The paper studies deformations of Hermitian Yang-Mills and Donaldson-Thomas connections on G2-manifolds.
problem Deformation theory of connections on G2-manifolds. method Introducing new coclosed G2-structures and analyzing elliptic complexes. result Moduli spaces of connections are shown to be tori under certain conditions.
First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of CP2 and cones over 3-Sasakian 7-manifolds. result First non-trivial examples of deformed Spin(7)-instantons.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
First non-trivial examples of deformed G_2-instantons, distinguishing nearly parallel G_2-structures.
problem Distinguishing between nearly parallel G_2-structures and isometric G_2-structures.
method Provided first non-trivial examples of deformed G_2-instantons and studied their deformation theory.
result Found non-trivial deformed G_2-instantons with obstructed deformation theory and moduli spaces of different dimensions.
Study deforms Hermitian metrics with positive curvature.
problem Deforming Hermitian metrics with positive curvature.
method Adapted conformal perturbation method to Hermitian setting.
result Hermitian metrics with quasi-positive curvature can be deformed to positive curvature.
The paper connects isomonodromic and isospectral deformations for sl2(C) connections.
problem Connecting isomonodromic and isospectral deformations for sl2(C) connections. method Explicitly constructing Lax pairs and Darboux coordinates to bridge isomonodromic and isospectral deformations.
result Explicit change of Darboux coordinates to match spectral invariants, solving an open issue.
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
New examples of deformed Hermitian-Yang-Mills connections found.
problem Constructing deformed Hermitian-Yang-Mills connections on manifolds.
method Constructed first higher rank, irreducible deformed Hermitian-Yang-Mills connections in both small and large radius regimes.
result Existence of solutions with any possible angle and ruling out some stability conditions.
Computes deformations of parabolic structures on Riemann surfaces.
problem Infinitesimal deformations of parabolic connections and opers.
method Computes infinitesimal deformations of quadruples (X, S, E*, D) and (X, S, D).
result Monodromy map is an immersion from the moduli space of triples to the character variety.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar resu…
We provide an infinite family of pared manifolds whose relative deformation spaces of hyperbolic structures on these manifolds are not locally connected. This is a natural extension of the recent result of Bromberg that shows the space of Kleinian punctured torus groups is not locally connected.
Based on the analogies between knot theory and number theory, we study a deformation theory for SL_2-representations of knot groups, following after Mazur's deformation theory of Galois representations. Firstly, by employing the pseudo-SL_2-representations, we prove the existence of the universal deformation of a given…
For any closed surface S of genus g≥2, we show that the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to S, AH(S×I), is not locally connected. This proves a conjecture of Bromberg who recently proved that the space of Kleinian punctured torus groups is not locally connected.…
Study geodesics on a modified cotangent bundle over Kählerian manifolds.
problem Investigate geodesics on a modified cotangent bundle.
method Introduced Berger-type deformed Sasaki metric, investigated Levi-Civita connections, and studied geodesics.
result Geodesic properties on modified cotangent bundles.
Defines curvature for spectral triples and applies to θ-deformations.
problem Defining curvature for noncommutative spectral triples.
method Using Levi-Civita connection, defines curvature tensors and derives Weitzenbock formula.
result Riemann and Ricci tensors transform naturally under θ-deformation, while scalar curvature is invariant.
Deformed holomorphic Chern-Simons theory yields new instantons.
problem Deforming classical holomorphic Chern-Simons theory on Calabi-Yau manifolds.
method Deformation of complex structure by a parameter \( h \) leading to new instanton solutions.
result Existence of instanton solutions invariant under re-scalings of \( h \) and their connection to \( G_2 \)-instantons.
The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.
problem Understanding singularities and deformations in meromorphic connections and quadratic differentials.
method Local formal invariants and jets of meromorphic quadratic differentials, universal isomonodromic deformation, unfolded Stokes phenomenon, horizontal and vertical foliations.
result Establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials, describing parameter spaces and moduli spaces.
We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid…
Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections…
Necessary and sufficient conditions for some deformation algebras to provide formal Frobenius structures are given. Also, examples of formal Frobenius structures with fundamental tensor that is not of the deformation type and examples of symmetric non-metric connections are presented.
We consider deformations of G-structures via the right action on the frame bundle in a base-point-dependent manner. We investigate which of these deformations again lead to G-structures and in which cases the original and the deformed G-structures define the same instantons. Further, we construct a bijection from conne…
Affine deformations serve as basic examples in the continuum mechanics of deformable 3-dimensional bodies (referred as homogeneous deformations). They preserve parallelism and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by …
The paper computes KV cochain differentials and their geometric implications.
problem Deformation theory of flat and torsion-free affine connections.
method Explicit computation of KV cochain differentials and their relations to geometric transformations.
result KV algebra with non-vanishing second cohomology group.
Quantizes functions on Kähler manifolds without formal deformation.
problem Deforming smooth functions on Kähler manifolds to non-formal quantization.
method Using Fedosov connections and prequantum line bundles.
result Quantizable functions form a sheaf of twisted differential operators.
Deformation quantization yields a new moment map on symplectic diffeomorphisms.
problem Formalizing moment maps on diffeomorphism groups of symplectic manifolds.
method Deformation quantization framework applied to extrmDiff0(M). result Obtained a deformation of the Donaldson moment map.
We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.
The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, S. Koch and I recently showed that the deformation space has in…
Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of $\Aut(G)$ on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev…
We establish that Hitchin's connection exist for any rigid holomorphic family of Kahler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin's connection induces a unique formal connection on smooth functions…
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kaehler manifold which respects the almost-Kaehler structure to construct a Fedosov-type deformation quantization on this manifold.
Study mSpin(7)-dDT connections on manifolds with mSpin(7)-structures.
problem Understanding moduli spaces of mSpin(7)-dDT connections. method Introduced and studied mSpin(7)-dDT connections using fully nonlinear PDEs. result Moduli space MmSpin(7)′ has finite expected dimension and smoothness under certain conditions. A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove…
We construct a simply connected minimal complex surface of general type with pg=0 and K2=2 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with pg=0 and K2=1. In order to construct the example, we combin…
New theorem connects minimal and maximal surfaces, affecting graphness.
problem Understanding graphness of minimal surfaces in different spaces.
method Introducing a new deformation family and proving Krust-type theorems.
result Graphness of minimal surfaces in isotropic 3-space affects deformed surfaces.
Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.
problem Understanding the geometry and topology of affine-orthogonal manifolds.
method Deformation of flat connections into Levi-Civita connections and analysis of Euler characteristic.
result Deformations force the Euler characteristic to vanish, supporting Chern's conjecture.
Study Rarita-Schwinger fields on nearly Kähler manifolds, finding coinciding spaces of fields and deformations.
problem Investigate Rarita-Schwinger fields on compact strict nearly Kähler manifolds.
method Clarify differential operator relationships, use deformation theory.
result Spaces of Rarita-Schwinger fields and Killing spinors coincide with specific eigenspaces and harmonic forms.
The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.
problem Deforming quasi-Hamiltonian spaces to Hamiltonian spaces.
method Introducing and proving examples of deformations, including Lie groups and conjugacy classes.
result Moduli space of flat G-connections deforms to T*G^r+g.
The paper proves unique Levi-Civita connections on noncommutative forms.
problem Existence and uniqueness of Levi-Civita connections on noncommutative differential forms.
method Combining Hilbert module and algebraic techniques, proving conditions for existence and uniqueness of Hermitian torsion-free connections.
result Existence and uniqueness of Levi-Civita connections on θ-deformations of compact Riemannian manifolds.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at quasiconformally rigid points.
Study symplectic structures in moduli spaces of meromorphic connections.
problem Understanding symplectic structures in moduli spaces of meromorphic connections.
method Explicit and infinite dimensional description of symplectic structures.
result Intrinsic symplectic description of isomonodromic deformation equations.
Geometric models for algebraic suspensions using affine deformation spaces.
problem Analyzing which motivic homotopy types have smooth scheme representatives.
method Affine deformation spaces of embeddings provide models for P1-suspensions of schemes. result Yields many examples of A1-connected and strictly quasi-affine schemes. Alternative definition of dDT connections for Spin(7) manifolds.
problem Defining deformed Donaldson-Thomas connections for Spin(7) manifolds.
method Real Fourier-Mukai transform applied to compute new connections.
result Alternative definition of dDT connections is compatible with existing connections.
B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flatte…
Spaces of circle embeddings in curved surfaces indexed by trees.
problem Classifying spaces of braided automorphism groups of trees.
method Indexed connected components with finite rooted trees, constructed strong deformation retract.
result Connected components are classifying spaces of braided automorphism groups.
In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannia…
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. Study deformations of G2-instantons on nearly G2 manifolds.
problem Deformations of G2-instantons on nearly G2 manifolds.
method Formulated in terms of spinors and Dirac operators, proved isomorphism of infinitesimal deformations to kernel of an elliptic operator.
result Proved abelian instantons are rigid and described the deformation space of the canonical connection on specific nearly G2 manifolds.