arXiv research

Study pseudoisotopies in 4-manifolds, finding specific elements of obstruction.

The paper constructs a nontrivial diffeomorphism in 4-manifold topology.

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…

Proves Massey's theorems on complex structure obstructions.

Study second-order obstruction to nearly $G_2$ structure deformations.

Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. T…

This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture of Boileau, Boyer, Cebanu, and Walsh. We also provide a second obstruction to a…

We prove that a compact log symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of b-log symplectic structures similar to those in symplectic geometry.

This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(π_1(E))) and, provided that this obstruction vanishes and one additional condition is verified, a second obstruction in Wh(π_1(E)) >. Both elements v…

Study on complex Grassmannians' rigidity using Einstein deformations.

Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.

The study constructs examples of pseudo-isotopic but not isotopic 4-manifold diffeomorphisms.

Study on deformation theory of nearly G2 manifolds with obstructions.

Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show…

On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\mathbb{C}^2$ which are diffeomorphic t…

We carry out the programme of R. Liouville \cite{Liouville} to construct an explicit local obstruction to the existence of a Levi--Civita connection within a given projective structure $[Γ]$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a poi…

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …

Torelli group's second homology is finite for large genus surfaces.

The study finds obstructions to certain Riemannian metrics using Lorentzian geometry.

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq κ\leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction fo…

Non-trivial Clifford bundle from loop space tangent bundle.

The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors"…

New bounds on Jones polynomial positivity for specific links.

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …

New bounds on Khovanov homology for positive links families.

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularit…

Extends Lie algebroids by Lie algebroids with specific conditions.

New links identified with Alexander polynomial signs to detect satellite links.

The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it p…

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while …

Study confoliations' symplectic fillability, finding obstructions.

Proves section conjecture for curves and surface bundles over various fields.

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topologi…

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casso…

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operat…

We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the t…

New obstruction prevents certain spacetimes with both big bang and big crunch.

We define an algebraic group comprising symmetric chain complexes which captures the first two stages of the Cochran-Orr-Teichner solvable filtration of the knot concordance group in a single obstruction. To achieve this we impose additional structure on each chain complex which puts extra control on the fundamental gr…

In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_θ\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}…

New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.

Paper characterizes foliated bundle classes via quasi-morphisms and studies their boundedness.

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions…

This research studies the smoothness of moduli spaces of self-dual contact instantons on Sasakian manifolds.

Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second fundamental form of the hypersurface and the vector field itself. We show how these inv…

This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between…

New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.

This paper studies torsion obstructions to complex sections on manifolds.