Paper finds topological block to certain Lagrangian submanifolds.
problem Existence of specific Lagrangian submanifolds in symplectic 4m-manifolds.
method Topological obstruction analysis.
result Certain Lagrangian submanifolds cannot exist in symplectic 4m-manifolds.
We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for δ-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second f…
Fewer obstructions for small graphs in knotless embedding.
problem Finite obstructions for knotless embedding of small graphs.
method Analysis of graph minor obstructions and knotless embedding.
result There are only three obstructions for size 23 graphs, significantly fewer than previously known.
Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.
problem Existence of topological solitons in Yang-Mills-Chern-Simons theories on compact manifolds.
method Cohomological formulations of the calculus of variations, focusing on Yang-Mills-Chern-Simons theories on compact manifolds in odd dimensions.
result Non-trivial obstructions leading to a strong non-existence theorem for topological solitons.
Study the topological information of map germs using Euler obstruction.
problem Capturing topological information from map germs with complex singular spaces.
method Investigates the Euler obstruction and its relation to local Euler obstruction and Brasselet number.
result Relates Chern number to the number of cusps in a perturbed map-germ.
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…
Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
problem Finding topological obstructions to compact gradient shrinking Ricci solitons in dimension four.
method Discussion of background material, introduction of new problem, exploration of limitations of current results.
result Introduction of new problem and limitations of current results in extending Hitchin-Thorpe inequality.
Study on symplectic structures and their deformations.
problem Preservation of complex symplectic structures under deformations.
method Analyzes various cohomologies and conditions for deformations.
result Obtains topological obstructions for compact complex symplectic manifolds.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
Study curvatures of submanifolds in space forms with topological obstructions.
problem Investigate intrinsic curvatures and their implications for submanifolds in space forms.
method Derive inequalities involving mean curvature and normal scalar curvature, derive topological obstructions.
result Prove existence of compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres.
We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen …
Topological obstructions to admissibility in σk-Loewner--Nirenberg problem
problem Admissibility condition for σk-Loewner--Nirenberg problem method Exhibit topological obstructions
result Illustrate with examples
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
A nonstandard (maximal) inclusion SO(3) in SO(5) associated with the irreducible representation ρ_5 of SO(3) in R^5 is considered. The topological obstructions for admitting the SO(3) structure on the frame bundle over 5-manifold are investigated. The necessary and sufficient conditions are formulated.
We use topological quantum field theory to derive an invariant of a three-manifold with boundary. We then show how to use this invariant as an obstruction to embedding one three-manifold in another.
New obstruction found for embedding Riemannian manifolds into Euclidean spaces.
problem Embedding Riemannian manifolds into Euclidean spaces with specific conditions.
method Motivated by incompressible Euler equations, a dynamical-topological obstruction is derived.
result Nontrivial first real homology and trivial center of fundamental group imply embedding violation.
The study explores surfaces with curvature satisfying a specific relation, leading to isometric immersions and topological obstructions.
problem Curvature relations on smooth Riemannian surfaces and their geometric implications.
method Properties of log-harmonic functions and isometric immersions theorems.
result Characterization of surfaces that locally admit minimal isometric immersions into constant curvature manifolds.
From Furuta's 810 theorem, we derive a smooth slicing obstruction for knots in S3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly sl…
New topological obstructions found for elliptic and quasiregularly elliptic manifolds.
problem Connection between ellipticity and quasiregular ellipticity.
method Proved new topological obstructions.
result Closed manifolds of both types must have virtually abelian fundamental group.
We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.
In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into C3 (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient co…
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
New obstructions show some 4-manifold homeomorphisms are pseudo-isotopic but not isotopic.
problem Tackling the difference between pseudo-isotopy and isotopy for 4-manifolds.
method Defining and showing realizable obstructions in Whitehead groups.
result Found homeomorphisms that are pseudo-isotopic but not isotopic.
The paper introduces new invariants to study topological properties of map germs.
problem Investigating topological invariants of map germs and their families.
method Introducing the Bruce-Roberts and relative Bruce-Roberts numbers as algebraic tools.
result Closed formulas for these invariants under certain conditions.
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into R2n for n=2, and it was recently shown to be incomplete for n=2. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embed…
Given a map f: M \to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposi…
Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, bk-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions…
Explains research on 3D dynamics and manifold topology.
problem Understanding obstructions for Anosov flows on 3-manifolds.
method Expository note on partially hyperbolic diffeomorphisms and Anosov flows.
result Margulis and Plante-Thurston's topological obstructions for Anosov flows.
We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…
Study deformations of compact Calabi-Yau conifolds with singularities.
problem Understanding deformations of compact Calabi-Yau conifolds with singularities.
method Analyzes the obstructions and local triviality of deformations under specific topological and geometric hypotheses.
result Obstruction to deformations concentrates at singularities, generalizing previous results.
We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have C1 robustly transitive endomorphisms with critical points on surfaces, and the only surfaces …
Study shows non-orientable manifolds restrict signature-changing metrics globally.
problem Global obstructions to signature-changing metrics on non-orientable manifolds.
method Explicit geometric constructions based on Möbius strip topology.
result Radical of signature-changing metrics cannot be everywhere transverse.
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
Study finds limits on curvature and shape of certain spacetimes.
problem Limits on curvature and shape of compact stationary spacetimes.
method Used Petersen & Wink '21 result on Lorentzian manifolds.
result Obstructions found to curvature and topology of compact spacetimes.
The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
problem Proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
method Rigorously proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories on Rd′imesCd. result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
New compact manifolds with closed G2 structures found.
problem Existence of torsion-free G2 structures on compact manifolds.
method Equivariant resolution procedure for G2 orbifolds with Z2 isotropy.
result Construction of new compact manifolds with closed G2 structures.
New action-angle coordinates found for singular symplectic manifolds.
problem Existence of action-angle coordinates for singular symplectic manifolds.
method Action-angle theorem for folded symplectic integrable systems.
result New topological obstructions found for global existence of action-angle coordinates.
We construct examples of nonresolvable generalized n-manifolds, n≥6, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed n-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology.
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
This paper provides a K-theoretic obstruction for higher kernel dimension for Dirac operators. For this we use a fibre-wise Dirac operator that gives rise to a family of Fredholm operators representing a class in topological K-theory. Then Chern classes of this K-class contain some information about the kernel of…
Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.
problem Understanding which 4-manifolds can have metrics with uniformly positive scalar curvature.
method Topological obstructions and metric constructions on specific 4-manifolds.
result Existence of uncountably many exotic R4's without such metrics and topological uniqueness of certain metrics. Study on prime knots, slice obstructions, and ribbon concordances.
problem Determining which prime knots are slice in smooth and topological categories.
method Use of a wide range of tools and techniques, including new or refined methods for probing these properties.
result About 1.6 million prime knots are smoothly slice (ribbon), and 350.5 million are not even topologically slice.