The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.
problem Classical rigidity results in scalar curvature geometry are extended to the spectral setting.
method Warped μ-bubble method is systematically employed to classify stable weighted minimal hypersurfaces and establish band width estimates. result Classification theorems and band width estimates for spectral Ricci and scalar curvatures are proven.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
Two proofs of Melrose-Piazza theorem on spectral sections.
problem Analytic index of families of Fredholm operators.
method Two independent proofs of the theorem, generalizing and clarifying the analytic index definition.
result Generalization and clarification of the Melrose-Piazza theorem on spectral sections.
The paper develops heat kernel comparison theorems and applies them to spectral geometry.
problem Developing mathematical tools for spectral geometry.
method Established weighted heat kernel comparison theorems for manifolds with bounded radial curvatures.
result Two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian.
The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.
problem Proving Strichartz and spectral projection theorems on curved surfaces.
method Using large negative curvature neighborhoods, the study proves theorems on asymptotically conic and Euclidean ends surfaces.
result The study proves theorems without loss of interval on specific types of curved surfaces.
Develops Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.
problem Bounding scalar curvature and metric on 3D and 4D bands simultaneously.
method Warped μ-bubble method result Establishes Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.
Sharp spectral theorem splits certain non-compact manifolds.
problem Proving spectral splitting for non-compact manifolds with specific curvature conditions.
method Sharp spectral analysis and geometric splitting theorem.
result Non-compact manifolds split as RimesN under given curvature constraints. The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
Formula for spectral flow connects manifold properties to index theorem.
problem Establishing a formula for spectral flow on manifolds.
method Reduction to Atiyah-Patodi-Singer index theorem for manifolds with boundary.
result Formula for spectral flow expressed in manifold and connection properties.
Constructs Serre spectral sequence for bounded cohomology.
problem No specific problem stated; focuses on a new mathematical construction.
method Constructs the Serre spectral sequence for bounded cohomology.
result Obtains a non-isometric generalization of Gromov's mapping theorem and partial results on simplicial volume.
Extends Einstein-Hilbert action to higher-order spectral triples.
problem No specific problem stated; focuses on extending action.
method Introduced two second-order spectral triples and computed their Einstein-Hilbert actions.
result Demonstrated applicability of the theoretical framework.
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
New proof of Llarull's rigidity theorem in odd dimensions via spectral flow.
problem Rigidity of smooth maps from compact spin manifolds to spheres.
method Spectral flow argument for odd dimensions, generalization to convex hypersurfaces.
result Generalization of Llarull's theorem to arbitrary smooth strictly convex hypersurfaces.
This paper is based on the paper "Locally free sheaves on complex supermanifolds" of A.L.Onishchik, E.G. Vishnyakova, where two classification theorems for locally free sheaves on supermanifolds were proved and a spectral sequence for a locally free sheaf of modules E was obtained. We consider another filtration of the…
Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.
problem Understanding the geometry and topology of manifolds with nonnegative Ricci curvature.
method New spectral inequalities and isoperimetric problems involving unequal weights and warped bubbles.
result Sharp spectral and isoperimetric bounds for manifolds with nonnegative Ricci curvature.
New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
Spectral sequence analysis for Sobolev mappings in Carnot groups.
problem Analyzing spectral sequences for Sobolev mappings in Carnot groups.
method Showed Pansu pullback induces a spectral sequence mapping.
result Pansu pullback induces a spectral sequence mapping.
Theorem proves spectral rigidity of warped product metrics.
problem Spectral rigidity of warped product metrics.
method Spinor and spacetime harmonic function methods.
result Proves spectral Llarull theorem for warped product metrics.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
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.
The Ray-Singer isospectral theorem (1971) is applied to a general spectral function for Laplacians of twisted p-forms (say) on homogeneous Clifford-Klein factors of the three-sphere. The inducing formulae necessary to express any spectral quantity for any twisting in terms of those for cyclic subgroups of the tetrahedr…
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
problem Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
method Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
result Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic opera…
A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.
Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.
problem Analyzing the behavior of random walks on relatively hyperbolic groups.
method Study of convergent random walks with finite derivative of Green function at spectral radius.
result Proves a local limit theorem for the probability of returning to the origin.
The spectral flow theorem is applied to operators on finite intervals.
problem Operators on finite intervals without boundary conditions are not Fredholm.
method Interpolation theory is used to define boundary conditions making the operators Fredholm. The spectral flow theorem is applied to find the Fredholm index.
result The Fredholm index is given by the spectral flow of the operator path.
Extended Gauss-Markov theorem for linear estimation with bounded bias.
problem Linear estimation with bounded bias operator.
method Derive optimal estimator formulas for Nuclear and Spectral norms, analyze generalization error.
result Cross-validated Nuclear and Spectral regressors outperform Ridge regression in simulations.
The paper extends a splitting theorem for a specific type of tensor in Riemannian geometry.
problem The extension of a splitting theorem for a new type of tensor in Riemannian geometry.
method The approach involves extending the spectral Cheeger-Gromoll splitting theorem to smooth metric measure spaces.
result The theorem allows for the isometric splitting of a manifold under certain conditions on the tensor and its eigenvalues.
Given a link in the three-sphere, Ozsváth and Szabó showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homolo…
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may …
The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.
problem Understanding criticality and splitting theorems for manifolds with spectral Ricci bounds.
method Proving criticality criteria and spectral splitting theorems for manifolds with more than one end and spectral Ricci bounds.
result New insights into Li-Wang's theory and applications to stable and δ-stable minimal hypersurfaces.
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
Study on spectral asymptotics of Toeplitz operators on CR manifolds.
problem Analyzing spectral properties of Toeplitz operators on CR manifolds.
method Full asymptotic expansion of functional calculus of Toeplitz operators.
result Established several CR analogues of complex geometry results.
Proves positive mass theorem for hyperbolic manifolds with ends.
problem Establishing positive mass theorem for complex initial data sets.
method Used spectral PSC, Jang equation, and quantitative shielding theorem.
result Proved positive mass theorem for asymptotically hyperbolic manifolds.
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
Paper solves long neck problem on odd-dimensional spin manifolds.
problem Long neck problem on odd-dimensional spin manifolds.
method Spectral flow of Callias operators.
result Complete answer to Gromov's long neck problem.
The paper proves a theorem about splitting manifolds with specific curvature properties.
problem Understanding the structure of manifolds with nonnegative Ricci curvature and mean-convex boundaries.
method Proving a splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary.
result The manifold is either isometric to a closed manifold with nonnegative Ricci curvature or has no interior ends.