arXiv research

Survey on L^2-invariants for 3-manifolds.

The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…

Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.

Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.

We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.

In this paper we discuss open problems concerning L^2-invariants focusing on approximation by towers of finite coverings.

New gradient Ricci solitons found for $SU(2)$ invariants.

We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, $U(1)^2$-invariant initial data sets on simply connected four dimensional manifolds $Σ$. Moreover, we extend the local mass angular momenta inequality result Ref [1] for $U(1)^2$ invariant black holes to the case with nonzero stre…

We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.

The universal sl_2 invariant of bottom tangles has a universality property for the colored Jones polynomial of links. Habiro conjectured that the universal sl_2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra U_h(sl_2) of the Li…

We give a survey on L^2-invariants such as L^2-Betti numbers and L^2-torsion taking an algebraic point of view. We discuss their basic definitions, properties and applications to problems arising in topology, geometry, group theory and K-theory.

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentu…

Studied $L^2-$invariants on stratified spaces, proving their stability.

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…

Study coclosed G2-structures on SU(2)²-invariant manifolds.

New steady gradient Ricci solitons found on specific four-manifolds.

We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl_2. This implies an integrality result on the…

We define extensions of the $L^2$-analytic invariants of closed manifolds, called delocalized $L^2$-invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spect…

New Einstein metrics constructed on complex line bundle over CP1.

We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provid…

New one-parameter families of $SU(2)^2$-invariant instantons found on Calabi-Yau 3-folds.

A link L is called Brunnian if every proper sublink of L is trivial. Similarly, a bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we give a small subalgebra of the n-fold completed tensor power of U_h(sl_2) in which the universal sl_2 invariant of n-component Brunnian bottom…

Let $X$ be a finite aspherical CW-complex whose fundamental group $π_1(X)$ possesses a subnormal series $π_1(X) \rhd G_m \rhd ... \rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invarian…

We show that the L^2-torsion and the von Neumann rho-invariant give rise to commensurability invariants of knots.

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-Kähler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…

This is an elementary observation that the symmetry properties of the Riemann curvature tensor can be (efficiently) expressed as SL(2)-invariance.

Study $G_2$-instantons on $\mathbb{R}^4 imes S^3$ with invariant properties.

Paper introduces new invariant for pairs of immersions.

New Thurston norm defined for a specific type of groups using $L^2$-invariants.

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

Locally symmetric metrics on 4-manifolds with non-negative curvature.

Paper is withdrawn due to errors (superseded by math.AG/0604303). Formula 6.5 is false. Section 6 is false, and the main statement is true only for bundles $B$ with $c_1(B)$ SU(2)-invariant.

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthe…

Paper withdrawn due to errors (superseded by math.AG/0604303). Proposition 11.4 is false, Section 12 is false, and the main statement is true only for bundles $B$ with $c_1(B)$ SU(2)-invariant.

We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.

We classify the SU(2)-invariant anti-self-dual metrics with a signature (+,+,-,-). The metrics are specified by a solution of Painleve VI, V, III or II. Moreover we show the geometric meaning of the metrics specified by each type of Painlevé functions.

New extremal Kähler metrics found on 4-manifolds with U(2) symmetry.

We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a TQFT.

We study the behavior of the Witten-Reshetikhin-Turaev SU(2) invariants of links in L(p,q) as a function of the level r-2. They are given by 1 over the square root of r times one of p Laurent polynomials evaluated at e to the 2 pi i divided by 4pr. The congruence class of r modulo p determines which polynomial is appli…

We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic four-manifolds with compatible but non-integrable almost complex structures.

The paper constructs special Lagrangian n-folds in arbitrary dimensions.

We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and diff…

In this paper, we prove that the $L^2$ Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, proving a conjecture that we made in an earlier paper. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant cla…

We consider the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a Kähler metric invariant by the $U(2)$ action on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and prove it has exactly one $U(2)-$invariant solution up to homothecies.

Biconservative hypersurfaces are hypersurfaces with conservative stress-energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3-dimensional Bianchi-Cartan-Vranceanu spaces,…

The study classifies Einstein metrics on a specific total space, revealing various behaviors and transitions.

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curve…

We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of sl_2 invariants of integral homology 3-spheres. Habiro originally derived these formulas using the quantum group U_q sl_2. As an application, we give a formula for the colored Jones polynomial of twist knots, g…