Related papers: Willmore surfaces in spheres via loop groups III: …

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$,…

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ of finite uniton number, from a Riemann surface $M,$ compact or non-compact, into inner symmetric spaces of compact or non-compact type. As…

In this paper we determine for relatively minimal elliptic surfaces with positive Euler number the image of the natural representation of the group of orientation preserving self-diffeomorphisms on $\Hbar$, the second homology group reduced…

In this paper, we establish a Willmore-type inequality for closed hypersurfaces in a complete Riemannian manifold of dimension $n+1$ with ${\rm Ric}\geq-ng$. It extends the classic result of Argostianiani, Fogagnolo, and Mazzieri in [1] to…

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in $\mathbb R^3$ with bounded image. The analogous result holds for holomorphic immersions…

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of…

We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be the unique…

In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture…

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function…

Explicit harmonic and wave maps are typically available only in highly symmetric or constant-curvature settings, where additional symmetry or integrability structures are present. We develop a reduction framework for pseudo-Riemannian…

We study symmetric minimal surfaces in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal…

A spacelike surface $S\subset \mathbb{S}^4_1$ is marginally trapped if its mean curvature vector is lightlike. On any oriented spacelike surface $S \subset \mathbb{S}^4_1$ we show that a choice of orientation of the normal bundle $\nu(S)$…

We prove existence results that give information about the space of minimal immersions of 2-tori into $ S ^ 3 $. More specifically, we show that \begin{enumerate} \item For every positive integer $ n $, there are countably many real $n…

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 discover a family of closed, embedded minimal surfaces in the three-dimensional round sphere which includes new examples with low genus. The existence proof relies on an equivariant min-max procedure applied to a novel sweepout which is…

For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $\gamma=\gamma^1\cup...\cup\gamma^\alpha$. This is…

Constrained Willmore surfaces are critical points of the Willmore functional under conformal variations. As shown in [5] one can associate to any conformally immersed constrained Willmore torus f a compact Riemann surface \Sigma, such that…

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian…

In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples…

We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of…