Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
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.
Proves better rigidity theorems for special solitons.
problem Understanding rigidity properties of specific solitons.
method Refined point-wise estimates for mean curvature.
result Stronger rigidity results for Lagrangian and symplectic translating solitons.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
New Witten rigidity theorems for elliptic genus in various dimensions.
problem Proving rigidity theorems for elliptic genus in different dimensions.
method Combining Liu's and Han-Yu's methods to prove Witten rigidity theorems for elliptic genus in even and odd dimensions.
result Several new Witten rigidity theorems for elliptic genus in even and odd dimensions have been established.
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundar…
In this note we present various extensions of Obata's rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity case we also chara…
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
New rigidity theorems for spin^c manifolds using modular invariance.
problem Establishing rigidity theorems for twisted Dirac and Toeplitz operators.
method Liu's modular invariance method and its odd-dimensional extension.
result New Witten rigidity theorems for even and odd-dimensional spin^c manifolds.
Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.
problem Proving rigidity for hypersurfaces with curvature restrictions.
method Analogue to Guan and Shen's theorem for Riemannian space forms.
result Rigidity theorem for locally isometric hypersurfaces in de Sitter space.
Proves rigidity of ancient solutions in mean curvature flow.
problem Rigidity of ancient solutions in mean curvature flow.
method Point-wise estimate for second fundamental form.
result Rigidity theorem of ancient solutions in codimension one.
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.
In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains…
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
We will present a new proof of the Gromoll-Grove diameter rigidity theorem.
Survey on mean curvature flow with sphere theorems and Yau rigidity theory.
problem Sphere theorems for submanifolds with arbitrary codimension.
method Recent developments on convergence theorems for mean curvature flow.
result Optimal convergence theorem for arbitrary codimension mean curvature flow.
Article strengthens initial data rigidity theorem to show unique spacetime extension.
problem Initial data rigidity in spacetime geometry.
method Showed initial data sets carry a lightlike parallel vector field, leading to unique spacetime extension.
result Local uniqueness of spacetimes extending initial data sets under dominant energy condition.
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.
The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.
problem Understanding the structure and behavior of Ricci shrinkers.
method Proving rigidity and ε-regularity theorems for Ricci shrinkers using entropy and curvature.
result Non-compact Ricci shrinkers are asymptotic to cones under certain curvature conditions.
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…
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler 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 rigidity in Penrose's singularity theorem with weakly trapped surfaces.
problem Understand the global structure of spacetimes with weakly trapped surfaces.
method Show foliation of MOTS generating totally geodesic null hypersurfaces.
result Obtain local or global rigidity results based on assumptions.
Handlebody groups are rigid under measure equivalence.
problem Proving handlebody groups are rigid under measure equivalence.
method Proving superrigidity for measure equivalence of handlebody groups.
result Every countable group measure equivalent to handlebody groups is virtually isomorphic to them.
The paper examines rigidity of special submanifolds in spheres with curvature constraints.
problem Rigidity of k-extremal submanifolds in a sphere under curvature conditions. method Proves pinching theorems for submanifolds with various curvature conditions.
result Various curvature conditions lead to rigidity of k-extremal submanifolds. We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of …
Investigates conditions for non-rigidity in extremal metrics involving scalar curvature.
problem Rigidity of extremal metrics involving scalar curvature.
method Analyzes sufficient conditions for non-rigidity and provides examples.
result Provides sufficient conditions for metrics not to be rigid.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
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.
Unified rigidity theorem for Plateau surfaces in Bn.
problem Rigidity of free-boundary minimal surfaces in Bn. method Analyzing conformal free-boundary minimal immersions of Plateau model cones.
result Every conformal free-boundary minimal immersion of the flat T-cone into Bn is congruent to the flat T-cone. 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. Rigidity theorem for critical points of Allen-Cahn equation on S³.
problem Rigidity of critical points with low Morse index on S³.
method Analysis of nullity and symmetries of critical points, Frankel-type theorem for nodal sets.
result Critical points with index five are symmetric and vanish on a Clifford torus, realizing the fifth width of the min-max spectrum.
In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
Rigidity theorem for flag manifolds in various dimensions.
problem Rigidity of flag manifolds under certain mappings.
method Rigidity theorem derived from quasiconformal homeomorphisms and Sobolev mappings.
result Quasiconformal homeomorphisms and Sobolev mappings are rigid for flag manifolds in dimensions n≥4. Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. method Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. result Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. New rigidity theorem on static manifolds with boundary.
problem Static metrics on manifolds with boundary.
method Obata-type rigidity theorem, sufficient geometric conditions.
result Scalar curvature map can be locally surjective at static metrics on manifolds with boundary.
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
New theorems compare Laplacian on Kähler manifolds.
problem Comparing Laplacian on Kähler manifolds.
method New curvature notions between Ricci and holomorphic bisectional curvatures.
result Established Laplacian comparison theorems and rigidity theorems.
For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.
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.
Green functions for GJMS operators on spheres derived, linking geometry and rigidity.
problem Deriving Green functions for GJMS operators on spheres.
method Explicit representation formulae derived using Gegenbauer polynomials.
result Spheres uniquely characterized by their Green functions, with strong rigidity theorems for n=3,4,5. Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
In this paper, we first establish a K-theory version of the equivariant family index theorem for a circle action, then use it to prove several rigidity and vanishing theorems on the equivariant K-theory level.