Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.
problem Maximally symmetric (2,3,5)-distribution of An-Nurowski surface rolling without slipping or twisting. method Calculated vector fields defining a split g2 Lie algebra and projected to an action of SL(3,R). result Obtained an action of SL(3,R) on the configuration space without a surface. The paper classifies surfaces in a 3-torus with maximal symmetry.
problem Classifying surfaces with maximal symmetry in a 3-torus.
method Analyzing group actions and their effects on surfaces.
result All group actions and surfaces achieving maximal symmetry are identified.
Study of minimal surfaces in a specific symmetric space with polynomial growth.
problem Asymptotic geometry of minimal surfaces in a symmetric space.
method Homeomorphism between Hitchin components and maximal surfaces, identification of convex embeddings, local limits of equivariant surfaces.
result Identification of planar maximal surfaces as local limits of equivariant surfaces.
Study fibrations of projective spaces for maximal representations.
problem Characterize maximal representations of surface groups.
method Analyze fibrations of projective spaces and use geometric structures.
result Maximal representations correspond to fibrations with specific properties.
We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdSn+1, any subset E of the boundary at infinity which is the boun…
Estimates spacelike surfaces' curvature in de Sitter space.
problem Estimating maximal curvatures of spacelike hypersurfaces.
method Obtained local estimates for k-symmetric curvature functions.
result Curvatures depend on interior and boundary data.
The paper describes conformal structures and Pfaffian systems for rolling surfaces.
problem Maximally symmetric rolling distributions and their conformal structures.
method Analyzes Nurowski's conformal structure and complexifies rolling distributions.
result Changes of coordinates map conformal structures to flat metrics.
Maximal representations in infinite dimensional Hermitian spaces are studied with boundary maps.
problem Characterizing maximal representations in infinite dimensional Hermitian symmetric spaces.
method Definition of Toledo number, study of boundary maps, geometric constructions.
result Existence and non-existence conditions for maximal representations.
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …
We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature S1×S2 Cauchy surface also contains a maximal Cauchy surface. Combining …
Holomorphic discs converge to maximal surfaces under specific flows.
problem Understanding the evolution of holomorphic discs under mean curvature flow.
method Mean curvature flow with boundary conditions in the space of oriented lines.
result Holomorphic discs converge to Bishop filling by holomorphic discs under certain conditions.
The article constructs Bolza-like surfaces for infinitely many genera and studies their properties.
problem Maximizing systole functions in Teichmüller spaces for genus two and higher.
method Defining and constructing Bolza-like surfaces with specific triangulations and properties.
result Global maximal surfaces can be constructed using Bolza-like surfaces, and systolic geodesics intersect at even points.
Symmetric minimal surfaces in spheres are constructed using symmetries of the Hopf fibration.
problem Constructing symmetric minimal surfaces in 3D spheres.
method Using Lawson's theorem and symmetries of the Hopf fibration.
result New minimal surfaces with genera 9, 25, 49, 121, 121, 361, and 841 are constructed.
We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces wit…
Classifies maximal antipodal sets in symmetric spaces.
problem Identifying maximal antipodal sets in symmetric spaces.
method Explicit classification for most irreducible compact symmetric spaces.
result Classification of maximal antipodal sets in most symmetric spaces.
Let S be a closed surface of genus at least 2. For each maximal representation ρ:π1(S)→Sp(4,R) in one of the 2g−3 exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric spa…
Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by…
Rotationally symmetric hypersurfaces converge to cylinders under area-preserving flow.
problem Convergence of rotationally symmetric hypersurfaces to cylinders under area-preserving mean curvature flow.
method Geometric properties and maximal principle used for gradient and curvature estimates, leading to long-time existence and convergence.
result Rotationally symmetric hypersurfaces converge to cylinders under area-preserving mean curvature flow.
Study maximal antipodal sets in exceptional symmetric spaces.
problem Classify maximal antipodal sets in exceptional symmetric spaces.
method Combining existing literature and new results, classify maximal antipodal sets.
result Complete classification of maximal antipodal sets in all exceptional compact symmetric spaces.
Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling (2,3,5)-distribution to the flat Cartan distribution. result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these repr…
We define a family of four-point invariants for Shilov boundaries of bounded symmetric domains of tube type, which generalizes the classical four-point cross ratio on the unit circle. This generalization, which is based on a similar construction of Clerc and Ørsted, is functorial and well-behaved under products; these …
Extends flat submanifold properties from hyperbolic plane to symmetric spaces.
problem Spectral asymptotics for orbital integrals in symmetric spaces.
method Generalizes geodesic properties to maximal flat submanifolds.
result Establishes geometric properties of maximal flat submanifolds in symmetric spaces.
Maximizes mixing efficiency in surface braids.
problem Finding the maximum mixing efficiency in surface braids.
method Introduced an efficient algorithm to compute topological entropy and TEPO for surface braids.
result Conjectured a novel candidate braid to have maximal mixing efficiency.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.
problem Understanding the measure of maximizing orbits in symmetric billiard tables.
method Introduced a closed invariant set of locally maximizing orbits and gave an effective bound on its measure.
result An effective bound on the measure of the invariant set in terms of the isoperimetric defect of the curve.
We give general sufficient conditions for the existence of trapped surfaces due to concentration of matter in spherically symmetric initial data sets satisfying the dominant energy condition. These results are novel in that they apply and are meaningful for arbitrary spacelike slices, that is they do not require any au…
Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.
problem Determining the maximal antipodal set in the outer 3-symmetric space S7imesS7. method Investigated the polar and maximal antipodal set P for the given 3-symmetric space S7imesS7. result The maximal antipodal set P has three elements. Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
Unified rigidity theorem for cyclic and alternating surfaces.
problem Infinitesimal rigidity of equivariant minimal maps.
method Unified Lie-theoretic framework connecting cyclic surfaces and cyclic harmonic bundles.
result Infinitesimal rigidity for irreducible cyclic surfaces under various variations.
Local equivalence shown between specific distributions and flat Cartan distribution.
problem Establishing local equivalence between specific distributions and flat Cartan distribution.
method Change of coordinates mapping specific distributions to flat Cartan distribution.
result Local equivalence between maximally symmetric (2,3,5)-distributions and flat Cartan distribution. We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of T2-symmetric initial data with positive cosmological constant Λ>0, in the vacuum or with Vlaso…
The paper shows spherical naked singularities are unstable under gravitational perturbations.
problem Stability of spherical naked singularities in a scalar field under gravitational perturbations.
method Characteristic initial value problem with initial data on intersecting null cones, focusing on the instability of singularities.
result The set of initial conformal metrics leading to maximal future developments without closed trapped surfaces is of first category, indicating instability.
The study classifies and characterizes totally symmetric sets in the general linear group.
problem Understanding the structure and properties of totally symmetric sets in the general linear group.
method Formulated a notion of irreducibility for totally symmetric sets in the general linear group and classified them.
result Classification of irreducible totally symmetric sets and those of maximal cardinality.
Solves equivalence problem for CR geometries with simple models.
problem Equivalence problem for 2--nondegenerate CR geometries with simple models.
method Uses homogeneous spaces G/H as maximally symmetric models for simple Lie groups. result Constructs local embeddings of these models into complex space.
Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compacti…
We study the canonical complexifications of non-compact Riemannian symmetric spaces G/K by the Grauert tube construction. We determine the maximal such complexification, a domain already constructed in another context by Akhiezer and Gindikin (Math. Ann., 1990), and show that this domain is Stein. We show there is an a…
New solutions to Chazy equations lead to Ricci-flat metrics.
problem Finding Ricci-flat metrics for a specific distribution.
method Conformal rescaling of Nurowski's class to find Ricci-flat metrics.
result Solutions to Chazy equations appear in the rescaling process.
Classifies R-spaces with a specific symmetric structure.
problem Classifying R-spaces with a natural Γ-symmetric structure.
method Classification and determination of maximal antipodal sets.
result Classification of R-spaces with a natural Γ-symmetric structure.
Diagonal complexes and symmetric complexes study surfaces with involution and punctures.
problem Understanding surfaces with involution and punctures through diagonal complexes.
method Construction and study of diagonal and symmetric diagonal complexes, their barycentric subdivisions, and homotopy equivalence.
result Symmetric diagonal complex is homotopy equivalent to a punctured symmetric surface.
Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.
problem Exploring conditions for maximality of Hilbert square of real surfaces.
method Analyzing Hilbert square of maximal real surfaces and examining specific examples.
result Hilbert square can be maximal even for surfaces with disconnected real locus.
The study finds that certain curved manifolds can be mapped to symmetric spaces.
problem Understanding singular Riemannian foliations in positively curved manifolds.
method Generalizing fixed point homogeneous actions to singular Riemannian foliations.
result Positively curved manifolds with point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces.
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
Characterizes circles in self-dual symmetric R-spaces using geometric properties.
problem Defines and characterizes special curves (circles) in self-dual symmetric R-spaces.
method Characterizes elements of the transformation group G and describes circles in Riemannian geometric terms.
result Describes circles in terms of maximal compact subgroups and geodesics.
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equ…
The Laplace spectrum uniquely identifies five out of eight metrically maximal three-dimensional geometries.
problem Characterizing compact locally homogeneous three-manifolds using their Laplace spectra.
method Analyzing geometric structures and their spectral properties.
result For five out of eight metrically maximal three-dimensional geometries, compact locally homogeneous three-manifolds are uniquely determined by their spectra.
New classification of conformal structures with maximal G2 symmetry.
problem Classifying conformal structures with maximal G2 symmetry. method Complete local classification of homogeneous 4D split-conformal structures.
result Established a complete local classification of conformal structures with maximal G2 symmetry.