Unified approach to global weak rigidity of Riemannian manifolds and their immersions.
problem Global weak rigidity of Gauss-Codazzi-Ricci equations and isometric immersions of Riemannian manifolds.
method Unified intrinsic approach, div-curl structure, compensated compactness theorem, global intrinsic div-curl lemma.
result Established global weak rigidity of GCR equations and isometric immersions of Riemannian manifolds with lower regularity.
The study establishes a new theorem for Banach spaces to solve geometric rigidity problems.
problem Weak rigidity of isometric immersions of manifolds with lower regularity.
method Compensated compactness theorem in Banach spaces.
result Global weak rigidity of Gauss-Codazzi-Ricci equations and isometric immersions.
We study weakened f-structures on manifolds, generalizing classical results.
problem Classical f-structures and their properties on manifolds. method Introduced and studied weakened f-structures, subclasses, and their properties. result Generalized known results on globally framed f-manifolds. New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Paper proves rigidity of weak solutions for anisotropic N-Laplacian equations with Neumann or Robin boundary conditions.
problem Rigidity of weak solutions for anisotropic N-Laplacian equations with boundary conditions.
method Established a key integral inequality involving anisotropic gradient and second fundamental form, proving rigidity under natural monotonicity assumptions.
result All weak solutions to Neumann boundary problems are constant without a priori boundedness assumption.
Embeds flag manifolds into classical ones, proving rigidity in Kähler geometry.
problem Rigidity phenomena in homogeneous Kähler manifolds.
method Holomorphic isometric embeddings and rigidity analysis.
result No weak-relative relationship among flag manifolds, flat spaces, and homogeneous bounded domains.
Survey of recent results in weak almost contact structures.
problem New geometric structures replacing complex structure in contact manifolds.
method Survey of recent findings in weak almost contact manifolds.
result Recent results on geodesic and Killing fields, rigidity and splitting theorems, etc.
Proves global rigidity of sphere packings on 3D manifolds.
problem Global rigidity of sphere packings on 3D manifolds.
method Combination of combinatorial scalar curvature and Andreev-Thurston rigidity theorem.
result Proves global rigidity of sphere packings on 3D manifolds.
Study the geometry of weak para-f-structures and subclasses.
problem Understand the geometry of weak para-f-structures and their subclasses.
method Express covariant derivative of f, prove Killing characteristic vector fields, show foliations, and demonstrate rigidity.
result Prove that characteristic vector fields are Killing and ker f defines a totally geodesic foliation.
Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.
problem Understanding unique spacetime extensions across null boundaries in 1+1 dimensions.
method Analyzing the C0- and C1-structures of continuous spacetime extensions. result Extensions can have the same C0-structure but different C1-structures. New spectral conditions ensure graph rigidity and global rigidity in the Euclidean plane.
problem Ensuring graph rigidity and global rigidity in the Euclidean plane.
method Improving algebraic connectivity bounds for graph rigidity and global rigidity.
result Every 6-connected graph is rigid and globally rigid if its algebraic connectivity exceeds specific thresholds.
Global weak solution found for harmonic map flow into Lorentzian manifolds.
problem Existence of global weak solutions for harmonic map flow into Lorentzian manifolds.
method Proved existence of a unique global weak solution with regularity except for finitely many singular points.
result Global weak solution found for harmonic map flow into Lorentzian manifolds.
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
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 automorphisms on procongruence curve and pants complexes.
problem Understanding automorphism groups of procongruence curve and pants complexes.
method Action on procongruence mapping class group and rigidity theorem for pants complex.
result Prove rigidity theorem for procongruence completion of pants complex.
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.
We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\T^N$.
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.
problem Investigate regularity of weak Kantorovich potentials on globally hyperbolic spacetimes.
method Apply insights from Riemannian and Lorentzian cases to study π-solutions. result Conclude existence, uniqueness, and structure of optimal transport maps.
Extends submanifold rigidity to include singularities, unifying and extending known theorems.
problem Global rigidity of submanifolds with singularities.
method Allowing mild singularities to unify and extend known theorems.
result Any compact n-dimensional submanifold of R^(n+p) is singularly genuinely rigid in R^(n+q) for q < min{5,n} - p.
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
Global solution found for string heat flow on surfaces.
problem Existence of global weak solutions for bosonic string heat flow.
method Proved existence with smallness conditions on form and potential.
result Global weak solution found, smooth with finitely many singular points.
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
Study proves rigidity of non-negative scalar curvature on Euclidean space perturbations.
problem Rigidity of non-negative scalar curvature on Euclidean space perturbations.
method Analysis of long time solutions of Ricci DeTurck flow.
result Proves a weak version of the positive mass theorem.
Paper proves rigidity of inversive distance circle packings.
problem Proving rigidity of inversive distance circle packings.
method Variational principles and combinatorial curvature study.
result Global rigidity of inversive distance circle packings proved.
The paper proves rigidity results for compact initial data sets.
problem Understanding the structure of compact initial data sets.
method Proving rigidity results under specific conditions.
result Global version of the main result in [15] is obtained.
Local and global rigidity results for Lie group actions on pseudo-Riemannian manifolds.
problem Rigidity of isometric actions of simple Lie groups on pseudo-Riemannian manifolds.
method Analyzing normal bundles and using Lie group properties.
result Local and global rigidity theorems for specific Lie groups and manifolds.
Global weak solutions found for Landau-Lifshitz equations into compact Lie algebras.
problem Existence of global weak solutions to Landau-Lifshitz equations into compact Lie algebras.
method Followed Arnold's ideas and used test functions and approximate equations.
result Existence of global weak solutions for the Cauchy problems of Landau-Lifshitz equations.
Study proves global existence of weak solutions for mean curvature flow with transport and forcing terms.
problem Existence of weak solutions for mean curvature flow with non-smooth terms.
method Used modified Allen-Cahn equation with properties like monotonicity formula to prove existence.
result Global existence of weak solutions for the mean curvature flow with transport and forcing terms.
In this paper we show that a substantial Riemannian submersion of S(15) with 7- dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of of the Diameter Rigidity theorem for the Cayley plane which is considerably stronger than the very recent Radius Rigidity Theo…
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
Paper solves boundary rigidity and lens rigidity problems for Riemannian manifolds.
problem Boundary and lens rigidity problems for Riemannian manifolds.
method Analysis of geodesic X-ray transform in normal coordinates.
result Metric can be determined from boundary distance function and lens relation.
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…
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.
New proof simplifies existing proofs of Bowers-Stephenson conjecture.
problem Proving the global rigidity of circle packing with inversive distance.
method Variational proof simplifying previous methods.
result Global rigidity of circle packing with inversive distance in (-1, +∞).
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.
We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension n=5, with non-empty totally geodesic boundary. More precisely, if M1n,M2n are any two such manifolds, we show that (1) ∂∞M~1n is homeomorphic to $\partial ^\infty…
Proves two non-trapping obstacles coincide if scattering rays have similar travelling times or scattering length spectra.
problem Identifying non-trapping obstacles based on scattering properties.
method Proves two obstacles coincide if their scattering rays have similar travelling times or scattering length spectra under weak non-degeneracy conditions.
result Two non-trapping obstacles coincide if their scattering rays have similar travelling times or scattering length spectra.
Global rigidity theorem for curve to abelian variety maps.
problem Characterizing maps between moduli spaces of curves and abelian varieties.
method Analyzing nonconstant holomorphic maps between moduli spaces.
result Holomorphic maps between specific moduli spaces are rigid, with only one possible form.
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…
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
Simple uniqueness proof for McKean-Vlasov systems with weak feedback.
problem Global and short-time uniqueness for McKean-Vlasov systems with small feedback.
method Simple probabilistic comparison argument robust to solution regularity.
result Global uniqueness for a broad class of McKean-Vlasov problems in the weak feedback regime.
The paper proves rigidity of Seiberg-Witten invariants for spin 4-manifold families.
problem Rigidity of Seiberg-Witten invariants for spin 4-manifold families.
method Mechanism of rigidity theorem and 10/8-type inequality.
result Existence of non-smoothable topological families of 4-manifolds.
Proves global existence for quasilinear wave equations with weak null condition.
problem Global existence for quasilinear wave equations with weak null condition.
method p-weighted energy method, hierarchical structure in semilinear terms, robust methods.
result Proves global existence for a larger class of quasilinear wave equations.
Einstein manifolds are rigid under certain metric deformations.
problem Characterizing Einstein manifolds that resist volume-preserving metric deformations.
method Various characterizations and constructions of mass-decreasing perturbations.
result Constructs mass-decreasing perturbations of specific metrics.
Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metri…