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.
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 how to increase Steklov spectral gaps on manifolds with fixed boundary.
problem Finding ways to increase Steklov spectral gaps on manifolds with fixed boundary.
method Constructing compact manifolds with fixed boundary geometry and applying localized conformal deformations.
result It is possible to make the spectral gap arbitrarily large using localized conformal deformations.
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.
Study on deformations of (p,q)-forms and spectral sequence degenerations.
problem Understanding deformations of (p,q)-forms under complex structure changes. method Analyzing Frölicher spectral sequence conditions for (p,q)-form deformations. result Unobstructed deformations of (p,q)-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.
Study on degeneration of spectral sequence in complex manifolds under deformations.
problem Behavior of spectral sequence degeneration in complex manifolds under small deformations.
method Deformation theory, pseudo-differential operators, Kodaira-Spencer techniques.
result Degeneration at second step is open under certain conditions but not without them.
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.
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 G2-orbifolds using spectral covers.
problem Deforming G2-orbifolds with coassociative fibrations. method Using spectral/cameral covers associated to Higgs bundles.
result Generalizes known deformations of Calabi-Yau threefolds.
Study of harmonic maps from 2-torus to S^3 using spectral curves and Whitham deformations.
problem Investigating the space of harmonic maps from a 2-torus to S^3.
method Using spectral curve correspondence and Whitham deformations.
result The space of harmonic maps is smooth and has dimension two in an open and dense subset of the parameter space.
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.
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 y-ified glN stable Khovanov--Rozansky homology of torus knots. We calculate bi-Hamiltonian cohomology groups and central invariants.
problem Deformation theory of semi-simple bi-Hamiltonian systems.
method Refined spectral sequence arguments.
result Central invariants are \( N \) smooth functions of one variable.
Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for (p,q)-forms and complex structures, using Frölicher spectral sequence. result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
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 h-∂∂ˉ-property. result Introduction of a positivity cone and its lower semicontinuity under deformations.
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.
The paper provides conditions for factoring equivariant spectral triples in unbounded KK-theory.
problem Factoring equivariant spectral triples in unbounded KK-theory.
method Sufficient conditions for factorization of equivariant spectral triples as a Kasparov product.
result Equivariant Dirac-type spectral triples on torus principal bundles always factorize.
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.
Study classifies deformations of star-shaped curves in n-dimensional space.
problem Classifying deformations of star-shaped curves in n-dimensional space.
method Using connections on vector bundles and cyclic D-modules, defining integral curves and classifying them via iso-spectral flows.
result Iso-spectral flows are described by equations from the n-KdV hierarchy.
In this paper we numerically construct CMC deformations of the Lawson minimal surfaces ξg,1 using a spectral curve and a DPW approach to CMC surfaces in spaceforms.
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 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.
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 F5 homology. result A spectral sequence connects instanton and Khovanov homologies.
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.
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.
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.
The paper bounds higher Steklov eigenvalues of graphs on surfaces.
problem Bounding higher Steklov eigenvalues of graphs on surfaces.
method Using metrical deformation via probability flows, the upper bound is derived.
result The upper bound of higher Steklov eigenvalues is established.
We associate a periodic two-dimensional Schrodinger operator to every Lagrangian torus in CP^2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov-Veselov hierarchy of eq…
New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.
problem Stability and deformation theory of Einstein metrics.
method Introduces Chen-Nagano gauge condition, linking Lichnerowicz Laplacian to shifted scalar operator.
result Chen-Nagano gauge collapses to classical transverse-traceless gauge under spectral pinching assumptions.
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…
Defines special classes of constrained Willmore surfaces using polynomial conserved quantities.
problem Characterizing and understanding constrained Willmore surfaces.
method Defines a hierarchy of special classes of constrained Willmore surfaces by polynomial conserved quantities.
result The hierarchy is preserved under spectral deformation and Baecklund transformation, leading to transformations of constant mean curvature surfaces.
In S2×R there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in S2×R by periodic harmonic maps $G : \…
We study the triple $(G,π,\prs)$ where G is a connected and simply connected Lie group, π and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on G such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the d…
This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in R4. This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geome…
The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…
A novel 3D shape registration method using spectral graph embedding and probabilistic matching.
problem Challenges in 3D shape analysis and registration, especially with large variability.
method Combining spectral graph matching with Laplacian embedding for large graphs, using commute-time embedding and PCA.
result A method to register shapes with different samplings and isometric deformations.
New spectral sequences link knot homology to instantons, revealing concordance invariants.
problem Understanding the relationship between knot homology and instantons.
method Established spectral sequences connecting Bar-Natan homology to instanton homology.
result Derived concordance invariants from instanton homology groups.
A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior a…
New spectral sequence connects link homology to Hochschild homology.
problem Computing Hochschild homology of link invariants.
method Uses spectral sequence with E2-page from Khovanov homology of links in S1imesS2. result Spectral sequence converges to Hochschild homology of bordered Floer invariants.
Paper proves spectral rigidity of hyperbolic cusped manifolds for compact deformations.
problem Spectral rigidity of manifolds with hyperbolic cusps.
method Extends microlocal calculus to invert pseudodifferential operators on Sobolev and Hölder-Zygmund spaces.
result Injectivity of X-ray transform on symmetric solenoidal 2-tensors.
Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In the recent years, significant attention has been devoted to descriptors obtained from the spectral decomposi…
Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…