Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
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…
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.
Maximal representations show strong entropy rigidity.
problem Entropy rigidity for maximal representations.
method Measurable hypertransversality, Gromov product, Bowen-Margulis-Sullivan measure.
result Strong entropy rigidity proved for maximal representations.
Study on compact strong HKT manifolds and their properties.
problem Characterizing the structure of compact strong HKT manifolds.
method Geometric analysis, rigidity theorems, classification, and properties of Ricci foliations.
result Compact strong HKT manifolds are Hopf fibrations over compact 4-dimensional orbifolds.
Solves the gauge problem for Ricci flow cylinders, proving strong rigidity.
problem Recognizing metrics in different coordinates and diffeomorphisms.
method Solves a nonlinear system of PDEs to produce a diffeomorphism fixing a gauge.
result Strong rigidity of cylinders in Ricci flow, proving all tangent flows are cylinders.
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
Profinite rigidity proven for many hyperbolic manifolds.
problem Profinite rigidity of hyperbolic manifolds.
method Geometric topology and bubble-drilling construction.
result Profinite rigidity of many cusped hyperbolic manifolds.
Study shows critical width for rigidity of equatorial zones on spheres.
problem Mean curvature rigidity of equatorial zones on spheres.
method Used tangency principle and trap-slice lemma for strong rigidity, and constructed nontrivial perturbations using Delaunay surfaces for non-rigidity.
result Critical width exists for rigidity, beyond which zones are non-rigid.
Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an n-sphere equipped with a certain Finsler metric, and vise versa.
Map quandle orders to actions, characterize isolated orders, and prove no isolated right orders.
problem Characterizing isolated orders on quandles and proving the absence of certain orders.
method Construct a continuous map from orders to actions, use strong rigidity, and analyze specific cases.
result No isolated right orders on free quandles, except for specific cases.
New proof shows all conformal fields are Killing on specific spaces.
problem Infinitesimal conformal rigidity on Damek-Ricci spaces.
method Formulated as PDEs, analyzed locally and directly.
result Constructive proof of rigidity without global methods.
Proves rigidity of boundaries with constant mean curvature in warped product manifolds.
problem Rigidity and compactness of boundaries with constant mean curvature in warped product manifolds.
method Distributional CMC-rigidity proof for rectifiable boundaries.
result Characterizes limits of boundaries with converging mean curvatures.
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.
Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
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. Let g be a Riemannian metric for Rd (d≥3) which differs from the Euclidean metric only in a smooth and strictly convex bounded domain M. The lens rigidity problem is concerned with recovering the metric g inside M from the corresponding lens relation on the boundary ∂M. In this paper…
Groups acting on CAT(0) spaces without 3-flats have rigid properties.
problem Understanding group actions on CAT(0) spaces without 3-flats.
method Analyzing the structure of CAT(0) spaces and group actions.
result Groups acting geometrically on such spaces have rigid properties.
Stability and rigidity of Ricci-flat ALE manifolds proven.
problem Stability and rigidity of Ricci-flat ALE manifolds.
method Proved stability and rigidity of ALE manifolds with a parallel spinor under Ricci flow, given initial metrics close in Lp∩L∞. result Strong decay rates prove positive scalar curvature rigidity in Lp for each p∈[1,n−2n). Sharp pinching conditions restrict the geometry and topology of submanifolds.
problem Understanding submanifolds under pinching conditions in arbitrary Riemannian manifolds.
method Analyzing submanifolds with pinching conditions involving second fundamental form and mean curvature.
result The pinching condition imposes strong geometric and topological restrictions on submanifolds.
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …
Normal forms prove dynamical results for magnetic fields on surfaces.
problem Existence and rigidity of Zoll flows on surfaces.
method Proved normal forms for strong magnetic fields and used them to derive dynamical results.
result Flow cannot be Zoll unless specific conditions hold.
We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any n≥4, we construct a finite subgraph Xn of the pants graph P(S0,n) of the n-punctured sphere S0,n with the following property. Any simplicial embedding of Xn into any pants graph P(S0,m) of a punctured …
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.
Study shows curvature constraints force submanifolds to have specific topology or geometry.
problem Curvature constraints on submanifolds in nonnegative curvature spaces.
method Investigates submanifolds with lower bounds on sectional curvature and mean curvature.
result Curvature constraints force submanifolds to have specific topology or geometry.
We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is…
This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…
Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, [CM1] showed that the only generic are round cylinders $\SS^k\times \RR^{n-k}$. We prove here that round cylinders are rigid in a very strong sense. Namel…
Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at −∞, generalizing a theorem for cylinders in [CM19b]. In the case of the m-covered circle, we apply this bound to prove a stron…
Maps between positively curved manifolds with non-increasing area are rigid.
problem Understanding maps between manifolds with positive curvature and non-increasing area.
method Exploring the graphical mean curvature flow and using Brendle's sphere theorem.
result Maps between certain positively curved manifolds are homotopy trivial, Riemannian submersion, local isometry, or isometric immersion.
We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…
Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weig…
The paper classifies solutions to semilinear equations on curved spaces.
problem Classifying solutions to semilinear equations on manifolds with nonnegative Ricci curvature.
method Proving classification results for subcritical and critical semilinear elliptic equations.
result Strong rigidity results for nontrivial solutions in the critical case.
The paper shows how certain Kähler groups are uniquely determined by their profinite completions.
problem Understanding the uniqueness of Kähler groups within residually finite groups.
method Holomorphic fibrations and profinite completions of fundamental groups.
result Aspherical smooth projective varieties are determined by their algebraic fundamental groups.
Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curva…
We show that, in many situations, a homeomorphism f of a manifold M may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing f. As an application, we relate the notions of {\em critical regularity} and of {\em differentiable rigidity}, give examples of groups…
In all possible cases, we prove that local embeddings between two curve complexes whose complexities do not increase from domain to codomain are induced by surface homeomorphism. This is our first main result. From this we can deduce our second, a strong local co-Hopfian result for mapping class groups.
The report presents the theory of harmonic maps from Kähler manifolds.
problem Understanding harmonic maps from Kähler manifolds.
method Reviewing and specializing the theory of harmonic maps between Riemannian manifolds, introducing pluriharmonic maps, and proving refined Bochner formulas.
result Strong rigidity results and applications to symmetric spaces of noncompact type.
New rigidity results for tensors on non-compact manifolds with curvature conditions.
problem Rigidity phenomena for tensors on non-compact Riemannian manifolds.
method Extending Bochner technique to non-compact settings, using Lichnerowicz Laplacian.
result Vanishing and rigidity of curvature tensors on Ricci-flat and Einstein manifolds.
Artin groups of hyperbolic type are boundary amenable and have rigid properties.
problem Characterizing rigidity and measure equivalence properties of Artin groups.
method Analyzing boundary amenability, measure equivalence, and fixed set graphs.
result Measure equivalent Artin groups of hyperbolic type have isomorphic fixed set graphs.
A new method to predict uncertainties in trained neural networks.
problem Reliability of trained neural networks outside their training domain.
method Prediction rigidities as a constrained optimization problem.
result Cheap uncertainties without modifying neural networks.
Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the norm square of the pull-back of the target volume form by such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic ave…
Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \fracπ{2}. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that whe…
In this paper, we firstly establish an Interpolating curvature invariance between the well known nonnegative and 2-non-negative curvature invariant along the Ricci flow. Then a related strong maximum principle for the (λ1,λ2)-nonnegativity is also derived along Ricci flow. Based on these, finally we obtain a rigid…
Effective rank rigidity proved for cubulated groups with factor systems.
problem Rank rigidity in cubulated groups with factor systems.
method Exhibiting special pairs of hyperplanes and curtains for skewering.
result Effective form of rank rigidity proved for cubulated groups.
The paper explores rigidity and proximality in dynamical systems, proving new results about C∗-algebras.
problem Understanding rigidity and proximality in dynamical systems and their algebraic counterparts.
method Analyzing crossed products of dynamical systems and their C∗-algebras, focusing on uniform rigidity and proximality. result Uniformly rigid systems are almost reflecting, and certain crossed products are reflecting.
The paper introduces Lagrangian vanishing cycles to prove obstructions for symplectic foliations.
problem Lack of complexity in foliation generalizations from 3D to higher dimensions.
method Introducing strong symplectic foliations and Lagrangian vanishing cycles.
result Lagrangian vanishing cycles prevent a symplectic foliation from being strong.
Survey of rigidity and gap phenomena in sphere-ball submanifolds.
problem Rigidity and gap phenomena in submanifolds of sphere and ball.
method Comparison of techniques, pinching and gap theorems, Morse index and topology.
result Free boundary condition in ball forces stronger rigidity than in sphere.