Develops deformation theory for symplectic foliations using L∞-algebras.
problem Deformation of symplectic foliations.
method Uses L∞-algebras to control deformation problems. result Establishes a correspondence between small deformations and Maurer-Cartan elements of L∞-algebra. Study how pairs of 1D foliations can be deformed into contact structures.
problem Understanding deformations of pairs of 1D foliations.
method Linear deformations of pairs of codimension one foliations into contact pairs.
result Main result provides applications and insights into foliation deformations.
Local study of foliation deformation cohomology.
problem Understanding deformations of singular foliations.
method Introducing and studying local deformation cohomology.
result Local deformation cohomology for singular foliations and related structures.
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on (C2,0). method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
Study deformations of Calabi-Yau foliations using Kuranishi spaces.
problem Deforming Calabi-Yau foliations and understanding their properties.
method Analysis of three types of deformations (unfoldings, holomorphic, transversally holomorphic) using Kuranishi spaces.
result Smoothness of Kf and product structure of Kh. Characterizes rigidity of compact foliations using deformations of Lie groupoids and algebroids.
problem Rigidity of compact foliations on manifolds.
method Combining stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids.
result Cohomological characterization for rigidity of compact foliations.
The paper extends Witten's deformation to foliations and Morse functions.
problem Extending Witten's deformation to foliations and Morse functions.
method Using deformation to the normal cone and C*-modules, the paper constructs the Witten deformation for generic functions on foliations.
result Establishes the compactness of the resolvent and Morse inequalities for foliations with invariant transverse measures.
Develops tools to construct Einstein 4-manifolds from conformal foliations.
problem Constructing Einstein metrics on 4-manifolds with conformal foliations.
method Calculates Ricci curvature transformation under biconformal deformations.
result Constructs Einstein 4-manifolds with specific asymptotic behavior.
The paper shows how foliations' cohomology remains unchanged under certain deformations.
problem Preserving geometric and topological properties of foliations under deformations.
method Analyzing equivariant basic cohomology and its invariance under deformations.
result Equivariant basic cohomology structure is preserved under deformations, leading to algebraic conditions for Betti numbers preservation.
Constructs deformations of Vaisman manifolds preserving foliations.
problem Deforming Vaisman manifolds while maintaining their canonical foliations.
method Uses a basic 1-form with specific properties to construct transverse deformations.
result Basic 1-forms exist in abundance for constructing deformations.
The paper shows how to deform foliations to prove geometric properties of manifolds.
problem Proving geometric properties of manifolds with Killing foliations.
method Deforming foliations to maintain transverse geometric properties and applying Riemannian geometry of orbifolds.
result Positive basic Euler characteristic for positively curved Killing foliations.
The number of Klein-bottle leaves in taut foliations is invariant under smooth deformations.
problem Invariance of Klein-bottle leaf counts in taut foliations.
method Proving invariance of the parity of Klein-bottle leaf counts under smooth deformations.
result The parity of Klein-bottle leaf counts is invariant under smooth deformations.
The study classifies nilpotent Lie foliations with cohomological obstructions.
problem Understanding rigidity of nilpotent Lie foliations under solvable deformations.
method Development of a cohomological framework and algebraic criterion for rigidity.
result Established a necessary and sufficient algebraic criterion for rigidity in generalized Heisenberg groups.
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using L∞-algebras. result Gauge equivalences for foliations and pre-symplectic structures are consistent.
Study on characteristic classes for foliation deformations.
problem Characterizing and understanding characteristic classes for foliation deformations.
method Introduced a differential graded algebra (DGA) to recover Bott vanishing and formulae, and discussed properties of its cohomology.
result Discovered new classes that cannot be described by existing classes like Godbillon--Vey and Fuks--Lodder--Kotschick.
Hodge numbers of Sasakian manifolds remain unchanged under deformations.
problem Invariance of Hodge numbers under deformations of Sasakian manifolds.
method Analysis of deformations of Sasakian structures and use of transversely elliptic operators.
result Hodge numbers are invariant under arbitrary deformations of the Sasakian structure.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (Ω^*[1](\fol), d_\fol…
In this note we observe, answering a question of Eliashberg and Thurston, that all contact structures on a closed oriented 3-manifold are C∞-deformations of foliations.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
The paper introduces new metric structures on g-foliations and uses a flow to deform them.
problem Developing new flexible metric structures on g-foliations. method Introducing new metric structures and using the partial Ricci flow to deform them.
result Deformation retraction of new structures with positive partial Ricci curvature onto classical structures.
Study shows how to deform foliated manifolds with flat leaves, leading to geometric characterization.
problem Deforming foliated manifolds with flat leaves while maintaining curvature bounds.
method Collapsing a manifold with a closed flat regular Riemannian foliation, keeping curvature uniformly bounded.
result For compact, simply connected manifolds, foliations are given by torus actions.
The study of geometric structures around transversals using deformation spaces.
problem Understanding local behavior of geometric structures around singular foliations.
method Using deformation spaces to study local behavior of geometric structures.
result Obtained normal form theorems around transversals for various geometric structures.
This is a note of the author's lectures at "Advanced courses in Foliation" in the research program "Foliation", which was held at the Centre de Recerca Mathematica in the May of 2010. In this note, we discuss about the relationship between deformation of actions of Lie groups and the leafwise cohomology of the orbit fo…
We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of represe…
The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.
problem Constructing foliations of minimal surfaces in negatively curved 3-manifolds.
method Deformations of totally geodesic foliations, using Grassmann bundle and negatively curved metrics.
result The foliations of minimal surfaces are deformations of totally geodesic foliations.
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.
Let f : Y -> X be a morphism of complex projective manifolds, and let F be a subsheaf of the tangent bundle which is closed under the Lie bracket, but not necessarily a foliation. This short paper contains an elementary and very geometric argument to show that all obstructions to deforming the morphism f along the shea…
Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.
The interior Kasparov product formula is extended for foliated ρ-classes on Riemannian bundles.
problem Extending the Kasparov product formula for foliated ρ-classes.
method Construction of asymptotic morphisms and adiabatic deformation groupoids.
result Interior Kasparov product formula for foliated ρ-classes on Riemannian bundles.
We realize the Reeb foliation of S^3 as a family of Legendrian submanifolds of the unit S^5 \subset C^3, Moreover we construct a deformation of the standard contact S^3 in S^5, via a family of contact submanifolds, into this realization.
We study the space of deformations of a smooth foliation of the 5-sphere by complex manifolds
We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the Álvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the Álvarez class under deformations shown in arXiv:1009.1098v2, …
New proof classifies orbit closures in Hodge bundle.
problem Classifying mGL+(2,R)-orbit closures in Hodge bundle. method Using deformations of flat pairs of pants.
result Short proof of absolute period foliation classification.
Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…
Sphere theorems extended to Riemannian foliations with new results on curvature and leaf spaces.
problem Sphere theorems for Riemannian foliations with transverse curvature constraints.
method Deformation theory and Gromov-Hausdorff limits to prove sphere theorems.
result Complete Riemannian foliations with quarter-pinched transverse sectional curvature develop to simple foliations.
Study on foliation automorphisms, finding non-Lie groups and ILH Lie groups.
problem Understanding the structure of diffeomorphism groups of foliations.
method Investigation of diffeomorphism groups of foliations, proving properties of automorphism groups.
result Found examples of foliations with non-Lie automorphism groups and proved properties of ILH Lie groups for certain foliations.
We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliati…
Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then w…
We derive simple forms for saddle-node singular points of analytic foliations in the real or complex plane just by gluing foliated complex manifolds. We give the versal analytic deformation of the simplest model. We also derive a unique analytic form for those saddle-node having a central manifold. By this way, we reco…
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. Constructs deformed G2-instantons on specific G2-manifolds.
problem Finding non-trivial deformed G2-instantons on G2-manifolds.
method Explicit construction of deformed G2-instantons on specified manifolds.
result First non-trivial examples of deformed G2-instantons on G2-manifolds.
The thesis explores integrable systems and rigidity in PDEs with symmetry.
problem Understanding the deformation theory and rigidity of PDEs with symmetry.
method The approach involves studying completely integrable systems, their equivalence relations, and the deformation theory of PDEs with pseudogroups of symmetries.
result A solution is rigid if its deformation cohomology vanishes and certain estimates hold.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
The paper explores symplectic foliations and their leaves on manifolds.
problem Which manifolds can be realized as leaves of codimension-1 symplectic foliations?
method Observations and deformations of symplectic structures; examples of manifolds.
result Examples of manifolds that can be realized as leaves but not as symplectic leaves.
We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the…