Spacelike intersection curve of three spacelike hypersurfaces in E14E_1^4

In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_1^4$

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve

A parametric finite element method for fourth order geometric evolution equations

Accepted versio

Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional

We study the holographic entanglement entropy of spatial regions having arbitrary shapes in the AdS4/BCFT3 correspondence with static gravitational backgrounds, focusing on the subleading term with respect to the area law term in its expansion as the UV cutoff vanishes. An analytic expression depending on the unit vector normal to the minimal area surface anchored to the entangling curve is obtained. When the bulk spacetime is a part of AdS4, this formula becomes the Willmore functional with a proper boundary term evaluated on the minimal surface viewed as a submanifold of a three dimensional flat Euclidean space with boundary. For some smooth domains, the analytic expressions of the finite term are reproduced, including the case of a disk disjoint from a boundary which is either flat or circular. When the spatial region contains corners adjacent to the boundary, the subleading term is a logarithmic divergence whose coefficient is determined by a corner function that is known analytically, and this result is also recovered. A numerical approach is employed to construct extremal surfaces anchored to entangling curves with arbitrary shapes. This analysis is used both to check some analytic results and to find numerically the finite term of the holographic entanglement entropy for some ellipses at finite distance from a flat boundary

Distributed branch points and the shape of elastic surfaces with constant negative curvature

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric (DDG) approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius $R$ containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c' R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e. the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.Comment: 59 pages, 20 figures. Major revisions including new proofs with weakened hypotheses, expanded discussion and additional references. Some images are not at their original resolution to keep them at a reasonable size. Comments are very welcome and much appreciate

Parametric AFEM for Geometric Evolution Equations and Coupled Fluid-Membrane Interaction

When lipid molecules are immersed in aqueous environment at a proper concentration they spontaneously aggregate into a bilayer or membrane that forms an encapsulating bag called vesicle. This phenomenon is of interest in biophysics because lipid membranes are ubiquitous in biological systems, and an understanding of vesicles provides an important element to understand real cells. Also lately there has been a lot of activity when different types of lipids are used in the membrane. Doing mathematics in such a complex physical phenomena, as most problems coming from the bio-world, involves cyclic iterations of: modeling and analysis, design of a solving method, its implementation, and validation of the numerical results. In this thesis, motivated by the modeling and simulation of biomembrane shape and behavior, new techniques and tools are developed that allow us to handle large deformations of surface flows and fluid-structure interaction problems using the finite element method (FEM). Most simulations reported in the literature using this method are academic and do not involve large deformation. One of the questions this work is able to address is whether the method can be successfully applied to more realistic applications. The quick answer is not without additional crucial ingredients. To make the method work it is necessary to develop a synergistic set of tools and a proper way for them to interact with each other. They include space refinement/coarsening, smoothing and time adaptivity. Also a method to impose isoperimetric constraints to machine precision is developed. Another use of the computational tools developed for the parametric method is mesh generation. A mesh generation code is developed that has its own unique features not available elsewhere as for example the generation of two and three dimensional meshes compatible for bisection refinement with an underlying coarse macro mesh. A number of interesting simulations using the methods and tools are presented. The simulations are meant first to examine the effect of the various computational tools developed. But also they serve to investigate the nonlinear dynamics under large deformations and discover some illuminating similarities and differences for geometric and coupled membrane-fluid problems