arXiv research

The set of maximal non-integrable structures $(SU(2)\times SU(2),B,I)$, where $B$ is Killing-Cartan metric is described as subset of $\mathbb{CP}^3$. The visualization of complex projective space $\mathbb{CP}^3$ as tetrahedron which edges and faces are $\mathbb{CP}^1$ and $\mathbb{CP}^2$ is used.

Researchers found the $\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_m$ is constant regardless of $m$.

Let ${\bf H}_{\mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${\rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${\rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $\partial {\bf H}_{\mathbb C}^n$. We classify ${\rm SU}(n,1)$ conju…

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

New families of Lie groups with special foliations discovered.

We show that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $Γ$ contains a loxodromic element $A$. If $A$ has eigenvalues $λ_1 = λe^{i\varphi},$ $λ_2 = e^{-2i\varphi}$, $λ_3 = λ^{-1}e^{i\varphi}$, we prove that $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SU}(2,1,\mathbb{Q}(Γ,λ)),$ where $\mat…

Unified framework counts knot representations into SU(2) and SL(2,R).

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

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes $SL_n(\mathbb{C})/SU(n),$ and extends the previously known spinorial representation of a surfa…

SU(n)-structures derived from Kähler manifolds with torus actions.

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski m…

Study geodesics and shortest arcs on Lie groups with specific metrics.

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

Study extended Bogomolny equations on curved space with special boundary conditions.

Hitchin shows that half-flat SU(3)-structures on a 6-dimensional manifold M can be lifted to parallel G_{2}-structure on the product $M\times\mathbb{R}$. We show that Hitchin's approach can also be used to construct nearly parallel G_{2}-structures by lifting so-called nearly half-flat structures. These SU(3)-structure…

Study on holonomy of Obata connection on Joyce hypercomplex manifolds.

Study geodesics and shortest arcs on Lie groups with specific metrics.

New knot classification based on SU(2) representations and instanton homology.

Study of a $G_2$-equivariant octonionic operator and its right spectrum.

The study explores SU(2) representations in 3-manifolds and knots with specific Heegaard genus constraints.

Let $Γ$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $Γ$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $Γ$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variet…

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

We show that $Γ< \textbf{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(γ) \in \R$ for all $γ\in Γ$ if and only if $Γ$ is conjugate to a subgroup of $\textbf{SO}(3,1)$ or $\textbf{SU}(1,1)\times\textbf{SU}(2)$.

New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.

Graph manifolds with small homology have non-trivial SU(2) representations.

Study conformal foliations on Lie groups, finding new families and harmonic morphisms.

The thesis examines the relationship between three-manifold invariants and knot theory, finding equalities and patterns.

New spectral triples defined for SU(1,1) using harmonic analysis.

Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.

Study shows unique Einstein metrics on $SU_{2n+1}$ and related spaces.

We construct globally-defined $SU(3)$ structures on smooth compact toric varieties (SCTV) in the class of $\mathbb{CP}^1$ bundles over $M$, where $M$ is an arbitrary SCTV of complex dimension two. The construction can be extended to the case where the base is Kähler-Einstein of positive curvature, but not necessarily t…

Proof that $SU(2)$ character variety of genus 2 surface is ${\mathbb C} P^3$.

We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained a…

The paper classifies nilmanifolds with specific SL(3,C) structures.

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. T…

Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.

Let $A\neq 0$ be a complex number with $|A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Ku…

We prove the existence of steady Kähler-Ricci solitons on equivariant crepant resolutions of $\mathbb{C}^n/G$, where $G$ is a finite subgroup of $SU(n)$.

We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ c…

New Lagrangian and special Lagrangian examples found in complex space.

Uniform doubling property proven for specific Lie groups.

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant,…

We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by $SU(2)$ or $SO(3)$. We show that their Euler characteristic agrees with that of the known examples, i.e. $S^6$, $\mathbb{CP}^3$, the Wallach space $SU(3)/T^2$ and the bi…

Proofs centerless unimodular contact Lie algebras.

Formula established for instanton homology of dual knots.

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form $\mathbb{R}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times\mathbb{R}^+$ generalised…

The dimension of the space of SU(n) and translation invariant continuous valuations on $\mathbb{C}^n, n \geq 2$ is computed. For even $n$, this dimension equals $(n^2+3n+10)/2$; for odd $n$ it equals $(n^2+3n+6)/2$. An explicit geometric basis of this space is constructed. The kinematic formulas for SU(n) are obtained …

Study on Yang-Mills heat flow on $\mathbb{R}^4$ bundles, showing infinite time bubbling.