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.
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.
We say that a subset S⊆FN is \emph{spectrally rigid} if whenever T1,T2∈cvN are points of the (unprojectivized) Outer space such that ∣∣g∣∣T1=∣∣g∣∣T2 for every g∈S then T1=T2 in $\cvn$. It is well-known that FN itself is spectrally rigid; it also follows from the result of Smil…
The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
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.
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.
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.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
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.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…
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.
Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.
problem Inverse spectral problem for Riemannian manifolds
method Proving near isospectrality implies full isospectrality
result Compact quotients of symmetric spaces have full isospectrality
Sharp spectral extension of rigidity theorem for mean-convex manifolds.
problem Rigidity and flexibility of manifolds with mean-convex boundary and nonnegative Ricci curvature.
method Spectral Ricci lower bounds and mean-convex boundary conditions.
result Sharp spectral extension of rigidity theorem for specific conditions.
The study examines spectral rigidity in Ricci solitons and Einstein-type manifolds.
problem Determining sectional curvature from eigenvalues of the p-Laplacian.
method Analyzes spectral rigidity under gradient shrinking Ricci soliton and cohomologically Einstein conditions.
result With some exceptions, sectional curvature can be determined by eigenvalues of the p-Laplacian.
We consider a rigidity problem for the spectral gap of the Laplacian on an RCD(K,∞)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive K. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a 1-dimensional G…
Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G,p) of graph G together with a map p of the vertices of G into the Euclidean pla…
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.
Anosov diffeomorphisms with integrable subbundles have coherent dynamics and spectral rigidity.
problem Characterizing Anosov diffeomorphisms with integrable subbundles.
method Joint integrability of strong stable and unstable subbundles leads to coherent dynamics and spectral rigidity.
result Anosov diffeomorphisms with integrable subbundles are dynamically coherent and have spectral rigidity.
The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.
problem Rigidity of self-shrinking hypersurfaces in mean curvature flow.
method Spectral upper-pinching theorem and weighted Poincaré estimate.
result Self-shrinking hypersurfaces are restricted to specific forms under certain conditions.
The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.
problem Certifying infinitesimal projective rigidity for hyperbolic once-punctured torus bundles.
method Using twisted Alexander polynomials of representations associated with the holonomy.
result The induced action on the tangent space of the character variety matches the group theoretic action.
High-dimensional curved diffusions show abrupt convergence at a critical time.
problem Understanding abrupt convergence in high-dimensional curved diffusions.
method Functional inequalities and spectral rigidity.
result Abrupt convergence (cutoff) occurs in high dimensions, linked to spectral rigidity.
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.
The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.
problem Understanding spectral encodings under Weyl growth conditions.
method Analyzing geometric Weyl bulk-density exponent and proving spectral rigidity.
result The geometric Weyl bulk-density exponent (d−2)/2 rigidifies spectral encodings in the O-regularly varying class, leading to unique admissible exponents and scaling laws. To a finite, connected, unoriented graph of Betti-number g>=2 and valencies >=3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another example where non-commutative geometry provides a rigid framework for classification.
The paper proves rigidity for certain product spaces and bounds for band widths.
problem Proving rigidity for product spaces and bounds for band widths.
method Combining stable weighted slicing with a spectral Dirac operator argument.
result Closed spin (Mn,g) is isometrically covered by Sn−mimesRm under certain conditions. 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 rigidity results for Anosov flows and their orbit equivalences.
problem Characterizing orbit equivalences of Anosov flows and their dynamics.
method Using hyperbolic-like dynamics, the paper proves a spectral rigidity theorem and gives efficient criteria for orbit equivalences.
result Characterizes orbit equivalent flows in terms of fundamental group elements represented by periodic orbits.
Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…
We extend Obata's rigidity theorem to free probability.
problem Establishing a free analogue of Obata's rigidity theorem.
method Analyzing self-adjoint n-tuples with Lipschitz conjugate variables under a non-commutative curvature-dimension condition. result The von Neumann algebra splits off a freely complemented semicircular component, revealing a rigidity mechanism under non-commutative curvature.
This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal…
Theorem proves minimal hypersurfaces in nonnegative scalar curvature manifolds are smooth.
problem Minimal hypersurfaces with singularities in manifolds of nonnegative scalar curvature.
method Singularity removal rigidity theorems, spectral PMT for AF manifolds.
result Smoothness of minimal hypersurfaces in nonnegative scalar curvature manifolds.
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
problem Homotopy rigidity of nearby Lagrangian cocores in Weinstein sectors.
method Spectral wrapped Donaldson-Fukaya category with orthogonal group coefficients.
result Inclusion followed by retract and quotient is null-homotopic.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
Our aim in this paper is to study local rigidity for metrics defined on a compact manifold M with boundary satisfying constant scalar curvature on M and constant mean curvature on ∂M. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case…
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.
It is well-known that a point T∈cvN in the (unprojectivized) Culler-Vogtmann Outer space cvN is uniquely determined by its \emph{translation length function} ∣∣.∣∣T:FN→R. A subset S of a free group FN is called \emph{spectrally rigid} if, whenever T,T′∈cvN are such that $||g||_T=||g||_…
To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose ``Riemannian'' aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove …
We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying Ric∞≥K>0. Assuming equality holds, we show that the 1-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap …
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.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.
problem Understanding the topological and geometric properties of foliations.
method Applications of higher Lefschetz theorems for foliations, involving Haefliger cohomology.
result The non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions.
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 prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…
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.
Paper resolves Chern conjecture for 4D minimal hypersurfaces in S5.
problem Chern conjecture for closed minimal hypersurfaces in S5.
method Constructing weighted 3-forms and proving global curvature estimates.
result Complete geometric rigidity achieved for constant Gauss-Kronecker curvature.