Related papers: Conformal submanifold geometry I-III

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral…

We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$…

We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We…

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by…

We use Clifford algebras to construct a unified formalism for studying constant extrinsic curvature immersed surfaces in riemannian and semi-riemannian $3$-manifolds in terms of immersed bilegendrian surfaces in their unitary bundles. As an…

We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…

We study conformal mappings in the Grushin plane and provide a number of their characterizations in terms of the Sobolev mappings and their geometry. Furthermore, we connect conformality on the Grushin plane with conformality on the complex…

We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit of Riemannian manifolds with non-negative Ricci curvature. Motivated by this analogy we propose the notion of non-commutative…

In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension $\geq 3$,…

We show that two properly embedded compact surfaces in an orientable 4-manifold are cobordant if and only if they are $\mathbb{Z}/2$-homologous and either the 4-manifold has boundary or the surfaces have the same normal Euler number. If the…

In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

It is shown that a superconformal surface with arbitrary codimension in flat Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is S-Willmore, the latter a well-known necessary condition to…

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant…

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such…

This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations.…

The main aim of this survey paper is to gather together some results concerning the Calabi type duality discovered by Hojoo Lee between certain families of (spacelike) graphs with constant mean curvature in Riemannian and Lorentzian…

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and…

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

Conformal geometry is studied using the unfolded formulation \`a la Vasiliev. Analyzing the first-order consistency of the unfolded equations, we identify the content of zero-forms as the spin-two off-shell Fradkin-Tseytlin module of…