Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answe…
Proves bijection between smooth conformal immersions and immersions.
problem Finding conformal immersions of closed Riemannian surfaces.
method Reformulated using h-principle and proved bijection on path connected components. result Induces a bijection between smooth conformal immersions and immersions.
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter …
Paper explores conformal immersions of Kaehler manifolds into Euclidean space.
problem Understanding conformal immersions of Kaehler manifolds.
method Used techniques from S. Chion and M. Dajczer for hyperbolic space immersions.
result Proved properties of conformal immersions into Euclidean space.
Study on isometric immersions in complex surfaces.
problem Understanding isometric immersions of locally conformally Kaehler manifolds.
method Investigation of isometric immersions into Hopf manifolds, focusing on compact complex surfaces.
result Characterization of Hopf-induced metrics on compact complex surfaces.
Proves conditions for conformal immersions of Riemannian products in Euclidean spaces.
problem Conditions for conformal immersions of Riemannian products in Euclidean spaces.
method Analyzes conditions for conformal immersions of Riemannian products in Euclidean spaces.
result Proves conditions for conformal immersions of Riemannian products in Euclidean spaces.
Introduces conformal Riemannian morphisms generalizing various Riemannian concepts.
problem None explicitly stated; generalizing Riemannian concepts.
method Introduces conformal Riemannian morphisms and proves properties.
result Every injective conformal Riemannian morphism is an injective conformal immersion, and similar results for surjective and bijective cases.
Every meromorphic function maps to a minimal surface in 3D.
problem Understanding the mapping properties of meromorphic functions to minimal surfaces.
method Proving that every meromorphic function on a Riemann surface is the Gauss map of a conformal minimal immersion into \(\mathbb{R}^3\).
result Meromorphic functions on Riemann surfaces are realized as the Gauss maps of conformal minimal immersions.
Study finds all conformal minimal immersions of 2-spheres in a complex Grassmann manifold with parallel second fundamental form.
problem Classifying conformal minimal immersions with parallel second fundamental form.
method Analyzing immersions in complex Grassmann manifold G(2,N;C). result Determined all conformal minimal immersions of 2-spheres with parallel second fundamental form.
The paper extends Weierstrass representation to non-minimal conformal immersions.
problem Understanding conformal immersions of Riemann surfaces into 3D space.
method Generalizing Weierstrass representation to arbitrary conformal immersions and studying spin structures and spinors.
result Immersions give rise to spin structures and spinors, allowing for new ways to study the immersions.
The paper extends a theorem and studies the convergence of branched conformal immersions with bounded areas and Willmore energies.
problem Analyzing the convergence of branched conformal immersions with bounded areas and Willmore energies.
method Extending a local convergence theorem and studying blowup behavior.
result The integral identity of Gauss curvature is proven.
Study of tractor bundles and spacelike immersions in Lorentzian manifolds.
problem Characterizing and understanding conformal tractor bundles and spacelike immersions.
method Extrinsic viewpoint, relating tractor bundles to spacelike immersions, reformulating equations in terms of spacelike immersion geometry.
result Every Riemannian conformal structure can be realized as a pullback of the tangent bundle of a Lorentzian ambient space.
The paper constructs conformal Willmore immersions on tori with specific properties.
problem Constructing conformal Willmore immersions on tori with specific properties.
method Constructing immersions with a single point of density and Willmore energy 4πk. result The infimum of the Willmore energy in every conformal class of tori is less than or equal to 8π. Study of minimal immersions from a sphere to a complex hyperquadric.
problem Characterizing minimal immersions from a sphere to a complex hyperquadric.
method Analysis through harmonic maps and linear full reducibility.
result Determination of reducible conformal minimal immersions with constant curvature.
New variational methods find close-to-conformal and isometric immersions of surfaces.
problem Finding conformal and isometric immersions of surfaces.
method Variational functionals for spinor fields on compact Riemann surfaces.
result Minimization of variational functionals can yield piecewise smooth isometric immersions.
New method constructs holonomic immersions from flat submanifolds.
problem Creating holonomic immersions from flat submanifolds.
method Ribaucour transformation and principal coordinate system.
result Holonomic immersions can be constructed using Ribaucour transformation.
Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.
problem Finding conformal superminimal surfaces in hyperbolic 4-space.
method Analysis of holomorphic Legendrian curves in the twistor space of H4. result Proper conformal superminimal immersions can be approximated by smooth ones.
In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into Rm. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach…
Every nonflat conformal minimal surface is homotopic to a proper one.
problem Proving homotopy of nonflat conformal minimal surfaces to proper ones.
method Analyzing immersions and fluxes of Riemann surfaces into \(\mathbb{R}^n\) and \(\mathbb{C}^n\).
result Every nonflat conformal minimal immersion is homotopic to a proper one.
Paper approximates continuous functions on Jordan arcs using conformal minimal immersions.
problem Approximating continuous functions on Jordan arcs using minimal immersions.
method Conformal minimal immersions and directed holomorphic curves.
result Continuous functions on Jordan arcs can be approximated by conformal minimal immersions.
In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in Rn with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minim…
Study on minimal two-spheres in complex hyperquadric, proving non-congruence of constant curvature spheres.
problem Classifying minimal two-spheres of constant curvature in complex hyperquadric.
method Construction of non-homogeneous constant curved minimal two-spheres and classification theorem.
result Minimal two-spheres of constant curvature in Q4 are not congruent. The paper studies biharmonic conformal hypersurfaces in Riemannian manifolds.
problem Characterizing biharmonic conformal hypersurfaces in Riemannian manifolds.
method Deriving biharmonic equations and proving properties of conformal immersions.
result Properties of biharmonic conformal hypersurfaces in space forms.
The paper studies conditions for minimal immersions to be injective and embedded.
problem Conditions for minimal immersions to be injective and embedded.
method A recent general criterion for injectivity of conformal immersions and its geometric applications.
result Conditions for minimal immersions to be embedded are derived, with extremal configurations only occurring on a catenoid.
The paper develops complex representations for spacelike surfaces in 4D Minkowski space and solves associated PDEs.
problem Developing complex representations for spacelike surfaces in 4D Minkowski space.
method Introducing complex valued functions and using holomorphic and anti-holomorphic theory to solve PDEs.
result Explicit solutions for PDEs and characterization of conformal totally umbilical immersions.
The family of Willmore immersions from a Riemann surface into Sn+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in Rn+2 and those which are not conformally equivalent to a minimal surface in Rn+2. On the level of their conformal Gauss maps…
The paper explores conditions for statistical manifolds to be immersed and dual to each other.
problem Conditions for statistical manifold structures to be dual and immersions into statistical manifolds.
method Obtained necessary and sufficient conditions for dual structures and immersion properties.
result Conditions for realizing and realizing in a dually flat statistical manifold of codimension two.
Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.
problem Analyzing spectra of magnetic Laplacians on non-simply connected manifolds.
method Definition of spectral varieties and construction of conformal invariants.
result New conformal invariants of immersions of surfaces into 3- and 4-dimensional spaces.
Abstract: Formula for Lorentzian surfaces in R^(2,2)
problem Representing Lorentzian surfaces in R^(2,2)
method Spinor and Lorentz number approach
result New spinor representation formula for flat Lorentzian surfaces in Anti-de Sitter space
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
The study proves a strong parametric h-principle for minimal surfaces.
problem Proving a parametric h-principle for minimal surfaces.
method Using a parametric h-principle due to Forstneric and Larusson.
result The space of complete nonflat conformal minimal immersions has the same homotopy type as the space of continuous maps.
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…
The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.
problem Understanding the space of Gauss maps of complete minimal surfaces and their homotopy types.
method Proves the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of meromorphic functions.
result The space of meromorphic functions on M that are the Gauss map of a complete full conformal minimal immersion has the same homotopy type as the space of all continuous maps from M to the 2-sphere. The paper explores generic properties of minimal surfaces in high dimensions.
problem Understanding the behavior of minimal surfaces in high-dimensional spaces.
method Analyzing the space of conformal minimal immersions using Baire category theory.
result A generic conformal minimal immersion is chaotic in various ways, such as being non-proper, almost proper, and g-complete. We prove that the conformal immersions of complex two tori into S3 which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…
The paper studies submanifolds of Euclidean space from Kaehler manifolds.
problem Understanding submanifolds of Euclidean space from Kaehler manifolds.
method Analyzing conformal immersions and isometric embeddings.
result Local constructions of submanifolds from Kaehler manifolds.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention and a cohomogeneity-one analytic system, proving local existence for nonconstant mean curvature and dilation.
result Local existence and rigidity results for biharmonic conformal immersions into anti-de Sitter space.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention, expressing biharmonic equation in terms of induced metric and curvature, deriving cohomogeneity-one analytic system, and solving scalar third-order ODE.
result Local existence and rigidity of biharmonic conformal immersions with nonconstant dilation.
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.
The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.
problem Obstructing conformal immersions of Riemannian manifolds.
method Defining a Chern-Simons invariant for stably trivial vector bundles and using it to derive an obstruction.
result An obstruction for conformally immersing a n-dimensional Riemannian manifold in a translation manifold of dimension n+1.
The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.
problem Understanding the flexibility of domains in Euclidean spaces for minimal surfaces.
method Investigates the concept of flexibility in terms of minimal surfaces contained in domains.
result Defines flexible domains and shows how they can be approximated by minimal immersions.
Exploring conjectures in constrained Willmore problem.
problem Understanding the Willmore functional over compact surfaces.
method Analyzing conjectures from partial results and numerical experiments.
result Ramifications for deeper understanding of the Willmore functional.
We study minimal immersions of closed surfaces (of genus g≥2) in hyperbolic 3-manifolds, with prescribed data (σ,tα), where σ is a conformal structure on a topological surface S, and αdz2 is a holomorphic quadratic differential on the surface (S,σ). We show that, for each t∈(0,τ0) for some $τ_0…
A Klein bottle is the unique minimizer in its conformal class.
problem Finding the unique minimizer of Klein bottles in Euclidean space.
method Stereographic projection and minimization analysis.
result Lawson's bipolar surface is the unique minimizer among immersed Klein bottles in its conformal class.
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy W=∫H2 under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…
In this paper, we prove that every confomal minimal immersion of an open Riemann surface into Rn for n≥5 can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into R5. One …
The paper proves uniform approximation for minimal surfaces with applications to a Mittag-Leffler theorem.
problem Approximating complete conformal minimal surfaces with finite curvature.
method Uniform approximation theorem with interpolation for minimal surfaces.
result Obtained a Mittag-Leffler type theorem for minimal immersions.