Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No C1-smooth slim limit set for higher rank semisimple algebraic groups. Study of Anosov representations with Lipschitz limit set and applications to rigidity.
problem Characterizing Anosov representations with specific limit set properties.
method Introducing an unstable Jacobian and analyzing its orbit growth rate.
result Many higher rank representations belong to the studied class.
Study rigid limit of hypermultiplet moduli spaces in string theory.
problem Understanding the rigid limit of hypermultiplet moduli spaces in Calabi-Yau compactifications.
method Performing a hyperkahler quotient of the Swann bundle over the moduli space, induced by a local limit of the Calabi-Yau.
result The resulting hyperkahler manifold is obtained as the target space of a non-linear sigma model for a five-dimensional gauge theory.
Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.
problem Rigidity of invariant Ricci flow blowdown limits on nilpotent bundles with zero curvature.
method Construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature.
result Proves blowdown limit is locally an expanding Ricci soliton for three-dimensional Heisenberg group structure group.
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
The paper proves rigidity theorems for Type II singularities in Lagrangian flows.
problem Understanding Type II singularities in Lagrangian flows with zero Maslov class.
method Rigidity theorems for blow-up limits of Type II singularities.
result Generalized previous results from 2D to arbitrary dimensions.
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.
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. We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the `achronal limits' introduced in [12]. This, in turn, allows for a broa…
The paper studies the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
problem Understanding the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
method Analyzes sequences of almost homogeneous RCD(K,N) spaces and their Gromov-Hausdorff limits.
result The Gromov-Hausdorff limit of a sequence of almost homogeneous RCD(K,N) spaces is a nilpotent Lie group with Ric ≥ K.
The study shows rigidity properties are lost in hyperbolic generalizations.
problem Lack of geometric and topological rigidity in hyperbolic groups.
method Observations and proofs on acylindrically hyperbolic and relatively hyperbolic groups.
result No well-defined limit set for acylindrical actions on hyperbolic spaces.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for certain amalgamations of free groups: the direct limits of two such diagrams are iso…
Mostow rigidity proven for special geometric manifolds.
problem Proving rigidity for specific geometric structures.
method Analyzing foliated bundles over closed hyperbolic manifolds with invariant measures.
result Mostow rigidity theorem extended to skew solenoidal manifolds.
We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G1 and a quasiconformal conjugate h−1G2h of a cocompact group G2. We show that if the conjugacy h is not conformal then this group contains a non-trivial one parameter subgroup. Th…
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
Ancient Ricci flows are identified without curvature sign condition.
problem Identifying type II ancient Ricci flows and their backward limits.
method Using a size condition of the sharp log Sobolev functional near infinity.
result Rigidity result for ancient Ricci flows without sign condition on curvatures.
For i=1,2, let Gi be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension n, n≥3. Let Hi⊂Gi be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of Hi is a codimension one topological sphere. 2) limit set of Hi is an e…
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.
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.
The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
New non-rigid discrete groups found in hyperbolic spaces.
problem Uniqueness of conformal or spherical CR structures on spheres.
method Nilpotent Sierpiński carpet and stretching to construct non-rigid groups.
result Discrete hyperbolic groups can have non-rigid deformations.
New insights on solutions to Allen-Cahn equation with degenerate minimal hypersurfaces.
problem Existence and rigidity of solutions to the Allen-Cahn equation.
method Analysis of degenerate minimal hypersurfaces as limit interfaces.
result New observations and examples of solutions to the Allen-Cahn equation.
Profinite rigidity studied for algebraic fibring of groups.
problem Profinite rigidity of groups through algebraic fibring.
method Introducing TAP groups and proving algebraic fibring is a profinite property.
result Algebraic fibring is a profinite property for LERF groups.
The study finds surface subgroups in specific types of groups.
problem Finding surface subgroups in certain groups.
method Analyzing graph pairs and using properties of fundamental groups and limit groups.
result Surface subgroups found in graph pairs and limit groups.
Rolling systems limit to billiard models with no-slip collisions.
problem Understanding how rolling systems behave as billiard models with no-slip collisions.
method Showed that no-slip billiards arise as limits of non-holonomic rolling systems.
result Rolling systems limit to billiard models with no-slip collisions.
Researchers refine local rigidity for marked length spectrum and introduce a new pressure metric.
problem Local rigidity of marked length spectrum and related metrics.
method Refined local rigidity result using geodesic stretch and Anosov flows, introduced new pressure metric.
result New pressure metric related to Weil-Peterson metric, reduces to it in Teichmüller space.
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
We derive a continuum model from discrete elastic models on smooth manifolds.
problem Modeling stress-free configurations in geometrically-incompatible elastic systems.
method Variational convergence of discrete models to a continuum model.
result No stress-free configurations unless the manifold is flat.
Study of Poisson homeomorphisms and rigidity of coisotropic submanifolds.
problem Rigidity and non-rigidity phenomena in Poisson geometry.
method Study of Poisson homeomorphisms, use of clean intersection points, and analysis of characteristic partitions.
result Poisson homeomorphisms preserve symplectic foliations and coisotropic submanifolds are flexible.
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.
In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.
Surface groups are uniquely identified by their profinite completions.
problem Identifying surface groups among limit groups.
method Using profinite completions and topology.
result The set of surface words is closed in the profinite topology.
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.
New insights into profinite rigidity of Kleinian groups and their subgroups.
problem Characterizing profinite completions of Kleinian groups and their subgroups.
method Analyzing profinite completions and using finite index subgroups to distinguish completions.
result Profinite completions of certain subgroups of finite index in Kleinian groups are not isomorphic.
Ancient mean curvature flows get codimension bounds from their tangent flow.
problem Understanding the limiting behavior of ancient mean curvature flows.
method Proving codimension bounds using the tangent flow at −∞. result Ancient mean curvature flows are rigid to their tangent flow at −∞. Study shows ortho spectrum doesn't fully determine systolic length but limits the number of possible structures.
problem Determining the systolic length of hyperbolic surfaces with boundary.
method Analyzing the ortho spectrum of hyperbolic surfaces with totally geodesic boundary.
result There are only finitely many possibilities for the ortho spectrum and corresponding hyperbolic structures.
Study of elastic models in non-Euclidean spaces via Γ-convergence.
problem Elasticity in non-Euclidean ambient spaces with incompatible local rest distances.
method Γ-convergence to derive a limit elastic model, relating minimum energy to curvature discrepancy.
result Linearized version of a conjecture in elasticity confirmed, linking energy to curvature.
Study rigidity of geodesic planes in geometrically finite manifolds with cusps.
problem Rigidity of geodesic planes in geometrically finite manifolds with rank 1 cusps.
method Analysis of horocycles recurrence and density criteria for immersions.
result Closed immersions give rise to surfaces with finitely generated fundamental groups.
Study on profinite rigidity of direct products of free and surface groups.
problem Profinite rigidity of direct products in residually free groups.
method Rank gradients of pro-p groups, virtual homology of limit groups. result Direct products of free and surface groups are profinitely rigid.
Optimizes PDE-constrained LDDMM for efficient non-rigid registration.
problem Inexact Newton-Krylov optimization in PDE-constrained LDDMM leads to poor geodesic paths.
method Band-limited vector field parameterization to optimize computational complexity.
result Optimized method shows competitive performance with reduced memory load and computational time.
Study shows only rotations can be approximated by Ginzburg-Landau critical points.
problem Proving not all harmonic maps can be approximated by Ginzburg-Landau critical points.
method Rigidity theorem applied to Ginzburg-Landau energy critical points.
result Only rotations can be approximated by Ginzburg-Landau critical points.
New energy functional bounds Ricci flows on ancient spaces.
problem Bounding Ricci flows on ancient spaces.
method Introducing a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows.
result Provides an upper bound for the ordinary λ-functional.
Study shows spherical hyperbolic manifolds almost rigidly converge to hyperbolic space.
problem Almost rigidity of positive mass theorem for spherical hyperbolic manifolds.
method Intrinsic flat distance to prove convergence.
result Spherically symmetric asymptotically hyperbolic manifolds converge to hyperbolic space if mass limit is zero.