Develops a framework for stuck knots with rigid constraints and invariants.
problem Rigidity constraints on knot diagrams restrict allowable isotopies.
method Formalizes stuck crossings as rigid configurations, introduces unstick move, and constructs invariants.
result Rigidity contributes independent information even when knot type remains fixed.
Bi-invariant metrics on Lie groups and homogeneous spaces are extremal and rigid.
problem Gromov's question on extremality of bi-invariant metrics on compact Lie groups.
method Proving rigidity of bi-invariant metrics on compact Lie groups and homogeneous spaces.
result Bi-invariant metrics on compact Lie groups and homogeneous spaces are extremal and rigid.
Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.
problem Identifying compact surfaces up to rigid transformations.
method Degree four polynomials in moments of delta function, effective inversion algorithm.
result Invariants and retrieval algorithm work on a comeagre subset of surfaces.
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…
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.
The study of rigidity theorems on 4-manifolds with boundary.
problem Understanding topological restrictions on 4-manifolds with boundary.
method Introducing new conformal and smooth invariants, studying Weyl functional, and analyzing the expansion of a smooth Riemannian metric near the boundary.
result Established several conformally invariant rigidity theorems for 4-manifolds with boundary.
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…
Survey on quasi-isometries of group pairs and their invariants.
problem Understanding quasi-isometries of group pairs and their invariants.
method Exploration of quasi-isometry and qi-characteristic collections of subgroups.
result New insights into phenomena observed in quasi-isometric rigidity.
The article discusses invariant measures outside homogeneous dynamics.
problem Understanding invariant measures in non-homogeneous settings.
method Comparing two recent results on rigidity and invariant measures.
result The importance of invariant and partially invariant measures in rigidity questions.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
We extend Forester's rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups).
Study proves structure results for homogeneous spaces supporting specific equations.
problem Proving structure results for homogeneous spaces supporting specific equations.
method Analyzing homogeneous spaces with non-constant solutions to two general classes of equations involving the Hessian and an invariant 2-tensor.
result Generalizes rigidity results for gradient Ricci solitons and warped product Einstein metrics.
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.
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwi…
Classifies non-arithmetic orbifolds in specific hyperbolic spaces.
problem Classifying non-arithmetic affine invariant orbifolds in Hodd(2, 2) and H(3, 1).
method Classification through Veech surfaces and rigidity results.
result Classification of non-arithmetic rank one orbifolds.
Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.
The Kauffman-Vogel polynomials are three variable polynomial invariants of 4-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented 4-valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with 2. Bataineh, Elha…
Projective structures are mostly rigid at the boundary but some are not.
problem Boundary rigidity of projective structures.
method Investigation of projective structures on manifolds with boundary.
result Existence of non-rigid projective structures and characterization of them.
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or ve…
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…
We prove integral rigidity for Seiberg-Witten invariants of 4-manifolds with specific hypersurfaces.
problem Integral rigidity of Seiberg-Witten invariants in 4-manifolds with non-separating hypersurfaces.
method Floer theoretic conditions and interplay between irreducible and reducible solutions to Seiberg-Witten equations.
result Sum of Seiberg-Witten invariants is determined cohomologically for specific 4-manifolds.
Groups with certain properties have invariant subalgebra rigidity.
problem Invariant subalgebra rigidity in groups with specific properties.
method Analyzing normal subgroups and invariant subalgebras in groups.
result Torsion-free acylindrically hyperbolic groups and hyperbolic groups have the relative ISR property.
This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
In this note we classify all homogeneous spaces G/H admitting a G-invariant G2-structure, assuming that G is a compact Lie group and G acts effectively on G/H. They include a subclass of all homogeneous spaces G/H with a G-invariant G~2-structure, where G is a compact Lie group. There are ma…
Study reveals how travel times on cylindrical boundaries can identify spacetime structure.
problem Determining spacetime structure from boundary travel times.
method Boundary rigidity problem on cylindrical domains with stationary metrics.
result Time separation function uniquely identifies spacetime up to diffeomorphisms.
New theorem shows metrics of certain groups are close if their lengths are identical.
problem Identifying metrics of relatively hyperbolic groups from their lengths.
method Proved rigidity for relatively hyperbolic groups using coarse marked length spectrum.
result Metrics of relatively hyperbolic groups are uniformly close if lengths are identical.
In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci cu…
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.
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the…
Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
Classifies solitons on invariant surfaces in solvable Lie group.
problem Classifying solitons on invariant surfaces in a specific Lie group.
method Analyzes solitons associated with Killing vector fields and proves rigidity results.
result Identifies the only solitons for specific invariant surfaces.
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
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.
We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and tota…
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new (∞,1)-category has universal mapping property. result The new category generates link invariants.
Survey on scalar curvature stability and related questions.
problem Understanding scalar curvature stability and rigidity phenomena.
method Survey and discussion of existing tools and questions.
result Survey of known results and open questions in scalar curvature stability.
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the technique…
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
New Gromov-Wasserstein metric controls rigidity and incorporates prior knowledge.
problem Inflexible Gromov-Wasserstein distance and lack of feature alignment.
method Augmented Gromov-Wasserstein distance with feature alignments and prior knowledge.
result Improved performance in single-cell multi-omic alignment and transfer learning.
Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
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.
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
Study measures rigidity for random walks and flows via generalized u-Gibbs states.
problem Measure rigidity for stationary measures of random walks and flows.
method Factorization method applied to generalized u-Gibbs states.
result Established extra invariance of generalized u-Gibbs states.
Rigidity theorem for special metrics on 4-manifolds.
problem Rigidity of Bach-flat metrics on manifolds with boundary.
method Critical point analysis of Weyl energy with boundary conditions.
result Rigidity of critical metrics on upper hemisphere.
In our earlier articles we studied tube hypersurfaces in C3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper…