We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property a…
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48 results for “balanced spectral deformation”
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…
Develops a new representation for constant mean curvature surfaces in hyperbolic 3-space.
problem Finding conformal immersions of constant mean curvature in hyperbolic 3-space.
method Uses a Weierstrass-Kenmotsu type representation based on the Hermitian model, balanced spectral deformation, and Iwasawa splitting of $\SL$.
result Establishes an explicit correspondence with Aiyama and Akutagawa's representation and interprets the construction in terms of Kokubu's adjusted normal Gauss map.
Study on spectral sequence of Iwasawa manifold and its deformations.
problem Properties of Frölicher spectral sequence on Iwasawa manifold and its deformations.
method Determination of successive pages of the Frölicher spectral sequence.
result New examples and counterexamples on spectral sequence properties.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.
problem Balanced metrics under deformations of complex structures.
method Necessary conditions for smooth curves of balanced metrics.
result Existence of smooth curves of balanced metrics starting from a fixed balanced metric.
In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.
We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to…
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.
The paper explores spectral sequences of complex manifolds with special metrics.
problem Understanding spectral sequences of compact complex manifolds with special metrics.
method Investigation of Frölicher spectral sequences and special metrics (balanced, SKT, Gauduchon) on manifolds.
result Found compact manifolds where spectral sequences do not degenerate at the second page, providing counterexamples and new families.
Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.
problem Parameterize spectral data of constant mean curvature tori in R^3.
method Use Whitham deformations, blowups, and spectral data analysis.
result Prove the Wente family is parameterized by the bisector of the right angle.
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…
Study on deformations of -forms and spectral sequence degenerations.
problem Understanding deformations of -forms under complex structure changes.
method Analyzing Frölicher spectral sequence conditions for -form deformations.
result Unobstructed deformations of -forms under specific spectral sequence conditions.
Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Constructs universal local deformations for curves and differential forms.
problem Local deformations of curves and differential forms under preservation of periods.
method Develops Kuranishi families for pairs of curves and meromorphic 1-forms, focusing on hyperelliptic cases.
result First paper in a series developing a deformation theory for spectral curve data of integrable systems.
In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…
Lindblad-Deformed Spectral Geometry: Heat-Kernel Asymptotics and Effective Spectral Dimensionquant-ph
Introduces a new spectral geometry framework with dissipative data.
problem Deforming spectral triples with dissipative Lindblad operators.
method Lindblad-deformed spectral geometry framework with heat-kernel asymptotics.
result First nontrivial dissipative effect appears at order gamma^4.
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
New deformations for -orbifolds using spectral covers.
problem Deforming -orbifolds with coassociative fibrations.
method Using spectral/cameral covers associated to Higgs bundles.
result Generalizes known deformations of Calabi-Yau threefolds.
Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined…
Deformations of Non-Kähler Hyperbolicity Notions and Modifications of Degenerate Balanced Manifoldsmath.CV
Study on new hyperbolicity notions for non-Kähler manifolds and their deformations.
problem Analyzing new hyperbolicity notions for non-Kähler complex manifolds.
method Introducing and analyzing two new notions of hyperbolicity for compact complex non-Kähler manifolds, and studying their behavior under smooth modifications.
result Established openness results for -HS hyperbolicity and -Kähler hyperbolicity under holomorphic deformations.
The paper studies deformations of symplectic groupoids using cohomology and spectral sequences.
problem Deformations of symplectic groupoids and their cohomology.
method Deformation cohomology, Moser path methods, Lie groupoids, multiplicative forms, de Rham models, spectral sequences.
result Computations and constructions of deformation cohomology for various types of symplectic groupoids.
On consistency of constrained spectral clustering under representation-aware stochastic block modelstat.ML
New method finds balanced clusters in graphs using auxiliary information.
problem Finding balanced clusters in graphs with population-level constraints.
method Proposes individual-level balancing constraint and develops spectral clustering algorithms.
result Establishes first statistical consistency result for constrained spectral clustering.
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
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…
Study reveals class disparities in balanced datasets through spectral imbalance.
problem Class disparities in balanced datasets are overlooked despite model performance gaps.
method Developed a theoretical framework and studied 11 encoders to diagnose spectral imbalance.
result Identified spectral imbalance as a source of class disparities in balanced datasets.
Study on existence of -Kähler structures on nilmanifolds with nilpotent complex structures.
problem Existence of -Kähler structures on nilmanifolds with nilpotent complex structures.
method Determine optimal for existence of -Kähler structures and analyze the relationship between balanced metrics and degeneracy steps of the Frölicher spectral sequence.
result No -Kähler structures exist for an optimal on nilmanifolds with nilpotent complex structures.
We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{à la Kodaira-Spencer} for the dimension o…
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…
Authors compute stable homology of torus knots using a new deformation technique.
problem Computing stable homology of torus knots.
method Link-splitting deformation (y-ification) of link homology.
result Explicit computation of -ified stable Khovanov--Rozansky homology of torus knots.
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…
Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for -forms and complex structures, using Frölicher spectral sequence.
result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3-sphere. We use Baker-Akhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and non-isospectral deformations.
Researchers prove spectral rigidity of Liouville tori under specific conditions.
problem Spectral rigidity of Liouville tori under generic conformal classes.
method Noncancellation of wave trace and analysis of second order variational formula for energy.
result Laplace isospectral deformations of Liouville metrics on torus are trivial.
Nonreversible Gauge Fields in Fokker--Planck Dynamics: Supersymmetric Hamiltonians and Learned Finite Forcescond-mat.dis-nn
New gauge fields modify Fokker-Planck dynamics without changing the stationary state.
problem Understanding and modifying nonreversible dynamics in Fokker-Planck models.
method Formulate nonreversible perturbations as gauge fields, mapping to supersymmetric Hamiltonians, and learning finite forces.
result Learned finite forces can recover the optimal Lyapunov-equation solution in nonconvex landscapes.
In this paper we numerically construct CMC deformations of the Lawson minimal surfaces using a spectral curve and a DPW approach to CMC surfaces in spaceforms.
Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfacesmath.GT
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more gene…
This work is dedicated to the study of the Moebius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Baecklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the so…
The paper connects isomonodromic and isospectral deformations for connections.
problem Connecting isomonodromic and isospectral deformations for 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.
Establishes a spectral sequence linking instanton and Khovanov homologies.
problem Connecting instanton and Khovanov homologies.
method Develops a spectral sequence specializing invariants from characteristic-2 homology.
result A spectral sequence connects instanton and Khovanov homologies.
Study uses spectral risk for learning with heavy-tailed data.
problem Learning with heavy-tailed loss distributions.
method Spectral risk with Lipschitz-continuous density, derivative-free learning.
result Excess risk guarantees and improved performance over traditional methods.
Study complex structures and curvature equations on compact manifolds.
problem Equations coupling scalar curvature with complex structure deformations.
method Infinite-dimensional Kaehler reduction, flat connections, variational characterization.
result Verification of conjecture in toric manifolds.
This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various ki…
New maps help understand deformations of modules over Lie algebroids.
problem Understanding deformations of modules over Lie algebroids.
method Introduce semiregularity maps and use DG-Lie algebra control.
result Semiregularity maps annihilate obstructions under certain conditions.
The paper explores moduli space of heterotic system using two deformation paths.
problem Exploring the moduli space of the heterotic system.
method Considering two dual deformation paths starting from a Kähler solution, one along Bott-Chern cohomology class and the other along Aeppli cohomology class. Using the implicit function theorem to prove local existence of heterotic solutions.
result Established an initial step to construct local moduli coordinates around a Kähler solution.
Study of double complexes on Iwasawa manifold yields 3 isomorphism types.
problem Understanding the structure of double complexes on the Iwasawa manifold.
method Used Stelzig and Qi-Khovanov's structure theorem for double complexes.
result Identified and described exactly 3 isomorphism types of double complexes.
We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.