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.
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.
Study extends isometric immersions using submanifolds.
problem General extension problem for isometric immersions.
method Establishing Cartan-Ambrose-Hicks theorems based on submanifolds.
result Geometric constructions of isometric extensions.
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.
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.
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. 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.
Constructs smooth isometric extensions for submanifolds.
problem Isometric immersions of submanifolds in codimension one.
method Method of convex integration for C1,α extensions. result Constructs C1,α isometric extensions for $α< rac{1}{n(n+1)+1}$. 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. 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.
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…
Estimates warping functions for isometric immersions on specific Riemannian manifolds.
problem Estimating warping functions for isometric immersions.
method Using results from \cite{P1}, estimates are derived by changing target manifolds to constant space forms and Hermitian symmetric spaces.
result Obtained estimates and equality cases for warping functions.
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.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
Flat isometric immersions in 3D are developable if they are C1,2/3 regular.
problem Characterizing flat isometric immersions in 3D with Hölder continuity.
method Weak second fundamental form, Gauss-Codazzi-Mainardi equations, and degenerate Monge-Ampère equation analysis.
result Isometric immersions of local C1,α regularity with α>2/3 are developable. Investigates totally umbilical immersions in R^3 with special metrics.
problem Existence of totally umbilical isometric immersions in R^3 with Liouville metrics.
method Examines isothermal parametrization and harmonic metrics.
result No existence of such immersions found.
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 method finds smooth isometric immersions for low regularity metrics, achieving full flexibility.
problem Finding smooth isometric immersions for metrics with low Hölder regularity.
method Techniques of convex integration to find isometric immersions of low regularity.
result Achieves full flexibility, reaching C1,1− for Cr,β metrics. Paper proves rigidity of certain manifold immersions into specific spacetimes.
problem Rigidity of isometric immersions into light cones.
method Analyzes Riemannian manifolds of dimension n-1 into light cones of n+1 spacetimes.
result Shows rigidity of isometric immersions for n ≥ 3.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
Paper discusses isometric immersion of surfaces with slowly decaying negative Gauss curvature.
problem Isometric immersion of complete surfaces with slowly decaying negative Gauss curvature.
method Gauss-Codazzi system, novel observations on decay properties of Riemann invariants, weighted Riemann invariants, comparison principle.
result Surface has a global smooth isometric immersion in three-dimensional Euclidean space if Gauss curvature decays slowly at infinity.
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.
New method approximates short immersions as C^{1,θ} isometric immersions for n ≥ 3.
problem Constructing C^{1,θ} isometric immersions of Riemannian metrics.
method Convex integration scheme with iterative integration by parts procedure.
result Uniform approximation of any short immersion by C^{1,θ} isometric immersions for θ < 1/(1+2(n-1)).
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 …
We smooth out the math of fitting surfaces into 3D space.
problem Fitting surfaces into 3D space isometrically.
method Introduce elliptic regularization to the PDE system.
result The regularization leads to a natural variational interpretation.
Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.
problem Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
method Using moving frames to demonstrate the impossibility of isometric minimal immersion.
result Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
Study proves uniqueness of corrugated negatively curved immersions in differential geometry.
problem Negatively curved immersions in differential geometry.
method Relative entropy method applied to Gauss-Codazzi system.
result Uniqueness of smooth isometric immersions within corrugated class.
Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.
problem Existence and regularity of isometric immersions in arbitrary dimensions.
method Proving W1,2-regularity for Pfaff system with antisymmetric L2-coefficient matrix. result Equivalence between W2,2-isometric immersions and weak solubility of Gauss--Codazzi--Ricci equations. Study rigidity of isometric immersions with same mean curvature in warped product spaces.
problem Rigidity of isometric immersions with the same mean curvature in warped product spaces.
method Analyzes isometric immersions in warped product spaces, proving rigidity under certain conditions.
result Two star-shaped hypersurfaces in specific manifolds differ only by rotation if they are isometric and have the same mean curvature.
Framework for isometric immersions of planar regions from framed curves.
problem Characterizing isometric immersions of planar regions with piecewise smooth boundaries.
method Develops a framework using framed curves and compatibility/regularity conditions.
result Exact dimensional reduction of bending energy to a line integral over the boundary curve.
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.
No isometric immersion of hyperbolic space into Euclidean space.
problem Isometric immersions of hyperbolic space into Euclidean space.
method Generalized Gauss-Bonnet formula for Riemannian polyhedra.
result Hilbert's Theorem generalized to higher dimensions.
Compact Riemannian manifolds embed as codimension-1 convex sets in space forms.
problem Embedding and rigidity of Riemannian manifolds in space forms.
method Analyzing the second fundamental form and curvature properties.
result Compact Riemannian manifolds are embeddings with codimension 1 in space forms.
Researchers find a list of non-isometric toric para-Kaehler-Einstein manifolds.
problem Finding a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form.
method Analytical approach to find mutually non-isometric toric para-Kaehler-Einstein manifolds.
result A list of mutually non-isometric toric para-Kaehler-Einstein manifolds analytically immersed in a finite-dimensional para-Kaehler space form.
We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into H-type groups.
Defines virtual immersions to characterize symmetric spaces.
problem Characterizing symmetric spaces using virtual immersions.
method Defines virtual immersions as a generalization of isometric immersions, proves characterization of symmetric spaces.
result A manifold admits a virtual immersion with skew symmetric second fundamental form if and only if it is a symmetric space.
Study isometric immersions in 3D Lie groups, proving new characterizations and classifications.
problem Characterizing and classifying isometric immersions in 3D Lie groups.
method Analytical models, fundamental theorems, and classification theorems.
result Isometric immersions are determined by their left-invariant Gauss maps up to certain angular companions.
Smoothly embed maps with controlled curvature errors.
problem Approximating short maps with smooth isometric immersions.
method δ-approximation of strictly short maps to C^∞-smooth isometric immersions with controlled curvature.
result Achieve isometric immersions with controlled curvature errors.
New equations for pseudo-spherical surfaces found, with unique isometric immersions.
problem Finding isometric immersions for pseudo-spherical surfaces described by k-th order evolution equations.
method Investigating the relationship between pseudo-spherical surfaces and k-th order evolution equations, proving the existence of unique isometric immersions.
result There is only one type of k-th order evolution equations that admit local isometric immersions, with universal coefficients of the second fundamental form.
The paper classifies biharmonic immersions and submersions in specific spheres.
problem Classifying biharmonic immersions and submersions in specific spheres.
method Analyzing biharmonic isometric immersions and Riemannian submersions from Berger 3-spheres.
result Complete classification of proper biharmonic Hopf tori in Berger 3-sphere.
This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.
The study establishes a new theorem for Banach spaces to solve geometric rigidity problems.
problem Weak rigidity of isometric immersions of manifolds with lower regularity.
method Compensated compactness theorem in Banach spaces.
result Global weak rigidity of Gauss-Codazzi-Ricci equations and isometric immersions.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of …
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.
Minimal hypersurfaces in Q^4(c) cannot be isometrically immersed in Q^4(≠c).
problem Characterizing minimal hypersurfaces in Q^4(c).
method Analyzing hypersurfaces with distinct principal curvatures.
result Minimal hypersurfaces with nonzero distinct principal curvatures cannot be isometrically immersed in Q^4(≠c).
Survey on recent developments in isometric immersions using PDE techniques.
problem Analyzing isometric immersions with low Sobolev regularity.
method Compensated compactness and Coulomb-Uhlenbeck gauges.
result Weak continuity and stability of Gauss-Codazzi-Ricci equations.
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold M2 which can be realized as isometric immersions into R3. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential…