Related papers: Analysis aspects of Willmore surfaces

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an…

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange…

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale…

We consider a closed Willmore surface properly immersed in ${\R}^m$ (m>2) with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the "conservative" reformulation of the…

This paper studies the regularity of constrained Willmore immersions into $\R^{m\ge3}$ locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions…

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on…

We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…

We study isometric immersions of a Riemannian surface $(\Omega,\frak{g})$, where $\Omega \subset \mathbb{R}^2$, into $\mathbb{R}^3$. We consider their bending energy, i.e., the square of the $L^2$-norm of their second fundamental form,…

We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(\sigma^{\cal…

We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling…

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the…

The unsigned p-Willmore functional introduced in \cite{mondino2011} generalizes important geometric functionals which measure the area and Willmore energy of immersed surfaces. Presently, techniques from \cite{dziuk2008} are adapted to…

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…

In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The…

We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal…

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean…

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric…

First introduced to describe surfaces embedded in $\mathbb{R}^3$, the Willmore invariant is a conformally-invariant extrinsic scalar curvature of a surface that vanishes when the surface minimizes bending and stretching. Both this invariant…

The Willmore Problem seeks the surface in $\mathbb S^3\subset\mathbb R^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |\mathbf{H}_{\mathbb{R}^4}|^2 = \operatorname{area} + \int H_{\mathbb{S}^3}^2$. The…

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with…