Related papers: Minimizers of the Willmore functional under fixed …

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 consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of…

We define and construct a conformally invariant energy for closed smoothly immersed submanifolds of even dimension, but of arbitrary codimension, in conformally flat Riemannian manifolds. This is a higher dimensional analogue of the…

We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerates in the moduli space or not. To do so, we translate the question on the conformal Gauss map's point of view.…

We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to…

Here we continue the investigation of the M\"obius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $\Sigma$ into $\mathbb{R}^n$ respectively…

Examples of aspherical closed symplectic 4-manifolds are presented whose Sullivan minimal models are (1,n)-formal for any n, without being formal. They have as cohomology algebra, signature, canonical class, those of a product of a closed…

We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this…

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with…

Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain…

We consider a free boundary problem for the Willmore functional. Given a smooth domain $\Omega$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial \Omega$ orthogonally along their…

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $$ \int_M(S-nH^2)^{\frac{n}{2}}dv, $$ where…

A foliated manifold (M,F) is minimizable if there exists a Riemannian metric g on M such that every leaf of F is a minimal submanifold of (M,g). Alvarez Lopez defined a cohomology class of degree 1 called the Alvarez class of (M,F) whose…

In this note two results are established for energy functionals that are given by the integral of $ W(\mathbf x,\nabla \mathbf u(\mathbf x))$ over $\Omega \subset\mathbb{R}^n$ with $\nabla \mathbf u \in BMO(\Omega;{\mathbb R}^{N\times n})$,…

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…

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma,…

We examine lower order perturbations of the harmonic map prob- lem from $\mathbb{R}^2$ to $\mathbb{S}^2$ including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform…

In this paper we present an inductive renormalizability proof for massive $\vp_4^4$ theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof…

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct…

We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a…