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…
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48 results for “Generic immersion”
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.
Paper introduces new invariant for pairs of immersions.
problem Understanding behavior of immersions through tangencies and triple points.
method Introduces -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.
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.
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…
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.
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.
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 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.
S. Blank solved the question of classifying immersed circles in that extend to immersed disks, and how many topologically inequivalent disks can be extended. The quetions of various cases in -dimension have already been solved by generalizing his method. In this paper, we give a new way, which is st…
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…
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.
New method characterizes surface quadrilateral layouts as special immersions.
problem Characterize surface quadrilateral layouts mathematically.
method Characterizes quadrilateral layouts as special immersions of a cut representation of the surface into the Euclidean plane.
result Mathematically describes and generalizes integer grid maps.
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.
A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…
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.
Study shows submanifolds can't be immersed in certain spaces.
problem Non-immersibility of submanifolds with infinite mean exit time.
method Not based on the weak maximum principle at infinity, generalizes previous results.
result Estimates for complete tower of moments for submanifolds with small mean curvature.
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provi…
Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic planemath.DG
We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As …
(CMC) 1-immersions in hyperbolic 3-manifolds often develop singularities.
problem Existence and uniqueness of (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds.
method Analysis of blow-up phenomena and orthogonality conditions.
result Existence and uniqueness of (CMC) 1-immersions for surfaces of any genus.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in space.
result Proves existence of differential structure from weak immersions with bounded second fundamental forms.
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in to other Lorentzian space forms. We also characterize immersions of Riemannia…
As an extension of the class of algebraic links, A'Campo, Gibson, and Ishikawa constructed links associated to immersed arcs and trees in a two-dimensional disk. By extending their arguments, we construct links associated to immersed graphs in a disk, and show that such links are quasipositive.
Abstract reviews recent Lagrangian analysis on immersions into higher dimensions.
problem Analyzing Lagrangians on immersions into higher dimensions.
method Reviews recent progress on Lagrangians on immersions with first and second fundamental forms and their derivatives.
result Recent progress in the analysis of Lagrangians on immersions into higher dimensions.
In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…
Study on constant curvature immersions of surfaces into flag manifolds.
problem Investigate constant curvature immersions of Riemann surfaces into flag manifolds.
method Investigate pseudoholomorphic maps and invariant metrics on flag manifolds.
result Unitarily equivalent primitive immersions of the two-sphere into full flag manifolds have constant curvature under all invariant metrics.
Unified invariant for immersed surface-links using biquandle cocycles.
problem Invariants for immersed surface-links in 4-space.
method Biquandle cohomology, Roseman moves, singular biquandle 3-cocycles.
result Invariance of state-sum invariant under all generating moves, including singular move (h).
Monodromy map from differential systems to character variety is generically immersive for complex -representations.
problem Characterizing when the monodromy map is immersive for differential systems.
method Analyzing the space of -differential systems on a compact Riemann surface and the character variety of -representations.
result The monodromy map is an immersion at the generic point when the complex dimension of is at least three.
We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)…
We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization of the fact that two plane curves are regularly homotopic if and only if they hav…
We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a -dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
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.
New results on hypersurfaces show no branch points, improving smoothness.
problem Analyzing area minimising hypersurfaces mod p without branch points.
method General analysis of immersed stable minimal hypersurfaces with alternating orientation.
result Area minimising hypersurfaces mod p do not admit immersed branch points.
We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh.…
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
Stability of hypersurface immersions in Riemannian manifolds proved for perturbations.
problem Stability of isometric immersions of hypersurfaces in Riemannian manifolds under perturbations of their fundamental forms.
method Young measure approach, relaxation of energy, regularity result for immersions.
result Sequence of immersions converges to an isometric immersion with the reference shape operator.
For any k<2n we construct a complete system of invariants in the problem of classifying singularities of immersed k-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.
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.
Study how invariants change under bifurcations of curves.
problem Understanding how invariants of curves change under bifurcations.
method Analyzing , , , and invariants of curves under -bifurcations.
result Invariant changes under bifurcations, preserving its essential properties.
The paper explores generic properties of minimal surfaces in high dimensions.
problem Understanding the behavior of minimal surfaces in high-dimensional spaces.
method Analyzing the space of conformal minimal immersions using Baire category theory.
result A generic conformal minimal immersion is chaotic in various ways, such as being non-proper, almost proper, and -complete.
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 -principle and proved bijection on path connected components.
result Induces a bijection between smooth conformal immersions and immersions.
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.
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…
In this paper, we extend a technique due to Romero, Rubio and Salamanca establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of …
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 for metrics.
The article proves the existence of horizontal immersions into fat distributions and contact structures.
problem Proving the existence of horizontal immersions in fat distributions and contact structures.
method Gromov's sheaf theoretic and analytic techniques of -principle.
result Existence of horizontal immersions of an arbitrary manifold into degree 2 fat distributions and quaternionic contact structures.
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to t…