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.
Holomorphic curves in moduli spaces are quasi-isometrically immersed.
problem Understanding the geometric properties of holomorphic curves in moduli spaces.
method Analyzing the quasi-isometric immersion of holomorphic maps from hyperbolic surfaces to moduli spaces.
result Holomorphic curves are quasi-isometrically immersed with parameters depending on surface and moduli space properties.
Sharp lower bound for curvature in Kähler manifolds.
problem Curvature bounds for real hypersurfaces in Kähler manifolds.
method Gauss equation for semi-isometric CR immersions.
result Proves $ rac12$-positivity of Tanaka-Webster scalar curvature.
Extremal Kähler submanifolds of complex projective spaces have natural extensions.
problem Understanding the structure and properties of extremal Kähler submanifolds.
method Analyzing the natural extensions and holomorphic isometric immersions of extremal Kähler submanifolds.
result Every connected extremal Kähler submanifold has a natural extension which is a complete Kähler manifold.
Minimal Kaehler submanifolds up to codimension four are studied.
problem Characterizing Kaehler submanifolds in high codimensions.
method Analyzing isometric immersions and compositions of submanifolds.
result Holomorphicity or composition of submanifolds with specific properties.
We present several local and global results on isometric immersions of Kaehler manifolds M2n into hyperbolic space $\Hy^{2n+p}$. For instance, a classification is given in the case of dimension n≥4 and codimension p=2. Moreover, as corollaries of general results, we conclude that there are no isometric imm…
Minimal Kaehler submanifolds in low codimension are often minimal.
problem Characterizing minimal Kaehler submanifolds in Euclidean space.
method Analyzing the second fundamental form and rank conditions.
result Generic rank conditions imply minimal submanifolds.
We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.
The paper proves existence and uniqueness of slant immersions in complex space forms.
problem Existence and uniqueness of slant immersions in complex space forms.
method Established existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
result Existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds into a complex space form.
Let f:M2n→R2n+p denote an isometric immersion of a Kaehler manifold of complex dimension n≥2 into Euclidean space with codimension p. If 2p≤2n−1, we show that generic rank conditions on the second fundamental form of the submanifold imply that f has to be a minimal submanifold. …
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.
A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem, itself a generalization of a Theorem of Takahashi, to generalize the theory of do Carmo and Wallach and to describe the moduli…
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…
Let M1 and M2 be two Kähler manifolds. We call M1 and M2 {\em relatives} if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h1:S→M1 and h2:S→M2. Moreover, two Kähler manifolds M1 and M2 are said to be …
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.
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. 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.
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.
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.
In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form μω, admits a holomorphic isometric immersion in the complex projective space, for a suitable $μ…
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.
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 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 extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Futhermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E∗(κ,τ) with 4-dimensional is…
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.