Characterizes affine vector fields on Finsler manifolds with rigidity results.
problem Understanding affine vector fields on Finsler manifolds.
method Utilizing the Jacobi type equation and spray characterization, proving rigidity theorems.
result Rigidity theorems for affine vector fields on Finsler manifolds with non-positive total Ricci curvature.
Short note explores conditions for Finsler functions to be uniquely metrizable.
problem Conditions for Finsler functions to be uniquely metrizable.
method Exploration of sufficient conditions for affinely rigidity.
result Discussion of open problems in Finsler function metrizability.
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.
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.
Study on projective and affine equivalence of sub-Riemannian metrics on manifolds.
problem Classifying projective and affine equivalence of sub-Riemannian metrics.
method Analyzing geodesics, integrals, and product structures on cotangent bundles.
result Generic sub-Riemannian metrics are projectively conformally rigid.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
Classifies rank 2 affine invariant subvarieties in H(6).
problem Classifying rank 2 affine invariant subvarieties in H(6).
method Extends Apisa's approach, reducing analysis to lower-genus strata.
result Complete description of rank 2 affine invariant subvarieties in H(6).
Comparison theorems in centro-affine differential geometry
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
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.
Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.
problem Rigidity of Kähler manifolds with nonnegative Ricci curvature.
method Analysis of Kähler manifolds with specific properties.
result Complete noncompact Kähler surface with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay is biholomorphic to the resolution of an affine algebraic variety.
Saddle connection complexes are rigid under affine equivalence.
problem Characterizing the rigidity of saddle connection complexes.
method Proving simplicial isomorphisms between saddle connection complexes are induced by affine diffeomorphisms.
result Saddle connection complexes are complete invariants of affine equivalence classes of half-translation surfaces.
Affine rigidity theorem for special surfaces without integration.
problem Understanding affine transformations of surfaces with zero Gaussian curvature.
method Pure affine geometry, avoiding analysis tools.
result Affine rigidity theorem for CR-flat surfaces.
Maps between certain configuration spaces are rigid and affine equivalent.
problem Rigidity of maps between configuration spaces.
method Homomorphisms of braid groups and irreducibility conditions.
result Holomorphic maps between configuration spaces are affine equivalent.
The paper proves rigidity of spacelike translating solitons in pseudo-Euclidean space.
problem Classifying and proving rigidity of spacelike translating solitons in pseudo-Euclidean space.
method Affine technique and classical gradient estimates to classify complete spacelike translating solitons.
result The only complete spacelike translating solitons in pseudo-Euclidean space are the spacelike m-planes. We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties o…
We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ∇ be …
Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.
problem Rigidity of Ricci curvature on Hessian manifold leaves.
method Analysis of Ricci curvature properties of Hessian metrics on foliation leaves.
result Non-negative Ricci curvature on a single leaf forces the Hessian metric to be flat and yields bounds on the first Betti number.
New method for flexible tubes and structures, enabling rigid-foldability.
problem Creating flexible tubes with rigid-foldability.
method Discrete, semi-discrete, and smooth construction of surfaces (T-hedra and profile-affine surfaces).
result Unified treatment of continuous flexible structures composed of tubes.
Study rigidifies torus bundles under first Betti number constraints.
problem Understanding the structure of torus fibrations under first Betti number restrictions.
method Established rigidity results and necessary/sufficient conditions for topological splitting.
result Classification of torus bundles under specific Betti number constraints.
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.
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
Curves in higher dimensions are either affine or have super-Euclidean energy growth.
problem Characterizing entire conformal curves in higher-dimensional spaces.
method Blow-down argument and interaction of generalized Cauchy--Riemann equations with calibrated geometries.
result Entire conformal curves are either affine or have super-Euclidean energy growth.
Study proves rigidity of specific self-shrinkers under certain geometric conditions.
problem Proving rigidity of self-shrinkers under geometric constraints.
method Analyzing complete self-shrinkers with specific tangent planes.
result Sphere, plane, and cylinder are the only self-shrinkers under the given geometric assumption.
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N=TM carries a canonical structure of (affine) special Kähler manifold. This gives a…
We consider a totally nonsymplectic Anosov action of Z^k which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is C^\infty--conjugate to an action by affine automorphisms. We also obtain similar global rigidity results for actions on an arbitrar…
Geodesic flows between hypersurfaces in Euclidean spaces using Lorentzian geometry.
problem Interpolation between hypersurfaces in Euclidean spaces.
method Lorentzian geodesic flow between tangent spaces of hypersurfaces.
result Geodesic flow is preserved by rigid transformations and homotheties.
We show that we can release the rigidity of the skew Howe duality process for sln knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine slm case, corresponding to looking at tan…
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
Global rigidity theorem for certain lattice actions on manifolds.
problem Volume-preserving actions of higher rank lattices on manifolds with dominated splitting.
method Proves standard conjugacy of actions with dominated splitting.
result Actions must be standard, manifold is flat torus with affine action.
Study quaternionic symplectic groups and their actions, proving rigidity and classifying actions.
problem Deformation rigidity of quaternionic symplectic group actions.
method Study a non-associative algebra and compute automorphism groups to classify actions.
result Uniqueness and classification of isometric actions of quaternionic symplectic groups.
Study of flows on complex manifolds with holomorphic properties.
problem Global rigidity of transversely holomorphic Anosov flows on smooth compact manifolds.
method Analyzing the integrability of unstable and stable distributions, proving uniqueness in low dimensions.
result For topologically transitive flows, they are either orbit equivalent to a hyperbolic automorphism or geodesic flow.
Let J be a semisimple Lie group with all simple factors of real rank at least two. Let Γ<J be a lattice. We prove a very general local rigidity result about actions of J or Γ. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of Γ by toral aut…
Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In…
The study translates Weinstein structures to Legendrian handlebodies for symplectic topology.
problem Detecting flexibility and rigidity in Weinstein manifolds.
method Systematic recipe for translating Weinstein Lefschetz fibrations to Legendrian handlebodies.
result Verification of Stein deformation equivalence and existence of closed exact Lagrangian submanifolds.
New proof shows affine manifolds with parallel volume are Riemannian-flat.
problem Characterize compact affine manifolds with parallel volume.
method Construct a representative metric with Levi-Civita connection, using Hessian of volume-normalized distance functions.
result Affine manifolds with parallel volume are Riemannian-flat.
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…
Let BS(1,n)= < a,b: aba^{-1}=b^n >. We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.
Characterizes how the shape of a polygon affects billiard dynamics.
problem Understanding how the shape of a billiard table influences its dynamics.
method New theorem linking Liouville current support to flat cone metrics.
result Only right-angled tables with affine differences have identical bounce spectra.
Study on non-Kähler Calabi-Yau manifolds and their geometric structures.
problem Holomorphic geometric structures on non-Kähler compact complex manifolds.
method Beauville-Bogomolov decomposition and weak Bochner principle.
result Rigidity of Vaisman Calabi-Yau manifolds implies they are Kodaira manifolds.
Improved non-squeezing theorem for calibrated geometries proved.
problem Proving an improved non-squeezing theorem for calibrated geometries.
method Two proofs: direct and reduction to classical case.
result Established an improved non-squeezing theorem for calibrated geometries.
The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metr…
All the actions considered here are (real) analytic. Consider a subgroup of finite index of SL(n, Z). We prove, in particular, the (global) homotopical rigidity, for both its standard affine action on the torus of dimension n > 2, and its standard projective action on the sphere of dimesnion n-1>3.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.
The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.
problem Understanding spectral encodings under Weyl growth conditions.
method Analyzing geometric Weyl bulk-density exponent and proving spectral rigidity.
result The geometric Weyl bulk-density exponent (d−2)/2 rigidifies spectral encodings in the O-regularly varying class, leading to unique admissible exponents and scaling laws. It is shown that any smooth strictly convex global solution of det(∂ξi∂ξj∂2u)=exp{−∑i=1ndi∂ξi∂u−d0}, where d0, d1,...,dn are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…
The paper establishes a correspondence between special Kähler manifolds and their deformations.
problem Mapping between affine and projective special Kähler manifolds.
method Formulation of a correspondence and application to r-maps in string theory.
result One-parameter deformations of projective special Kähler manifolds correspond to perturbative α'-corrections.