Solves Besse conjecture on 3D manifolds, proving metric rigidity.
problem Besse conjecture on 3D compact manifolds
method Analytical proof of critical point equation
result Proves rigidity of Miao-Tam metric
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48 results for “Rigidity Conjecture”
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
A minimal hypersurface in a sphere is uniquely determined.
problem Characterizing closed minimal hypersurfaces in spheres.
method Proving strong rigidity of closed minimal hypersurfaces.
result Closed minimal hypersurfaces in spheres are uniquely determined.
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
The Novikov Conjecturemath.GT
We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .
Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
CAT(0) spaces of higher rankmath.MG
Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.
problem CAT(0) spaces of higher rank with geometric group actions.
method Proving rigidity for spaces containing periodic flats and geodesics in flats.
result CAT(0) spaces of higher rank are rigid if they contain a periodic -flat.
Study rigidity of real moment-angle manifolds using cubical geometry.
problem Topological rigidity of real moment-angle manifolds.
method Cubical geometry and surgery theory.
result Real moment-angle manifolds of dimension at least five satisfy the Borel Conjecture.
Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
method Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
result Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
On the fluid ball conjecturemath.DG
Paper proves a rigidity result for static perfect fluids.
problem Proving a rigidity result for static perfect fluids.
method Robinson's divergence formula and boundary conditions.
result Rigidity result for static perfect fluids.
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
Paper proves Horowitz-Myers conjecture in 3-7 dimensions.
problem Proving Horowitz-Myers conjecture in specific dimensions.
method Alternative proof and local isometry demonstration.
result Metrics achieving conjecture equality locally match Horowitz-Myers metrics.
Proof of Gromov's conjecture in 3D simplifies existing methods.
problem Gromov's dihedral rigidity conjecture on scalar curvature in 3D.
method Self-contained proof avoiding technical complications, shorter and more accessible than general case.
result Simplified proof of Gromov's conjecture in 3D.
Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.
problem Proving Doyle conjecture for hexagonal lattice circle packings.
method Using Liouville theorem of discrete harmonic functions based on logarithmic radii ratio observation.
result Proves rigidity of Doyle spirals in hexagonal lattice circle packings with bounded radii ratios.
Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the globa…
Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…
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.
Study uses Zilber-Pink conjecture and dynamical methods to solve rigidity problems.
problem Rigidity problems for arithmetic hyperbolic lattices.
method Zilber-Pink conjecture and dynamical methods.
result New results about reconstructing Hodge structures from their loci.
Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…
Proves Green function rigidity for specific operators and obtains new ADM mass formula.
problem Proving Green function rigidity for specific operators and obtaining new ADM mass formula.
method Positive mass theorem and positive energy theorem for Paneitz operator.
result Obtained new formula for the ADM mass of asymptotically flat hypersurfaces.
Counterexample disproves key index computation in Gromov's conjecture paper.
problem Disproving an index computation in Gromov's conjecture paper.
method Constructing a counterexample to an index computation.
result Counterexample disproves the main result of the paper.
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…
Polyhedra rigidity theorem in hyperbolic space proved.
problem Dihedral rigidity conjecture in hyperbolic 3-space.
method Comparison theorem for polyhedra in a 3-manifold with scalar curvature bounded below.
result Confirms Gromov dihedral rigidity conjecture in hyperbolic 3-space.
Proofs non-realizability of mapping class group via homeomorphisms, resolves Thurston's conjecture.
problem Non-realizability of mapping class group via homeomorphisms
method Short and elementary proof, rigidity results for actions on Euclidean spaces
result Proof of non-realizability of mapping class group via homeomorphisms
Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.
problem Confirming the Hopf conjecture on compact Riemannian manifolds of even dimension.
method Decomposing the curvature operator into Hermitian components and developing eigenvalue criteria for sectional curvature.
result Prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor and confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature.
In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that the transformation Lie group of all CR automorphisms associated with each of the …
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.
In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and -Lipschitz rigidity theorem.
result Positive answers to Gromov's open questions on scalar curvature.
Discussing rigidity properties of conics, inspired by billiards in ellipses.
problem Rigidity properties of conics and billiards in ellipses.
method Analog of polar duality and circle map properties.
result Two rigidity properties of conics.
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 -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.
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.
Paper proves rigidity of certain minimal hypersurfaces in a 5D sphere.
problem Characterizing closed minimal hypersurfaces in a 5D sphere.
method Analyzes mean curvature and principal curvatures to prove rigidity.
result Closed minimal hypersurfaces with constant 3-mean curvature and distinct principal curvatures are isoparametric.
Paper studies a new curvature system and proves rigidity and gap theorems.
problem Extending CPE conjecture to manifolds with specific structures.
method Introduces system and proves rigidity and gap theorems.
result Proves rigidity and gap theorems for solutions.
Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
Paper proves Simon's third gap conjecture for minimal surfaces in spheres.
problem Investigating the third gap problem in Simon's conjecture for minimal surfaces in unit spheres.
method Developed refined third-order Simons-type integral identities and established new lower bounds for curvature terms.
result Obtained positive gap results for the squared norm of the second fundamental form throughout the interval \(\left[\frac{5}{3},\frac{9}{5}
ight]\).
A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length of their Levi-Tanaka algebra are {\em rigid}, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the comple…
In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).
We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.
Study geometric properties of generalized vacuum static spaces.
problem Estimating geometric properties of generalized -vacuum static spaces.
method Proving estimates for -scalar curvature and first eigenvalue of the Jacobi operator, and rigidity under various geometric assumptions.
result Proved a result related to the Cosmic no-hair conjecture.
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
If is the fundamental group of a complete finite volume hyperbolic -manifold, Guilloux conjectured that the Borel function on the -character variety of should be rigid at infinity, that is it should stay bounded away from its maximum at ideal points. In this paper we prove Guilloux'…