The study constructs immersions with controlled curvatures between manifolds and identifies obstacles.
problem Creating immersions with manageable curvatures between Riemannian manifolds.
method Develops techniques to construct immersions with controlled curvatures and identifies obstructions.
result Identifies conditions under which immersions with specific curvature constraints are possible.
Proves bijection between smooth conformal immersions and immersions.
problem Finding conformal immersions of closed Riemannian surfaces.
method Reformulated using h-principle and proved bijection on path connected components. result Induces a bijection between smooth conformal immersions and immersions.
An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…
Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answe…
Study shows some contractible complexes can't have certain immersions.
problem Understanding non-positive immersions in contractible complexes.
method Provided counterexamples to a conjecture by Wise.
result Some contractible complexes do not have non-positive immersions.
We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.
Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.
problem Characterize and classify biharmonic conformal immersions into a conformally flat 3-space.
method Characterization of totally umbilical surfaces, method to produce biharmonic immersions, classification of maps, construction of examples.
result Construct many examples of biharmonic conformal immersions, including proper immersions and isometric ones.
Extends cobordism groups of immersions to projections with new results.
problem Understanding cobordism groups of projected immersions.
method Classifying space construction by Szűcs and Terpai, Salomonsen's exact sequence.
result Obtains results on cobordism groups in small and large codimensional cases.
We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew symmetric second fundamental form, if and onl…
Method constructs harmonic immersions in R^3 using Enneper-type representation.
problem Constructing harmonic immersions in R^3.
method Enneper-type representation approach.
result Any harmonic immersion in R^3 can be obtained using this approach.
The paper studies a relative version of non-positive immersion for 2-complex pairs and shows conditions under which a transitivity law holds.
problem The study of collapsing non-positive immersion for 2-complex pairs and its implications.
method Introduced a relative version of collapsing non-positive immersion for 2-complex pairs (L,K) and proved a transitivity law under certain conditions. result Under certain conditions, a transitivity law holds: If (L,K) has relative collapsing non-positive immersion and K has collapsing non-positive immersion, then L has collapsing non-positive immersion. Immersions of graphs to the projective plane are studied. A classification of immersions up to regular homotopy is given. A complete invariant of immersions up to regular homotopy is constructed. Equivalence classes are described.
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
In this paper we study fundamental geometric properties of doubly warped product immersion which is an extension of warped product immersion. Moreover, we study geometric inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds.
New examples show immersions not homologous to embeddings.
problem Tackles the question of immersions vs. embeddings in manifolds.
method Provides examples showing immersions not homologous to embeddings.
result Double points represent non-trivial homology classes.
This paper studies the regularity of constrained Willmore immersions into Rm≥3 locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion, its first, and its second derivatives, given in terms of resi…
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.
The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
Invariant detects triple points in sphere immersions.
problem Detecting regular homotopy classes of triple-point-free sphere immersions.
method Defining an invariant using a directed tree and integer-valued function.
result Space of triple-point-free spheres has infinitely many regular homotopy classes.
Uniform negative immersions prove coherence of one-relator groups.
problem Proving coherence of one-relator groups.
method Using uniform negative immersions and linear-programming techniques.
result One-relator groups with uniform negative immersions are coherent.
The paper defines conditions for para-Kaehler immersions and classifies them between forms.
problem Existence and classification of para-Kaehler immersions.
method Necessary and sufficient conditions for para-Kaehler immersions in para-Kaehler space forms.
result Local para-Kaehler immersions cannot be globally extended on simply connected para-Kaehler manifolds.
We construct two infinite sequences of immersions of the 3-sphere into 4-space, parameterized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We compute their Smale invariants and bordism classes of im…
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter …
The paper extends Weierstrass representation to non-minimal conformal immersions.
problem Understanding conformal immersions of Riemann surfaces into 3D space.
method Generalizing Weierstrass representation to arbitrary conformal immersions and studying spin structures and spinors.
result Immersions give rise to spin structures and spinors, allowing for new ways to study the immersions.
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 immersions of pseudo-Riemannian manifolds into para-complex projective space.
problem Isometric immersions of pseudo-Riemannian manifolds into para-complex projective space.
method Lift to a quadric hypersurface, frame equations, compatibility conditions.
result Characterization of surface immersions with special properties in terms of primitive harmonicity of Gauss maps.
Determines regular homotopy classes for link immersions of simple singularities.
problem Classifying immersions of link singularities.
method Computing complete invariants of immersions and comparing with Dynkin diagrams.
result Inclusion map of link into 5-sphere is regularly homotopic to immersion associated with Dynkin diagram.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces J2+-invariant for oriented pairs of immersions, invariant under inverse tangencies and triple points. result Invariant changes under direct tangencies but remains invariant under orientation change and inverse tangencies.
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.
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…
Study shows spherical embedding and immersion components are related to homotopy groups.
problem Understanding the connected components of spherical embeddings and immersions.
method Analyzing the spaces of spherical embeddings and immersions modulo immersions, and relating them to homotopy groups.
result The set of connected components of spherical embeddings and immersions modulo immersions is isomorphic to π_{n+1}(SG,SG_q).
Algorithm decides if odd-dimensional maps can be immersed.
problem Deciding if maps can be immersed homotopic to a given map.
method Algorithm for odd codimension manifolds.
result Algorithm decides immersibility of maps in odd codimension.
Study nearly geodesic immersions in symmetric spaces, leading to explicit fibrations.
problem Understanding geodesic immersions in higher rank symmetric spaces.
method Define and analyze nearly geodesic immersions with a bound on fundamental form.
result Explicit fibrations of domains of discontinuity for certain representations.
Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.
problem Finding the minimal set of moves for surfaces in 4D.
method Derived minimal generating set of spatial moves, translated into planar moves.
result Minimal generating set of spatial moves for surfaces in 4D.
S. Blank solved the question of classifying immersed circles in R2 that extend to immersed disks, and how many topologically inequivalent disks can be extended. The quetions of various cases in 2-dimension have already been solved by generalizing his method. In this paper, we give a new way, which is st…
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. 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.
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.
Study immersions of Sasakian manifolds into Sasakian space forms.
problem Characterize Sasakian manifolds that can be immersed into Sasakian space forms.
method Proved Calabi's rigidity results, characterised homogeneous Sasakian manifolds.
result Characterized homogeneous Sasakian manifolds that can be embedded into the standard sphere.
We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…
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.
We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of …
Let $(V, \Om)$ be a symplectic vector space and let $φ: M \ra V$ be a symplectic immersion. We show that φ(M)⊂V is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of φ is parallel. Furthermore, we show that any symmetric spa…
Alexandrov immersed surfaces maintain their properties under mean curvature flow.
problem Preserving Alexandrov immersivity in mean curvature flow.
method Mean curvature flow techniques adapted for Alexandrov immersed, 2D surfaces.
result Mean curvature flow properties hold for Alexandrov immersed surfaces.
We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…
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 …