arXiv research

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…

New surfaces described that are symmetric and solve a specific equation.

Mathematical model describes how red blood cells return to equilibrium.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a sp…

Study of red blood cells using elastic surface theory.

For every $g\in\mathbb{N}_0$ and $ε>0$, we construct a smooth genus $g$ surface embedded into the unit ball with area $8π$ and Willmore energy smaller than $8π+ ε$. From this we deduce that a minimising sequence for Willmore's energy in the class of genus $g$ surfaces embedded in the unit ball with area $8π$ converges …

Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.

Study connects hyperbolic geometry to membrane shapes.

Study on membranes under confinement, proving existence and regularity of minimizers.

Lowered regularity assumption for a phase-dependent Helfrich energy equation.

Study the stability of membranes using Helfrich energy and second variation formula.

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solut…

Discretizes Helfrich-type energies on surfaces using triangular complexes.

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures…

We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers…

In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …

Delaunay tori minimize Willmore energy under isoperimetric constraints.

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…

Flow preserves isoperimetric ratio for immersed surfaces.

Study on surface configurations with curvature and elasticity.

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

In this paper we consider the evolution of regular closed elastic curves $γ$ immersed in $\R^n$. Equipping the ambient Euclidean space with a vector field $\ca:\R^n\rightarrow\R^n$ and a function $f:\R^n\rightarrow\R$, we assume the energy of $γ$ is smallest when the curvature $\k$ of $γ$ is parallel to $\c = (\ca \cir…

Symmetric hypersurfaces and boundaries in R^n+1 with group actions.

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'.…

Study on evolving interfaces with complex curvature and density effects.

Generates tubular and membranous shapes using curvature functionals.

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We develop the multicomponent functionalized Cahn-Hilliard (mFCH) free…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the exist ence of a $C^{1,1}$-regular minimizer for general geometric functionals and c…

For a bounded smooth domain in the plane and smooth boundary data we consider the minimisation of the Willmore functional for graphs subject to Dirichlet or Navier boundary conditions. For $H^2$-regular graphs we show that bounds for the Willmore energy imply area and diameter bounds. We then consider the $L^1$-lower s…

Recent years have witnessed a trend that advanced mathematical tools, such as algebraic topology, differential geometry, graph theory, and partial differential equations, have been developed for describing biological macromolecules. These tools have considerably strengthened our ability to understand the molecular mech…

New research finds 145 infinite families of CS spheres are standard.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

New theory proves infinite homology 3-spheres in homology 4-spheres.

Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.

Reduces weak reducing pairs to spheres in 3-sphere Heegaard surfaces.

The study shows how to construct $d$-spheres from $(d-1)$-spheres and $d$-balls without additional vertices.

New proof for sphere recognition algorithm.

Proves stability of convex spheres with similar geodesic lengths.

Soliton spheres are immersed 2-spheres in the conformal 4-sphere S^4=HP^1 that allow rational, conformal parametrizations f:CP^1->HP^1 obtained via twistor projection and dualization from rational curves in CP^{2n+1}. Soliton spheres can be characterized as the case of equality in the quaternionic Pluecker estimate. A …

Study on sphere immersions and their stability indices.

The paper constructs biharmonic maps between spheres using polynomial maps.

Characterizes a specific type of convex curves on a 3-sphere.

Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.

Author provides an alternate proof of the free ribbon lemma.

Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.

Sharp convergence theorem for sphere submanifolds proved.