arXiv research

This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.

Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.

The paper proves a factorization theorem for harmonic maps between Riemann surfaces and manifolds.

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and the…

We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si…

The CR $δ$-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR $δ$-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…

It is well known that the twisters, section of twister space, classify the almost complex structure on even dimensional Riemannian manifold $X$. In this paper, it will be proved that a harmonic and anti-holomorphic twister is equivalent ti a symplectic structure on $X$.

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichmü{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a funct…

New connections on symmetric spaces with invariant properties.

The study of real Einstein submanifolds in Kähler geometry.

Harmonic extension of Weil-Petersson circle homeomorphisms

Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Γ\,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $ρ\, :\, π_1(X, x_0)\,\longrightarrow\, Γ$ a homomorphism such that the adjoint action of $ρ(π_1(X, x_0))$ on $\text{Lie}(G)$ is comple…

We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give n…

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $φ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_…

Real slices of parabolic opers on Riemann surfaces are studied.

We study complex Lagrangian submanifolds of a compact hyper-Kähler manifold and prove two results: (a) that an involution of a hyper-Kähler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is co…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

Study Higgs bundles on smooth projective varieties and their restrictions to curves.

Study of Gauss maps for minimal surfaces in a specific 3D model.

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric sp…

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound i…

We find all m-spin structures on Klein surfaces of genus larger than one. An m-spin structure on a Riemann surface P is a complex line bundle on P whose m-th tensor power is the cotangent bundle of P. A Klein surface can be described by a pair (P,tau), where P is a Riemann surface and tau is an anti-holomorphic involut…

The Witten class is derived from equivariant cohomology of a conformal loop space.

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the gro…

We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constit…

Let $Γ$ be a finitely generated group, $G_\mathbb{C}$ be a classical complex group and $G_\mathbb{R}$ a real form of $G_\mathbb{C}$. We propose a definition of the $G_\mathbb{R}$-character variety of $Γ$ as a subset $\mathcal{X}_{G_\mathbb{R}}(Γ)$ of the $G_\mathbb{C}$-character variety $\mathcal{X}_{G_\mathbb{C}}(Γ)$.…

This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting part…

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal …

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…

Through the action of anti-holomorphic involutions on a compact Riemann surface, we construct families of (A,B,A)-branes in the moduli spaces of G_c-Higgs bundles on the Riemann surface. We study the geometry of these (A,B,A)-branes in terms of spectral data and show they have the structure of real integrable systems.

Maximal discs in Anti-de Sitter space linked to Teichmüller space.

We investigate representations of Kähler groups $Γ= π_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor--Wood inequality similar to those found by Burger--Iozzi and Koziarz--Maubon. Thanks…

The work of Ray and Singer which introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting t…

Study of algebraic curves and surfaces in flag manifold using twistor geometry.

The paper proves properties of Bergman spaces and Schiffer operators on Riemann surfaces with quasicircles.

15 Einstein 4-manifolds with positive conformal curvature are classified.

Generalization of twistor spinors to Kähler manifolds which are called Kählerian twistor spinors are considered. We find the differential equation satisfied by the bilinear forms of Kählerian twistor spinors. We show that the bilinear form equation reduces to Kählerian conformal Killing-Yano equation under special cond…

Study sigma-models on flag manifolds with curvature zero and relate to chiral models.

We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to f…

We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dime…

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the ant…

We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ h…

We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic $(2n+1)$-spheres. We proved that a nearly Kählerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold $M$ of dimension $2m\geq…

Study real slices of SL(r,C)-opers via Riemann surface involution.

Study gauge theory of real and quaternionic parabolic bundles over real curves.

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…