arXiv research

Constructs a homotopy Loday algebra from symplectic 2-manifolds.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,φ,Λ), with the calibration 3-form φand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. …

The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions $n=4r-2$. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…

Investigates counting associative 3-folds in G2-manifolds, analogous to holomorphic curves in Calabi-Yau.

The purpose of this paper is to introduce Harvey-Lawson manifolds and review the construction of certain mirror dual Calabi-Yau submanifolds inside a G_2 manifold. More specifically, given a Harvey-Lawson manifold HL, we explain how to assign a pair of tangent bundle valued 2 and 3-forms to a G_2 manifold (M,HL, \varph…

Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectom…

Study of K3 surfaces with special involutions and singularities.

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

We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. …

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and co…

The main result is the identification of the orthogonal complement of the subalgebra of conformal vector field inside the algebra of all vector fields of a compact flat 2-manifold. As a fundamental tool, the complete Hodge decomposition for manifold with boundary is used. The identification allows the derivation of gov…

Study classifies 2-manifolds with special homeomorphism groups.

Complex $G_2$ manifolds and their associative submanifolds are studied via Seiberg-Witten equations.

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Survey on distinguishing G_2-manifolds using metrics and constructions.

Equi-affine curvature of curves in 2-manifolds linked to Frenet curvature.

The study provides homological characterizations for $Q$-manifolds and $l_2$-manifolds.

The paper connects $G_2$-manifolds to metric dynamics.

We define deformations of $RP^2$-manifolds.

Game theory applied to splitting surfaces of compact 2-manifolds.

Conditions for Penrose-Ward transformation on specific manifolds.

The study explores homeomorphism groups of self-similar 2-manifolds, including the 2-sphere and Cantor set.

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

The paper constructs complex hyperbolic 2-manifolds with one cusp.

Formula derived for $G_2$-manifolds, showing moduli spaces are incomplete.

In this note we generalize a result by Alekseev and Strobl for the case of $p$-branes. We show that there is a relation between anomalous free current algebras and "isotropic" involutive subbundles of $T\oplus \wedge^p T^*$ with the Vinogradov bracket, that is a generalization of the Courant bracket. As an application …

We consider the minimum Yang-Mills energy on the complete $G_{2}$-manifolds and Calabi-Yau 3-folds,the connection $A$ is a stability Yang-Mills connection on the $G$-bundle $E$.We prove that the connection must be a $G_{2}$-instanton on $G_{2}$-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $…

Classifies manifolds with dense conjugacy classes in their mapping class groups.

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

This is a very short and elementary introduction to G_2 manifolds. We stress the similarities and the differences with Kähler manifolds in general and with Calabi-Yau manifolds in particular.

On a projective complex manifold, the Abelian group of Divisors maps surjectively onto that of holomorphic line bundles (the Picard group). On a $G_2$-manifold we use coassociative submanifolds to define an analogue of the first, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the …

Study of $U(1)$ gauge theories on $G_2$ manifolds, including instantons and Chern-Simons.

Study on holomorphic discs in bundles over compact surfaces, proving Fredholm regularity under certain conditions.

Study geometry on $G_2$-manifolds, focusing on $Spin(7)$ and $G_2$.

Computes a $ν$-invariant for Joyce's $G_2$-manifolds.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

The paper constructs diffeomorphisms on a $G_2$-manifold achieving entropy bounds.

We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of co…

Surveying $G_2$-instantons on noncompact manifolds, including examples and open problems.

The study constructs $G_2$-manifolds from K3 surfaces with a specific action.

We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.