Related papers: On conformal Gauss maps

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…

Let X be a closed surface of genus two embedded in the 3-sphere. Then X inherits a metric and an orientation, which give an almost complex structure, which automatically integrates to a genuine complex structure, making X a Riemann surface.…

We study expanding maps and shrinking maps of subvarieties of Grassmann varieties in arbitrary characteristic. The shrinking map was studied independently by Landsberg and Piontkowski in order to characterize Gauss images. To develop their…

It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional…

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…

This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars.…

Let $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and…

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…

In this paper, we study the Gauss map of a holomorphic codimension one foliation on the projective space $\mathbb{P}^n$, $n\ge 2$, mainly the case $n=3$. Among other things, we will investigate the case where the Gauss map is birational.

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…

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal…

An algorithm for the computation of global discrete conformal parametrizations with prescribed global holonomy signatures for triangle meshes was recently described in [Campen and Zorin 2017]. In this paper we provide a detailed analysis of…

Surface parameterizations are widely applied in computer graphics, medical imaging and transformation optics. In this paper, we rigorously derive the gradient vector and Hessian matrix of the discrete conformal energy for spherical…

We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map $Q$ and a collection of meromorphic functions and their derivatives, we…

We define a class of maps between holomorphically embedded null curves which generalize conformal transformations, and can be defined in any complex dimension. In four dimensions, we can also define a similar map between self-dual surfaces,…

In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a…

In the paper, we study the Gauss map of a completely immersed anisotropic minimal surface with respect to convex parametric integrand in $\mathbb{R}^3$. By a local analysis, we prove the discreteness of the critical set of the Gauss map of…

We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving…

A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce Self…

We discuss the construction of higher-dimensional surfaces based on the harmonic maps of $S^2$ into $CP^{N-1}$ and other grassmannians. We show that there are two ways of implementing this procedure - both based on the use of the relevant…