Related papers: Helfrich's Energy and Constrained Minimisation

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent M\"obius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we…

The existence of minimizers of the Canham--Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham--Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss…

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The…

We prove an enumerative min-max theorem that relates the number of genus g minimal surfaces in 3-manifolds of positive Ricci curvature to topological properties of the set of embedded surfaces of genus $\leq g$, possibly with finitely many…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to…

We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the…

We study global Mumford-Shah minimizers in $\R^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone.…

We study the constrained minimum energy problem with an external field relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant…

Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus…

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to H\'elein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence…

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell…

Smooth axially symmetric Helfrich topological spheres are either round or else they must satisfy a second order equation known as the reduced membrane equation [17]. In this paper, we show that, conversely, axially symmetric closed genus…

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…

Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$, and that $\Gamma$ is a smooth submanifold of $N$. This paper proves that for a generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$, if…

We consider an energy functional motivated by the celebrated $K_{13}$ problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional…

given two minimal surfaces embedded in $\S3$ of genus $g$ we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in $\S3$ of genus $g$ that converges in $C^{2,\alpha}$ to a compact embedded minimal surface…

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(\kappa,\tau)$ with isometry group of…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the…

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is…