Study automorphism groups of Artin groups, proving rigidity and classification results.
problem Understanding the structure and automorphisms of Artin groups.
method Computed automorphism groups of intersection graphs, deduced rigidity and classification results.
result Computation of outer automorphism groups and other rigidity properties.
A rigid submanifold result in contact geometry.
problem Rigidity of coisotropic submanifolds in contact geometry.
method Study coisotropic deformations and characteristic foliations.
result Compact regular coisotropic submanifolds are rigid among nearby ones.
Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in Rn with n≥4. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a b…
Paper proves new rigidity results for biconservative hypersurfaces.
problem Rigidity of non-negatively curved compact biconservative hypersurfaces.
method Alternative proofs and new estimates of the squared norm of the shape operator.
result New rigidity results for biconservative hypersurfaces in space forms.
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to MP-systems. result New rigidity results for stationary manifolds.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
Spinors prove rigidity for polyhedral spacetime data.
problem Rigidity of polyhedral spacetime data sets.
method Extending rigidity analysis from spacetime positive mass theorem.
result Dihedral rigidity connects mass theorem, trapped surfaces.
Extends rigidity results to non-homogeneous manifolds.
problem Measure and topological rigidity in dynamical systems.
method From homogeneous to general manifolds.
result Measure and topological rigidity results extended.
The paper proves rigidity results for Serrin-type problems in manifolds.
problem Proving rigidity for Serrin-type problems in Riemannian manifolds.
method Integral identities and Soap Bubble theorem.
result Rigidity results for annular regions in Einstein manifolds.
Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.
problem Reconstructing a Riemann surface from boundary geodesic lengths.
method Re-casting lens data as generalized Riemannian circles and solving a system of equations.
result Essentially optimal results on boundary and lens rigidity for 2D manifolds.
New rigidity results for scalar curvature with stabilized conditions.
problem Establishing rigidity for scalar curvature with stabilized conditions.
method Construction of foliations and development of a monotone quantity using Ricci flow and heat equation.
result Generalized classical scalar curvature rigidity results to the \(T^{
times}\)-stabilized setting.
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity. By using this splitting, we refine some existing results about the local rigidit…
Rigidity results for initial data sets related to the positive mass theorem.
problem Rigidity of initial data sets in general relativity.
method Establishing conditions for weak outermost marginally outer trapped surfaces and rigidity results for Riemannian manifolds.
result Marginally outer trapped surfaces are weakly outermost under certain conditions.
Plane Delaunay triangulations are rigid under Luo's discrete conformal change.
problem Rigidity of Delaunay triangulations under discrete conformal changes.
method Developed discrete Schwarz lemma and Liouville theorem, used conformal modulus and extremal length.
result Discrete analogue of conformal rigidity of the plane.
Rigidity proven for a specific type of solitons with harmonic curvature.
problem Proving rigidity of a specific class of solitons.
method Proof of rigidity for compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature.
result Compact three-dimensional Heterotic solitons with vanishing torsion and harmonic curvature are rigid.
Non-rigidity of hyperbolic manifold under scalar curvature constraints.
problem Non-rigidity of hyperbolic manifold under scalar curvature constraints.
method Compactly supported deformations, topological constraints.
result Non-rigidity under scalar curvature constraints, rigidity under topological constraints.
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
New findings extend rigidity results to broader classes of manifolds.
problem Extending rigidity results to non-warped product spaces.
method Establishing rigidity theorems for manifolds conformal to those with nonnegative curvature.
result New families of manifolds exhibit scalar-mean rigidity.
The paper proves rigidity results for manifolds with nonnegative scalar curvature.
problem Rigidity of manifolds with nonnegative scalar curvature.
method New proof and tricks to establish compactness and rigidity results.
result Optimal 2-systole inequality and rigidity results for manifolds with positive scalar curvature.
Researchers solve boundary and scattering rigidity problems for magnetic systems.
problem Recovering magnetic systems from boundary or scattering data.
method Reduced to magnetic systems and applied results from [DPSU07].
result Recovering MP-system up to a gauge. Paper finds a graph Steklov eigenvalue estimate with rigidity results.
problem Estimating Steklov eigenvalues on graphs.
method Lichnerowicz-type estimate for the first Steklov eigenvalues.
result Rigidity results for the Steklov eigenvalues on graphs.
Boundary rigidity proven for non-reversible Finsler metrics.
problem Recovering non-reversible Finsler metrics from boundary distance data.
method Sum of reversible Finsler norm and closed 1-form, boundary rigidity results.
result 1-form can be uniquely recovered from boundary distance data.
Alternative approach to rigidity of high-dimensional isometric immersions.
problem Rigidity of high-dimensional isometric immersions between compact manifolds.
method Quantitative rigidity estimates, reducing to Euclidean setting and applying Friesecke-James-Müller rigidity estimate.
result Quantitative results showing close proximity to isometric immersions for small stretching and bending energy.
Paper proves rigidity for Einstein metrics in high dimensions.
problem Einstein metrics on high-dimensional manifolds.
method Liouville type rigidity result for asymptotically hyperbolic metrics.
result Established a rigidity theorem for d≥5. Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
We define a gradient Ricci soliton to be rigid if it is a flat bundle where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci s…
New rigidity results for specific hypersurfaces in spacetimes.
problem Characterizing maximal hypersurfaces in Generalized Robertson-Walker spacetimes.
method Applying rigidity results under geometric assumptions and the Null Energy Condition.
result New parametric uniqueness and nonexistence results for maximal hypersurfaces.
The study proves rigidity and non-rigidity of spherical caps in mean curvature.
problem Understanding mean curvature rigidity and non-rigidity on spherical caps.
method Used a Tangency Principle to prove rigidity and constructed counterexamples for non-rigidity.
result Contrast between rigidity and non-rigidity phenomena on spherical caps.
In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(−1)-spaces that can be seen as a metric analog to the "entrop…
Proves rigidity of geodesic balls in spheres under certain deformations.
problem Rigidity of geodesic balls in spheres under smooth deformations.
method Real Killing connection and solution of Dirac operator boundary value problem.
result Rigidity result for geodesic balls in spheres fails for hemispheres.
Symplectic solitons rigid if bounded, study shows.
problem Understanding symplectic translating solitons with bounded second fundamental form.
method Using complex phase map to prove rigidity.
result Symplectic translating solitons rigid without bounded second fundamental form assumption.
For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving L2n-norm of the Weyl curvature, the traceless Ricci curvature and the Yamabe invaria…
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results…
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Two rigidity results for surfaces in Schwarzschild spacetime.
problem Understanding surfaces in Schwarzschild spacetime.
method Proving rigidity results for surfaces satisfying specific mean curvature equations.
result Established two new rigidity results for surfaces in Schwarzschild spacetime.
New rigidity results for a generalized Ricci-Hessian equation on manifolds.
problem Understanding rigidity in generalized Ricci-Hessian equations on manifolds.
method Proving new rigidity results related to a generalized Ricci-Hessian equation.
result New rigidity results for the generalized Ricci-Hessian equation on Riemannian manifolds.
Plane triangulations remain rigid under discrete conformal changes.
problem Rigidity of acute triangulations under discrete conformal changes.
method Maximum principles, discrete Liouville theorem, extremal lengths, Euclidean to hyperbolic discrete conformality.
result Uniformly acute triangulations are rigid under Luo's discrete conformal change.
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
Totally geodesic subvarieties in moduli space are locally rigid.
problem Understanding rigidity of subvarieties in moduli space.
method General rigidity result for orbifold maps to moduli space.
result Covering constructions and totally geodesic subvarieties are locally rigid.
The paper explores rigidity of hypersurfaces with constant curvature in Euclidean spaces.
problem Rigidity of hypersurfaces with constant mean and scalar curvature.
method Characterizations and rigidity results under various conditions of Gaussian-Kronecker and r-th mean curvatures. result Rigidity theorems for hypersurfaces in dimensions 4, 5, and 6, and general dimensions under pinching conditions.
The paper proves rigidity results for compact initial data sets.
problem Understanding the structure of compact initial data sets.
method Proving rigidity results under specific conditions.
result Global version of the main result in [15] is obtained.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.
Local rigidity proved for convex hypersurfaces in spaces of constant curvature.
problem Proving rigidity of convex hypersurfaces in spaces of constant curvature.
method Analyzing isometric convex hypersurfaces in spaces of constant curvature of dimension n≥4. result Two convex isometric hypersurfaces are congruent locally around their corresponding under strict convexity isometries.
Study on mappings in Carnot groups, proving rigidity results.
problem Understanding mappings in Carnot groups and proving rigidity.
method Structural results for Sobolev mappings, proving rigidity or regularity.
result Establishes partial rigidity and partial regularity theorems.
We prove that simple, thick hyperbolic P-manifolds of dimension >2 exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension >2. The key tool in the proofs of these rigidity results is a strong form of the Jordan separation th…