arXiv research

The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.

Classifies smooth isotopy of a specific Lagrangian embedding.

New method constructs explicit $p$-harmonic functions on Lie groups.

Sharp constants in curl-Sobolev inequalities on spheres determined.

Study on holonomy of Obata connection on Joyce hypercomplex manifolds.

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of…

The Gaiotto locus for Sp(2n) is shown to lie in the nilpotent cone.

The paper constructs paths of SL2(R) representations for pretzel knots and shows left-orderability conditions.

For a surface group, a new bound is given for conjugator length function.

The study of Seifert linking forms for punctured n-manifolds in (2n-1)-space.

The paper defines Dirichlet domains for Anosov subgroups in Lie groups.

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces wit…

We study generalized special cycles on Hermitian locally symmetric spaces $Γ\backslash D$ associated to the groups $G=\mathrm{U}(p,q)$, $\mathrm{Sp}(2n,\mathbb{R})$ and $\mathrm{O}^*(2n)$. These cycles are (covered by) locally symmetric spaces associated to subgroups of $G$ which are of the same type. Using oscillator…

The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.

This paper completes proofs for left orderable slopes of double twist knots.

Study on left orderability of specific knot covers.

The paper reduces the required dimension for a $\mathbb{Z}_2$-torus action on positively curved manifolds to half the dimension.

New geometry based on Siegel upper half-space with volume formula.

Maps between classifying spaces for certain groups are studied, with rational cohomology results.

The h-principle fails for prelegendrians in fat distributions of corank 2.

Extends Euler class result to symplectic group.

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of con…

The paper proves nonvanishing cohomology for ball quotient fundamental groups.

The coarea formula is proven for Heisenberg group maps, addressing open questions.

Proof shows volumes of certain geometric representations are always integers.

Constructs Kleinian groups from free groups via hyperbolization.

We find all extremal Lagrangian tori in symplectic unit balls and some toric domains.

Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Betti number $b_{i}^{(2)}(G)=0,$ $\$when $G=\mathrm{GL}_{n}(R)$ the general linear group, $\mathrm{SL}_{n}(R)$ the special linear group, $$ the group generated by…

In this paper we construct complex contact structures on $\mathbb{C}^{2n+1}$ for any $n\ge 1$ with the property that every holomorphic Legendrian map $\mathbb{C}\to \mathbb{C}^{2n+1}$ is constant. In particular, these contact structures are not globally contactomorphic to the standard complex contact structure on $\mat…

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ o…

We classify generalized distance-squared mappings of $\mathbb{R}^{n+1}$ into $\mathbb{R}^{2n+1}$ ($n\ge 1$) having generic central points. Moreover, we show that there does not exist a universal bad set $Σ\subset (\mathbb{R}^{n+1})^{2n+1}$ in the case of this dimension-pair.

New methods prove non-squeezing in locally conformal symplectic geometry.

Holonomy invariants from $\mathrm{SL}_2(\mathbb{C})$ link complements detect link geometry.

Minimal Kaehler submanifolds up to codimension four are studied.

The study finds surface subgroups in cocompact lattices of $H^{2n}$ for $n\geq2$.

The paper classifies topological properties of specific geometric forms on manifolds.

If $Γ<\mathrm{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $Γ\rightarrow \mathrm{PSL}(n,\mathbb{C})$ using the Borel class $β(n)\in \mathrm{H}^3_\mathrm{c}(\mathrm{PSL}(n,\mathbb{C}),\mathbb{R})$. We show that the invariant is bounded and its maximal value is attained by conjugation of t…

Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.

We find a presentation of symplectic Steinberg modules and show vanishing cohomology for certain groups.

Let $M^{2n}$ denote a closed $(n-1)$-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\overlineΘ_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_c(M^{2n})=0$ for $…

For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a nat…

The paper studies submanifolds of Euclidean space from Kaehler manifolds.

The study proves non-existence of hypercomplex structures on SL(3,R) and finds one on SL(2n+1,C).

Positive paths connect diffeomorphisms on contact manifolds.

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le α$, we prove that $\left(\sharp_{i=1}^α M_{i}\right) \sharp (α{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $α$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we…

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…

The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.