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.
We briefly review selected contributions to immersion-theoretic topology, from S. Smale's immersion theory for spheres to M. Gromov's convex integration theory, during the early "golden" period from about 1959-1973. Historical remarks are included and technical concepts are presented informally.
We develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature; extr…
Computes immersions of C2-projective spaces using K-theory.
problem Computing immersions of equivariant projective spaces.
method Geometric filtration and localized slice spectral sequence.
result Obtained equivariant analogue of James periodicity.
Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.
problem Proving equivalence between two Floer theories.
method Using pearly tree discs and local Hamiltonian flows.
result Equivalence relation between immersed Lagrangian Floer theory and Hamiltonian immersed Lagrangian Floer theory.
Spinorial approach characterizes submanifolds in product spaces of constant curvature.
problem Characterizing submanifolds in product spaces of constant curvature.
method Spinorial characterization and fundamental theorem proof.
result Spinorial approach provides new characterizations and proofs in specific cases.
The paper proves conditions for 4-manifolds to be parallelizable and non-existence results.
problem Conditions for 4-manifolds to be parallelizable and non-existence results.
method Homotopy theory, particularly obstruction theory.
result Compact, oriented 4-manifolds with vanishing Euler characteristic and signature admit coassociative-free immersions.
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…
We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…
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…
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…
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.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
Algorithmic study of manifold immersibility and embeddability.
problem Decidability of manifold immersibility and embeddability.
method Algorithmic analysis of immersion and embeddability questions for m-dimensional CAT-manifolds in Rn. result Undecidability of smooth embeddability when n−m is even and 11m≥10n+1. This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.
In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into R^5 in a geometric manner. The pair (c(f),i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theor…
Survey of complex analytic methods in minimal surface theory.
problem Global theory of minimal surfaces in Euclidean spaces.
method Complex analytic methods including Oka theory, holomorphic sprays, and Riemann-Hilbert boundary value problem.
result New constructions and results on minimal surfaces in various contexts.
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. We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
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.
Optimal regularity theory for stable minimal hypersurfaces with small singular set.
problem Optimal regularity of stable minimal hypersurfaces with small singular set.
method Analysis of stable minimal hypersurfaces in a specific domain with small singular set.
result Optimal size assumption on the non-immersed singular set guarantees optimal regularity.
This paper constructs a functor preserving unobstructedness in symplectic geometry.
problem Generalizing unobstructedness in symplectic geometry for arbitrary manifolds and Lagrangian submanifolds.
method Using filtered A-infinity categories and Lagrangian Floer theory, the paper constructs a 2-functor.
result The geometric transformation preserves unobstructedness of Lagrangian Floer theory.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
New invariant shows fifth move is unique and extends biquandle theory for surface-links.
problem Reproduce Yoshikawa's fifth move using other moves and isotopies.
method Develops a semi-invariant and biquandle-based invariant for surface-links.
result Yoshikawa's fifth move is unique and cannot be reproduced by other moves.
The paper studies variational functionals for submanifolds using the Lepage form.
problem Variational functionals for submanifolds in Grassmann fibrations.
method Introduces the fundamental Lepage form and uses it to study variations of submanifolds.
result Proves the first infinitesimal variation formula and Euler-Lagrange equations.
We develop the intersection theory associated to immersed, oriented and mea- sured solenoids, which were introduced in arXiv:0910.2836.
We derive a dimensionally-reduced limit theory for an n-dimensional nonlinear elastic body that is slender along k dimensions. The starting point is to view an elastic body as an n-dimensional Riemannian manifold together with a not necessarily isometric W1,2-immersion in n-dimensional Euclidean space. The…
Study knots with immersed crosscap number 1 and find nonorientable representatives of homology classes.
problem Understanding knots and their nonorientable representations in 3-manifolds.
method Analyzing knots with immersed crosscap number 1, studying differences between immersed and embedded crosscap numbers, and constructing specific 3-manifolds.
result Knots with immersed crosscap number 1 are specific types of torus or cable knots. Immersed crosscap number can differ from embedded crosscap number by large amounts and is not bounded.
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.
Extends pillowcase homology for immersed curves in a 3-ball.
problem Constructing algebraic versions of Lagrangian Floer homology.
method Associate algebra A and A-infinity modules M(L) to immersed curves in the pillowcase, proving isomorphism with algebraic module pairings.
result Lagrangian Floer homology is isomorphic to a suitable algebraic pairing of modules.
A new method is presented for solving the Gauss-Codazzi equations for a compact Riemann surface to be immersed in a 3-manifold of constant curvature. In the negative curvature case, the moduli for such embeddings are cohomology classes of (0,2) forms.
We apply the general theory of codimension one integrability conditions for G-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternion…
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
Study immersions of surfaces into SL(2,C) and geodesics space.
problem Classical theory of hypersurfaces in pseudo-Riemannian space forms.
method Holomorphic Riemannian space forms and geodesics space of hyperbolic space.
result Characterization and construction of immersions.
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…
We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the b…
Recover manifold homology from a sample in Euclidean space.
problem Recovering the homology of an immersed manifold from a sample.
method Estimate tangent bundle, apply measure-based filtrations for persistent homology, use normal reach.
result Construction consistent and stable, does not require manifold dimension.
Proves a conjecture about concordance invariant simplifying its relation to Rasmussen's invariant.
problem Concordance invariant and its relation to Rasmussen's invariant.
method Bar-Natan's tangle version of Khovanov homology, distilled into immersed curve theory.
result Simplifies the relation between ϑ and Rasmussen's s-invariant. Geometric interpretation of tangle invariants using immersed curves.
problem Understanding tangle invariants in Khovanov homology.
method Geometric interpretation of Bar-Natan's invariant using immersed curves.
result Recovery of Khovanov homology from immersed curves.
Relation between generalized Weierstrass representation for conformal immersion of generic surfaces into three-dimensional space and Lax-Phillips scattering theory for automorphic functions is considered.
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).
New theory for area of Legendrian surfaces, proving smoothness and variational results.
problem Understanding the area of Legendrian surfaces under constraints.
method Introducing PHSLVs, proving sequential compactness, regularity, and variational results.
result Generalized regularity theory for Legendrian surfaces, achieving variational minima.
The study examines the regularity of branched immersions using special coordinate systems.
problem Understanding the regularity of branched immersions and their fundamental elements.
method Development and use of special coordinate systems to express maps with branch points, proving existence and regularity conditions for mean curvature vectors.
result Characterization and existence of special coordinate systems for branch immersions, proving regularity conditions for mean curvature vectors.
We classify all order one invariants of immersions of a closed orientable surface F into R^3, with values in an arbitrary Abelian group G. We show that for any F and G and any regular homotopy class A of immersions of F into R^3, the group of all order one invariants on A is isomorphic to G^\aleph_0 \oplus B \oplus B w…
We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of solenoids with a trapping region.
Stability of hypersurface immersions in Riemannian manifolds proved for Lp perturbations.
problem Stability of isometric immersions of hypersurfaces in Riemannian manifolds under Lp 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.
New knots not slice in rational 4-balls found.
problem Identifying knots not slice in rational homology 4-balls.
method Using generalized Mazur patterns and immersed Heegaard Floer homology.
result Infinitely many examples of pattern knots P not slice in any rational homology 4-ball.