Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Study on controllability and groups of manifolds with boundaries.
problem Controllability of vector fields on manifolds with boundaries.
method Establish controllability results for diffeomorphism groups of manifolds with smooth boundaries.
result Diffeomorphism groups of manifolds with smooth boundaries form fibre bundles and are generated by the exponential map.
Exotic diffeomorphisms found on complex surfaces and 4-manifolds.
problem Finding exotic diffeomorphisms on 4-manifolds.
method Minimal complex surfaces and spin 4-manifolds with S3 boundary. result First known instances of exotic diffeomorphisms of irreducible 4-manifolds.
The paper proves conditions for smooth convergence of hyperkaehler 4-manifolds with boundary.
problem Compactness of hyperkaehler 4-manifolds with boundary.
method Analyzes sequences of hyperkaehler triples under topological and curvature conditions.
result Smooth convergence of hyperkaehler triples up to diffeomorphisms if boundary restrictions converge.
Study shows mean action of periodic orbits in annuli is bounded by their Calabi invariant.
problem Understanding the average distortion of periodic orbits in area-preserving annuli.
method Analyzes action functions and Calabi invariants of diffeomorphisms near annulus boundaries.
result Infimum of mean action of periodic orbits is bounded by their Calabi invariant.
Study of 3-manifolds with non-trivial boundary groups, showing they cannot be corks.
problem Investigating non-trivial boundary groups of 3-manifolds and their implications.
method Combination of 3 and 4-manifold techniques.
result Every diffeomorphism of the boundary extends to the full manifold, proving they cannot be corks.
Smooth solutions found for hydrodynamic equations.
problem Geodesic equations on diffeomorphism groups.
method Proving smoothness of solutions and constructing exponential maps.
result Smooth solutions match boundary conditions.
Global EQG sums boundary states over manifold diffeomorphism classes.
problem Summing boundary states over manifold diffeomorphism classes.
method Formulated as classical statistical physics, weights determined by general principles.
result Hartle-Hawking state as a probability measure.
Exponential map proof for area-preserving diffeomorphisms on surfaces.
problem Proving the properties of the exponential map for a specific group of diffeomorphisms.
method Analyzing the Riemannian exponential map of the right-invariant L2 metric. result The exponential map is a nonlinear Fredholm map of index zero.
Study shows finite topological rank for manifold diffeomorphisms.
problem Determining the finite topological rank of diffeomorphism groups.
method Analyzing compact smooth manifolds with boundary, proving finiteness for all k≥1. result Topological rank of diffeomorphism groups is finite for all dimensions and orders.
Study constraints on diffeomorphisms and homeomorphisms of 4-manifolds with boundary.
problem Constraints on smooth families of 4-manifolds with boundary.
method Use Manolescu's Seiberg-Witten Floer stable homotopy type.
result Inclusion map between diffeomorphisms and homeomorphisms is not a weak homotopy equivalence.
New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.
problem Finding non-smoothable homeomorphisms of 4-manifolds with boundary.
method Constructing specific examples of homeomorphisms that fix the boundary and act trivially on homology.
result First examples of non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary.
This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
Study controllability of diffeomorphisms of simple polytopes.
problem Controllability of diffeomorphisms of simple polytopes.
method Lie group structure and controllability results for diffeomorphisms of simple polytopes.
result Identity component of the diffeomorphism group is generated by the exponential image.
We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.
Study local structure of Einstein metrics with boundary conditions.
problem Understanding the local structure of Einstein metrics with boundary constraints.
method Analysis of moduli space of compact Einstein metrics, focusing on boundary conformal metric and mean curvature.
result For three dimensions, the map from Einstein metrics to boundary data is generically a local diffeomorphism.
The paper studies diffeomorphisms of a specific foliation on a Klein bottle.
problem Computing homotopy types of diffeomorphism groups for a specific foliation.
method Analyzes a Morse-Bott foliation on a solid Klein bottle and its twisted bundle.
result Computes homotopy types of foliated and leaf-preserving diffeomorphism groups.
Proves condition for 4-manifolds with sphere boundary to be standard.
problem Determining when acyclic 4-manifolds with sphere boundary are standard.
method Uses Turaev's shadows to provide a sufficient condition for diffeomorphism to the standard 4-ball.
result If a compact, smooth, acyclic 4-manifold with sphere boundary has shadow-complexity at most 2, it is diffeomorphic to the standard 4-ball.
New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
Study shows cohomological limits on extending actions on 3-manifold boundaries.
problem Limits to extending group actions on 3-manifold boundaries.
method Cohomological obstructions for C0-actions on 3-manifolds. result Cohomological obstructions prevent extending actions on certain 3-manifold boundaries.
Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.
problem Characterizing exotic diffeomorphisms on Milnor fibers.
method Compared family relative Bauer--Furuta invariants of mapping tori.
result Boundary Dehn twist is an exotic diffeomorphism under given conditions.
Models for self-equivalences and diffeomorphisms of manifolds.
problem Classifying spaces of self-equivalences and diffeomorphisms of manifolds.
method Construct rational models using equivariant algebraic methods.
result Formula for rational cohomology of classifying spaces.
Study reveals how travel times on cylindrical boundaries can identify spacetime structure.
problem Determining spacetime structure from boundary travel times.
method Boundary rigidity problem on cylindrical domains with stationary metrics.
result Time separation function uniquely identifies spacetime up to diffeomorphisms.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
The paper examines uniform perfectness of diffeomorphism groups on open manifolds.
problem Uniform perfectness of diffeomorphism groups on open manifolds.
method Study of uniform perfectness, boundedness, and simplicity of diffeomorphism groups of compact and open manifolds.
result Obtained upper bounds of diameters for commutator length, balls, and conjugation-generated norm.
R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism betwee…
The paper proves homological stability for high-dimensional manifold diffeomorphism groups.
problem Understanding the homology of high-dimensional manifold diffeomorphism groups.
method Proving homological stability for diffeomorphism groups of high-dimensional manifolds with boundary.
result The homology of the colimit of diffeomorphism groups is identified with that of a specific infinite loop space.
Non-realizable braid groups cannot be lifted to diffeomorphisms of surfaces.
problem The non-realizability of braid groups by diffeomorphisms on surfaces.
method Analyzing the group structure and applying techniques to spaces of codimension-2 embeddings.
result For sufficiently large n, no lift of the surface braid group Bn(S) to Diff(S,n) exists. New method distinguishes 4-manifold types using trisections.
problem Distinguishing different 4-manifold types.
method Capping operation to transform relative trisections into closed 4-manifold diagrams.
result Examples of non-diffeomorphic relative trisections of the same 4-manifold.
Two non-diffeomorphic minimal genus trisections found for a 4-manifold.
problem Existence of non-diffeomorphic minimal genus trisections of the same 4-manifold.
method Introduced a simple operation to create a trisection diagram from a relative trisection diagram.
result Existence of two non-diffeomorphic minimal genus trisections of the same (g,k;p,b)-type 4-manifold. Contractible diffeomorphism groups on lens spaces derived from Morse-Bott foliations.
problem Understanding the structure of diffeomorphism groups on lens spaces.
method Analyzing Morse-Bott foliations and their diffeomorphisms.
result Contractible diffeomorphism groups on lens spaces.
We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and…
New method to separate and create different 4-manifolds.
problem Creating and separating different 4-manifolds.
method Using a single cork and boundary diffeomorphisms.
result Finite families of corks can be separated.
Paper proves convergence of MDL to Einstein-Hilbert with boundary term.
problem Proving convergence of discrete MDL to continuous Einstein-Hilbert action.
method Proves \(Γ\)-convergence using diffeomorphism-natural discrete MDL-type functional.
result Identifies Carathéodory densities and obtains \(\liminf/\limsup\) bounds.
Optimizes maps with controlled distortion for geometric tasks.
problem Free-boundary diffeomorphism optimization in geometric modeling.
method Least-squares quasiconformal (LSQC) operator and Spectral Beltrami Network (SBN).
result LSQC minimizer well-posed under mild conditions, stable under mesh refinement.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
Geometrically constructs Virasoro-Bott group from circle diffeomorphisms.
problem Constructing the Virasoro-Bott group from circle diffeomorphisms.
method Analogous to loop group construction, using disc diffeomorphisms with special boundary conditions.
result Identifies Virasoro-Bott group as a quotient of disc diffeomorphisms with identified Lie algebra.
Study submersions with definite folds on manifolds with boundary into Euclidean spaces.
problem Understanding differential-topological properties of manifolds with boundary under submersions with definite folds.
method Analyzing submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on restrictions to the boundary and using results for m-functions.
result Obtained restrictions on the diffeomorphism types of the source manifolds and studied the diffeomorphism types and Euler characteristics of manifolds admitting such maps.
Study of diffeomorphisms groups on lens spaces with Morse-Bott foliations.
problem Computing homotopy types of diffeomorphism groups of specific foliations.
method Analysis of leaf-preserving and foliated diffeomorphisms on lens spaces.
result Inclusion of leaf-preserving groups into foliated groups is a homotopy equivalence.
We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is Rd or compact without boundary. The main result is that for s>dimM/2+1, the group Ds(M) is geodesically and me…
Finite type for 3-manifolds with boundary.
problem Finite type for 3-manifolds with boundary.
method Contractible space of reducing spheres, homotopy type of finite CW complex.
result B Diff(M rel boundary) has the homotopy type of a finite CW complex.
Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms o…
Symplectic structures on Teichmüller spaces for surfaces with ideal boundary.
problem Understanding symplectic structures on Teichmüller spaces for surfaces with boundary.
method Defined natural symplectic structures and proved Hamiltonian properties.
result Explicit formula for Hill potential and global Darboux coordinates.
Maps between Sobolev spaces become smoother under equivariance.
problem Regularity of maps between Sobolev spaces.
method Equivariance of maps allows trading regularity in the domain for regularity in the image space.
result Maps between Sobolev spaces are well-defined and smoother in the image space.
We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
problem Proving a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
method Homological approach to show finitely many nonzero homology groups, each finitely generated.
result BDiff(M, rel ∂) has finitely many nonzero homology groups, each finitely generated, for connected sums of irreducible 3-manifolds with nontrivial and non-spherical boundaries.
This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions.
Study calculates homotopy groups and derivatives for disc diffeomorphisms.
problem Understanding the homotopy groups of diffeomorphisms of discs.
method Computes rational homotopy groups and uses Weiss' orthogonal calculus.
result Determines optimal rational concordance stable range for high-dimensional discs.
The paper extends torus surgery results in 4-manifolds and shows diffeomorphisms.
problem Extending torus surgery results in 4-manifolds.
method Analyzing 4-manifolds diffeomorphic to S1 times exterior of fibered knots and T2imesD2 bundles over T2. result The result of torus surgery in S1imesY along S1imesK fibers over S1.