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.
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.
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…
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 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. 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.
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. 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 …
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. 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.
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.
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.
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.
In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method provides a geometric construction of such an isometric immersion.
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 investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.
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.
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.
We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This resu…
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 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.
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 …
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.
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…
We introduce an elliptic regularization of the PDE system representing the isometric immersion of a surface in R3. The regularization is geometric, and has a natural variational interpretation.
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space Rn in terms of an integral representation of Weierstrass type. Restricting to…
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H2xR and S2xR. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c>0 into H2xR and a unique one into S2xR when c>1, up to isometries of the ambient space. Mor…
Wextend the results obtained recently by G. D'Ambra and A. Loi towards the proof of a conjecture of M.Gromov on isometric immersions via non-free maps.
We show that any isometric immersion of a flat plane domain into R3 is developable provided it enjoys the little Hölder regulairty c1,2/3. In particular, isometric immersions of local C1,α regularity with α>2/3 belong to this class. The proof is based on the existence of a weak notion of second …
Extends Nash-Kuiper theorem to higher Hölder exponents.
problem Constructing isometric immersions beyond Borisov's exponent.
method Novel corrugation ansatz, integration by parts, and algebraic decomposition.
result Flexibility of C1,α isometric immersions beyond Borisov's exponent. Smooth low-regular connections lead to smooth immersions with controlled regularity.
problem Smoothability of Lp-connections and existence of isometric immersions with low regularity. method Adapting S. Mardare's work on surface theory, using Hodge decomposition and fixed point theorems.
result Low-regular connections can be approximated by smooth connections of the same curvature.
Study shows how compact shapes can be rigidly mapped into complete manifolds.
problem Rigidity of isometric immersions in complete manifolds.
method Local quantitative rigidity estimates, reduced to Euclidean setting.
result Subsequence of immersions converges to an isometric immersion.
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.
Extends submanifold theorem to general spaces.
problem Tackles submanifolds in general ambient spaces.
method Uses development of curves in positive codimension and generalizes Cartan-Ambrose-Hicks theorem.
result Provides a geometric construction of isometric immersions.
In this paper, we study the general extension problem for isometric immersions by establishing Cartan-Ambrose-Hicks theorems based on submanifolds. Our method also provides geometric constructions of such extensions.
We investigate isometric immersions of disks with constant negative curvature into R3, and the minimizers for the bending energy, i.e. the L2 norm of the principal curvatures over the class of W2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls i…