Study shows upper limit for torical band width with spectral curvature bounds.
problem Understanding the band width of torical bands with spectral curvature constraints.
method Used the warped \( μ\)-bubble method with spectral curvature bounds.
result Upper bound for the band width of torical bands is established.
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.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …
Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.
problem Closed orientable surfaces with nonnegative curvature in spectral sense.
method Spectral condition and associated conformal metrics to prove inequalities and bounds.
result Isoperimetric inequalities, area growth theorems, and diameter bounds for surfaces.
Estimates spectral projections restricted to uniformly embedded submanifolds.
problem Estimating spectral projections on submanifolds of manifolds with nonpositive curvature.
method Estimates the L2(M)oLq(Σ) norm of spectral projection operators. result Sharp spectral projection estimates for small spectral windows.
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.
Equivalence of norms on manifolds with curvature bounds established.
problem Establishing equivalence of norms on manifolds with bounded sectional curvature.
method Using spectral projector and thickness condition for subsets.
result Constant in equivalence depends only on manifold dimension, curvature bounds, and frequency threshold.
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.
Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.
problem Proving upper bounds for the width of compact manifolds with boundary.
method Utilizes spacetime harmonic functions, μ-bubbles, and spinorial Callias operators.
result Generalizes Schoen-Yau black hole existence theorem to higher dimensions.
The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.
problem Analyzing spectral properties of graph Laplacians on manifolds with curvature constraints.
method Quantitative bounds on eigenvalues and eigenfunctions of graph Laplacians constructed from random variables on manifolds with uniform lower Ricci curvature bounds.
result Spectral convergence of graph Laplacians on manifolds with curvature bounds and in non-collapsed spaces.
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.
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.
We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curv…
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 bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.
problem Finding a sharp lower bound for the first Dirichlet eigenvalue of geodesic balls.
method Quantitative explicit inequality linking the width of geodesic balls to the spectral gap.
result A quantitative inequality relating the width of geodesic balls to the spectral gap between the first Dirichlet eigenvalue and its lower bound.
For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…
Study shows how certain metrics can be split into warped products.
problem Understanding conditions under which metrics can be split as warped products.
method Investigating warped area-minimizing hypersurfaces and spectral Ricci/scalar curvature bounds.
result Metrics can be locally split as warped products under specific curvature conditions.
Simple proof that stable minimal hypersurfaces in R^4 are hyperplanes.
problem Proving stable minimal hypersurfaces in R^4 are hyperplanes.
method Using spectral Ricci curvature bounds and Green kernel estimates.
result Complete, two-sided stable minimal hypersurfaces in R^4 are hyperplanes.
We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collecti…
For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization by using the mean transition proba…
We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the numbe…
Study rigidity of spectral gap on Finsler manifolds with specific curvature bounds.
problem Rigidity of spectral gap on Finsler manifolds with Ricci curvature bound.
method Analysis of spectral gap, splitting phenomena, and needle decomposition.
result Rigidity results for spectral gap, logarithmic Sobolev, and Bakry-Ledoux inequalities.
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.
Graph curvature measured by inverse resistance distance.
problem Defining and analyzing curvature in graphs.
method Defining curvature via inverse resistance distance and proving properties.
result Graphs with positive curvature have controlled diameter and spectral properties.
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.
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.
Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.
problem Quantifying unique continuation for Schrödinger operators on manifolds with specific curvature conditions.
method Proving quantitative unique continuation estimates for Schrödinger operators on manifolds with Ricci curvature bounded below.
result Upper bound for energy range and constant in terms of Ricci curvature and parameters of relatively dense set.
We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an upper bound on the error that depends on upper bounds on the diameter and the sect…
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Study spectral properties on manifolds with conical singularities, proving new inequalities.
problem Analyzing spectral properties and geometric inequalities on manifolds with conical singularities.
method Develops new inequalities for manifolds with conical singularities, not covered by existing methods.
result Proves a Bakry-Émery inequality, Hardy inequality, and spectral gap estimate.
Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
problem Spectral rigidity of hypercube graphs
method Interplay between global spectral embedding and local curvature analysis
result Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.
We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show…
Paper bounds the A-hat genus using curvature and isoperimetric constants.
problem Bounding the A-hat genus of Riemannian manifolds.
method Spectral analysis of the Dirac operator, scalar curvature lower bounds, and isoperimetric constants.
result Proves an upper bound on the A-hat genus using manifold properties.
The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is …
In deep neural networks, the spectral norm of the Jacobian of a layer bounds the factor by which the norm of a signal changes during forward/backward propagation. Spectral norm regularizations have been shown to improve generalization, robustness and optimization of deep learning methods. Existing methods to compute th…
We give three lower bounds for the Morse index of a constant mean curvature torus in Euclidean 3-space in terms of its spectral genus g. The first two lower bounds grow linearly in g and are stronger for smaller values of g, while the third grows quadratically in g but is weaker for smaller values of g.
We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…
We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that L+2Ric is a positive operator where L is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small …
The spectrum of certain manifolds matches that of hyperbolic space if the bottom spectrum is maximal.
problem Investigating spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum.
method Analyzing the spectrum of the Laplacian on manifolds with specific Ricci curvature bounds.
result The spectrum of the manifold coincides with that of hyperbolic space if the bottom spectrum is maximal.
Study shows no new eigenvalues in specific finite coverings.
problem Proving the absence of new eigenvalues in finite coverings.
method Analyzing spectral stability of finite coverings with specific conditions on Ricci curvature and representation theory.
result Non-existence of new eigenvalues in a specific range.
We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
Short note proves Poincaré inequality for 4-manifold forms.
problem Quantifying Poincaré inequality for one forms on 4-manifolds.
method Hodge theory on orbifolds, comparison of fundamental groups, spectral convergence, degeneration to orbifolds.
result First non-trivial global Poincaré inequality without higher curvature assumptions.
The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.
problem Proving a necessary condition for observability of the heat semigroup on manifolds.
method Propagation of smallness estimates of Carleman and Logunov-Malinnikova type.
result Established spectral estimates for surfaces with hyperbolic ends, proving the thickness condition is necessary.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
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.