were distorted. The highly deformable red blood cell (erythrocyte; RBC) responds to mechanically imposed shape changes with enhanced glycolytic flux and cation transport. Such morphological changes are produced experimentally by suspending the cells in a gelatin gel, which is then elongated or compressed in a custom apparatus inside an NMR spectrometer. A key observation is the extent to which the maximum and minimum Principal Curvatures are localized symmetrically in patches at the poles or equators and distributed in rings around the main axis of the strained RBC. Changes on the nanometre to micro-meter scale of curvature, suggest activation of only a subset of the intrinsic mechanosensitive cation channels, Piezo1, during experiments carried out with controlled distortions, which persist for many hours. This finding is relevant to a proposal for non-uniform distribution of Piezo1 molecules around the RBC membrane. However, if the curvature that gates Piezo1 is at a very fine length scale, then membrane tension will determine local curvature; so, curvatures as computed here (in contrast to much finer surface irregularities) may not influence Piezo1 activity. Nevertheless, our analytical methods can be extended address these new mechanistic proposals. The geometrical reorganization of the simulated cytoskeleton informs ideas about the mechanism of concerted metabolic and cation-flux responses of the RBC to mechanically imposed shape changes.