arXiv research

No exact G₂-structures on compact Lie group quotients.

Compact quotients of homogeneous spaces are studied, leading to new findings about sphere bundles.

Study compact plane waves, showing they are essentially standard.

New manifold types defined on quotient spaces.

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $Γ$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $Γ\backslash G/H$ is the integral, over a certain homology class of $Γ…

Study compact symplectic solvmanifolds' hard Lefschetz property.

Paper studies Hessian quotient equations in warped product manifolds.

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space $G/H$ if $H$ is non-compact. The first half of the article elucidates general machinery …

Geodesics spiral around compact subsets in CAT(0) spaces.

Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)

Holomorphic curves found in compact quotients of SL(2,C).

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space R^{4d} by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration o…

The paper explores $p$-Kähler structures on Lie group quotients.

Let $K$ be a compact group. For a symplectic quotient $M_λ$ of a compact Hamiltonian Kähler $K$-manifold, we show that the induced complex structure on $M_λ$ is locally invariant when the parameter $λ$ varies in $\mathrm{Lie}(K)^*$. To prove such a result, we take two different approaches: (i) by using the complex geom…

The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these paramete…

Study on curves on specific arithmetic quotients of hyperbolic 2-ball.

In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients $\frac{\mathcal{H}_{2k+1}}{\mathcal{H}_{2k}}$ in the warped product manifolds. Here $\mathcal{H}_{2k}$ is the $k$-th Gauss-Bonnet curvature and $\mathcal{H}_{2k+1}$ arises from the first variation of the total integration of $…

Balanced metrics found on Lie groups and their quotients.

In the current article our primary objects of study are compact complex submanifolds of quotient manifolds of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms. We are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifo…

Knot groups of hyperbolic 2-bridge knots are uniquely identified by their finite quotients.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite q…

Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.

We consider the action of a noncompact torus H on the compact quotient G/L, where G is a Lie group containing H and L is a uniform lattice in G. Using harmonic analysis on G we prove a formula relating the compact orbits of H to the action of H on the (infinite dimensional) tangential cohomology. The formula may be vie…

If X is a proper CAT(-1)-space and $Γ$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$Γ$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wanderin…

Holomorphic curves found in compact quotients of SL(2,C).

Paper examines properties of specific solutions to Yamabe flow.

Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.

The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kaehler structure having twice the kinetic energy of the metric as its potential, and Kaehler reduction with reference to the adjoint action yields a stratified Kaehler structure on the resulting adjoint quotient. …

Paper generalizes sub-slope definition and solves complex equations on compact manifolds.

Study Dolbeault harmonic forms on Lie group quotients with specific structures.

Rational ellipticity proven for $G$-manifolds with specific quotient properties.

We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.

Let $M$ be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $\mathsf G$ in a way that the quotient space $M/\mathsf G$ has nonempty boundary. Let $π: M \to M/\mathsf G$ denote the quotient map and $B$ be any boundary stratum of $M/\mathsf G$. Via a specific soul co…

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural …

We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…

This paper classifies all admissible quotients of a surface braid group up to order 127.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

The paper classifies quasi-Einstein 3-manifolds and their properties.

In this paper, we give a complete classification of $κ$-solutions of Kähaler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

Embeds CR manifolds into complex spaces using equivariant actions.

Study of 4D symmetric spaces with (2,2) signature.

In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…