New CR-structures lemma simplifies CR-manifold deformation proof.
problem Deformation unobstructedness of CR-manifolds.
method New Tian-Todorov lemma applied to CR-manifolds.
result Reproved deformation unobstructedness of CR-manifolds.
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48 results for “deformation lemma”
New Morse lemma handles functions going to negative infinity.
problem Deforming functions with negative infinity behavior.
method Generalized Morse Theory lemma for functions going to negative infinity.
result Valid for functions with suitable growth condition.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying -lemma but with non-vanishing Aeppli-Bott-Chern-Massey product.
We have one more look at the (homological) perturbation lemma and we point out some non-standard consequences, including the relevance to deformations.
Compact complex manifolds with trivial canonical bundle and -Lemma have smooth deformations and a surjective Albanese map.
problem Understanding the structure and deformations of compact complex manifolds with specific properties.
method Analyzing the Kuranishi space, showing smooth deformations, and studying the Albanese map.
result Compact complex manifolds with trivial canonical bundle and -Lemma have smooth deformations and a surjective Albanese map.
Study canonical deformations of complex forms and their cohomology properties.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
Study cohomologies of complex manifolds with symplectic forms and their stability.
problem Analyzing cohomologies of complex manifolds with symplectic forms.
method Investigate the Hard Lefschetz Condition on Dolbeault cohomology groups using a double complex.
result Stability of the -Lemma under small deformations of but not under complex structure.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
A singular Poincare lemmamath.SG
We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of . This result has a natural interpretation in terms of the cohomology associated to the inf…
In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under t…
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…
Local stability of p-Kähler structures studied.
problem Stability of p-Kähler structures under deformations.
method Natural extension map and power series method.
result Local stability theorem for p-Kähler structures.
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of Kä…
We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particula…
This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…
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.
We introduce a property of compact complex manifolds under which the existence of balanced metric is stable by small deformations of the complex structure. This property, which is weaker than the -Lemma, is characterized in terms of the strongly Gauduchon cone and of the first $\partial\overl…
Logarithmic -lemma and several geometric applications (with an Appendix joint with Sheng Rao)math.AG
The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.
problem Proving a logarithmic partial derivative lemma for compact Kähler manifolds.
method Developed a new -type lemma for logarithmic differential forms.
result Confirmed a conjecture by X. Wan and derived several geometric applications.
Deformation Openness and Closedness of Various Classes of Compact Complex Manifolds; Examplesmath.AG
We review the relations between compact complex manifolds carrying various types of Hermitian metrics (Kähler, balanced or {\it strongly Gauduchon}) and those satisfying the -lemma or the degeneration at of the Frölicher spectral sequence, as well as the behaviour of these properties under h…
Given a compact complex -fold satisfying the -lemma and supposed to have a trivial canonical bundle and to admit a balanced (=semi-Kähler) Hermitian metric , we introduce the concept of deformations of that are {\bf co-polarised} by the balanced class $[ω^{n-1}]\in H^{n-1,\,n-1…
In this paper, we will study the existence problem of minmax minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of minmax minimal torus in Theorem 5.1. Firstly we prove a strong uniformization result(Proposition 3.1) using method of [1]. Then we use…
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of -curves, and two splitting lemmas for the functionals on Banach manifolds of -curves. Two results on critical groups of iterated closed geodesics a…
The paper extends holomorphic forms on generalized Hermitian manifolds.
problem Understanding holomorphic forms on generalized Hermitian manifolds.
method Developed a criterion for holomorphic forms and used it to extend -closed forms.
result Invariance of generalized Hodge numbers in deformations of compact generalized Hermitian manifolds.
Study on complex manifolds' Gauduchon metrics and spectral sequences under deformations.
problem Understanding Gauduchon metrics and spectral sequences in complex manifold deformations.
method Two approaches: partial degeneration of Frölicher spectral sequence and --property.
result Introduction of a positivity cone and its lower semicontinuity under deformations.
Study on cut locus of submanifolds in Finsler geometry.
problem Characterizing the cut locus of submanifolds in Finsler manifolds.
method Deformation and characterization of the cut locus, extending previous results.
result Generalization of Klingenberg's lemma for -geodesic loops in reversible Finsler setting.
The Bott-Chern cohomology of 6-dimensional nilmanifolds endowed with invariant complex structure is studied with special attention to the cases when balanced or strongly Gauduchon Hermitian metrics exist. We consider complex invariants introduced by Angella and Tomassini and by Schweitzer, which are related to the $\pa…
The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of -algebras for deformations.
result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.
New method solves -equations for logarithmic forms on Kahler manifolds.
problem Solving -equations for logarithmic forms on Kahler manifolds.
method Using harmonic integral theory for currents on Kahler manifolds.
result Constructs the extension for logarithmic -forms on the central fiber.
Study of split Nakamura manifolds and their automorphisms.
problem Understanding the structure and automorphisms of Nakamura manifolds.
method Analyzing cohomology, spectral sequences, and automorphism groups.
result Detailed description of the automorphism group of split Nakamura manifolds.
In this short note, we investigate some features of the space $\Inject{d}{m}$ of linear injective maps from $\bbR^d$ into $\bbR^m$; in particular, we discuss in detail its relationship with the Stiefel manifold , viewed, in this context, as the set of orthonormal systems of vectors in $\bbR^m$. Finally, we…
We investigate representations of Kähler groups to a semisimple non-compact Hermitian Lie group 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…
We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold , and they allow us to construct a countable family of compact complex non-$\partial\overline…
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
Whitehead's Lemma is a simple result of Stallings folds.
problem None explicitly stated; Whitehead's Lemma is observed as a consequence.
method Observation of Whitehead's Lemma as a consequence of Stallings folds.
result Whitehead's Lemma is a simple result of Stallings folds.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Study several weak forms of a lemma on compact complex manifolds.
problem Understanding weak forms of a lemma on compact complex manifolds.
method Complete unified study of weak forms of the $\ddb-$Lemma.
result Unified understanding of various weak forms of the $\ddb-$Lemma.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
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 -closed extensions and foliated cases.
result Local stabilities of transversely -Kähler structures and new theorems on Hodge numbers.
Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packingsmath.DG
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.