Isometric embeddings of Teichmüller spaces are derived from branched coverings.
problem Characterizing isometric embeddings of Teichmüller spaces.
method Holomorphic isometric embeddings induced by branched coverings.
result All isometric embeddings of Teichmüller spaces of dimension at least 2 arise from branched coverings.
Proof that specific groups are quasi-isometrically rigid.
problem Quasi-isometric rigidity of specific groups.
method Proof of quasi-isometric rigidity for Kleinian and three-manifold groups.
result Finitely generated Kleinian and three-manifold groups are quasi-isometrically rigid.
Paper finds isometric timelike minimal surfaces with unique properties.
problem Rigidity of isometric timelike minimal surfaces in Lorentz-Minkowski space.
method Analyzes symmetries and deformations of timelike minimal surfaces.
result Existence of isometric timelike minimal surfaces not congruent to associated family.
The paper explores rigidity and flexibility of isometric extensions with critical Hölder exponent.
problem The critical Hölder exponent in isometric extensions and its implications.
method Convex integration and construction of isometric extensions.
result The Hölder exponent $θ_0= rac12$ is critical, with extensions violating the tangential connection for $θ< rac12$.
Paper classifies pillow box isometric deformations preserving crease patterns.
problem Classifying pillow box isometric deformations preserving crease patterns.
method Continuous isometric deformations from pillow boxes to double rectangles, preserving crease patterns.
result Such deformations necessarily change pillow box topology.
In this paper we consider the Cauchy problem for isometric immersions. More precisely, given a smooth isometric immersion of a codimension one submanifold we construct C1,α isometric extensions for any α<n(n+1)+11 via the method of convex integration.
In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-k-curved metrics}. Quasi-k-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity.
Study on isometric submanifolds with preserved Gauss map metrics.
problem Investigating isometric immersions with preserved Gauss map metrics.
method Local analysis of isometric immersions and deformations in Euclidean space.
result Local characterization of non-minimal and non-reducible isometric immersions.
Heat kernels map RCD spaces to Riemannian manifolds.
problem Mapping RCD spaces to Riemannian manifolds using heat kernels.
method Using heat kernels to map RCD spaces into L2 space and then normalizing to achieve isometric immersions. result Compact RCD spaces with isometrically heat kernel immersions are isometric to unweighted smooth Riemannian manifolds.
We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.
In trying to generalize Bianchi's Bäcklund transformation of quadrics to Bäcklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the Bäcklund transformation with isometric correspondence of leaves of a general nature (…
The paper improves the approximation of isometric immersions in high codimension.
problem Isometric immersions in high codimension.
method Uniform approximation of short immersions by C1,θ isometric immersions. result Achieved C1,θ regularity for isometric immersions in local settings. The paper examines isometric timelike surfaces in 4D Minkowski space.
problem Analyzing geometric properties of isometric timelike surfaces.
method Study of Bour's theorem for four kinds of timelike helicoidal surfaces, analysis of geometric properties, presentation of parametrizations.
result Introduction of isometric pairs of timelike surfaces with same Gauss map.
Explains isometric immersions and their applications.
problem Isometric immersions and their applications in math and physics.
method Historical overview and applications.
result Explains the importance and applications of isometric immersions.
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
New findings on minimal isometric immersions of flat n-tori into spheres.
problem Conditions for minimal isometric immersions of flat n-tori into spheres.
method Analyzes rationality conditions and derives upper bounds for algebraic irrationality degree.
result Upper bound for algebraic irrationality degree of minimal isometric immersions is sharp and equals 4 for n=3.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…
Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.
problem Isometric immersions of S² into 3D Riemannian manifolds with non-negative Gauss curvature.
method Utilizes the framework of J-holomorphic curves developed by Labourie.
result Exhibits a sufficient condition for the existence of global C¹¹ isometric immersions.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
Stability result for nearly isometric subspaces and Finsler surfaces.
problem Stability of normed spaces and Finsler surfaces under near-isometric conditions.
method Refined topological argument and explicit quantification using Banach-Mazur distance.
result A 2-dimensional surface with near-monochromatic Finsler metric is approximately Riemannian.
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip a…
Characterizes quasi-isometric embeddings in coarsely Lipschitz category.
problem Understanding quasi-isometric embeddings in geometric terms.
method Formalizes quasi-isometric embeddings as regular monomorphisms in coarsely Lipschitz category.
result Quasi-isometric embeddings are equivalently characterised as effective, strong, or extremal monomorphisms.
In this article, we study holomorphic isometric embeddings between bounded symmetric domains. In particular, we show the total geodesy of any holomorphic isometric embedding between reducible bounded symmetric domains with the same rank.
We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.
Study minimal Kähler submanifolds in product of space forms.
problem Existence and properties of minimal isometric immersions of Kähler manifolds into product spaces.
method Analyse obstruction conditions and prove existence results for minimal immersions into Sm−1imesR and Hm−1imesR. result Only minimal isometric immersions of Riemannian surfaces into Sm−1imesR and Hm−1imesR exist. Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing C1 isometric embeddings. result Long spacelike embeddings can be C0-approximated by C1 isometric embeddings. We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. More…
We present some results on reductions and the copolarity of isometric group actions, which we obtained in our thesis. We also describe a resolution construction for isometric actions with respect to a reduction and give examples.
Research shows arc complex is not quasi-isometric to sphere complex.
problem Comparing quasi-isometry of arc complex and sphere complex.
method Simple proof of quasi-isometric rigidity of arc complex.
result Arc complex is not quasi-isometric to sphere complex.
First explicit isometric immersion of a flat Klein bottle in 3D space.
problem Finding an isometric embedding of a Klein bottle in 3D.
method Piecewise-linear map from a Klein bottle to Euclidean 3-space.
result Explicit numerical data for a flat Klein bottle isometrically immersed in 3D.
The paper extends isometric embedding results to null cones and spheres.
problem Isometric embedding of metrics on spheres in null cones.
method Extending Li-Wang's result to compact manifolds, specializing to 2D, developing existence and uniqueness theorems, and proving foliations.
result Existence and uniqueness of isometric embeddings in null cones.
The paper explores almost paracomplex structures on 4-manifolds and their properties.
problem Existence and properties of isometric and anti-isometric almost paracomplex structures on pseudo-Riemannian 4-manifolds.
method Analyzes the conditions under which these structures are parallel and their implications for the scalar curvature and Einstein metrics.
result If an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, the scalar curvature of the metric must be zero.
The paper proves isometric embeddings for smooth manifolds.
problem Embedding smooth manifolds isometrically into Euclidean spaces.
method Nash-Kuiper's convex integration construction and gluing technique.
result Infinitely many global isometric embeddings into 2n-dimensional Euclidean space.
We describe the quasi-isometric classification of fundamental groups of irreducible non-geometric 3-manifolds which do not have "too many" arithmetic hyperbolic geometric components, thus completing the quasi-isometric classification of 3--manifold groups in all but a few exceptional cases.
Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.
problem Existence of isometric immersions for surfaces with negative Gaussian curvature.
method Reformulated Gauss--Codazzi equations into hyperbolic conservation laws, applied theories of invariant regions and compensated compactness.
result Established existence of W2,p-isometric immersions for various families of metrics. New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
Study on embedding properties of Riemannian manifolds with specific geometric constraints.
problem Embedding Riemannian manifolds with certain geometric properties into Euclidean spaces.
method Utilizing a known trick to find embeddings with specific dimensions.
result Existence of isometric embeddings with specified dimensions for Riemannian manifolds.
Let Σ be a codimension one submanifold of an n-dimensional Riemannian manifold M, n⩾2. We give a necessary condition for an isometric immersion of Σ into Rq equipped with the standard Euclidean metric, q⩾n+1, to be locally isometrically C1-extendable to M. Even if this cond…
We study isometric embeddings of C2 Riemannian manifolds in the Euclidean space and we establish that the Hölder space C1,21 is critical in a suitable sense: in particular we prove that for α>21 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, wh…
Proves isometric embeddings in Euclidean spaces for RCD spaces.
problem Isometric immersions of RCD spaces in Euclidean spaces.
method Analyzes regular isometric immersions and eigenmaps of compact non-collapsed RCD spaces.
result Eigenmaps of compact non-collapsed RCD spaces are locally bi-Lipschitz embeddings to spheres.
We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…
New obstruction found for embedding Riemannian manifolds into Euclidean spaces.
problem Embedding Riemannian manifolds into Euclidean spaces with specific conditions.
method Motivated by incompressible Euler equations, a dynamical-topological obstruction is derived.
result Nontrivial first real homology and trivial center of fundamental group imply embedding violation.
Due to Janet-Cartan's theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M,g), it is in general hard to determine the least dimensional Euclidean space into which (M,g) can be locally isom…
We prove that every proper n-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space R3n+6,1. By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.
In this work we prove the fact that, for a short time, it is possible to construct a smooth parametrized family of isometric embeddings of an arbitrary smooth parametrized family of Riemannian metrics on a smooth closed manifold into an Euclidean space. In order to prove this statement we work out stability estimates w…
We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces f:M→S4. We prove that the space of all isometric minimal immersions of M into S4 with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that …
Alternative approach to rigidity of high-dimensional isometric immersions.
problem Rigidity of high-dimensional isometric immersions between compact manifolds.
method Quantitative rigidity estimates, reducing to Euclidean setting and applying Friesecke-James-Müller rigidity estimate.
result Quantitative results showing close proximity to isometric immersions for small stretching and bending energy.