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.
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.
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.
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.
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 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.
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…
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.
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.
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…
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 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. Proves rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on S2×S2 D.Gromoll and K.Tapp in [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on S2×R3): "The boundary $S^2\times S^2…
The thesis explores integrable systems and rigidity in PDEs with symmetry.
problem Understanding the deformation theory and rigidity of PDEs with symmetry.
method The approach involves studying completely integrable systems, their equivalence relations, and the deformation theory of PDEs with pseudogroups of symmetries.
result A solution is rigid if its deformation cohomology vanishes and certain estimates hold.
Let P⊂R3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assu…
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and Lp-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scal…
The paper proves rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.
problem Proving rigidity for self-shrinkers and surfaces with parallel weighted mean curvature.
method Using a new generalization of Cauchy's Theorem in complex analysis.
result Rigidity results for self-shrinkers and surfaces with parallel weighted mean curvature.
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner prop…
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. This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration …
Analyzes metric spaces homeomorphic to manifolds, proving rigidity and inequalities.
problem Analyzing metric spaces homeomorphic to manifolds.
method Geometric and analytic approaches, proving existence of integral currents, establishing rigidity and inequalities.
result Metric manifolds admit non-trivial integral currents and satisfy isoperimetric inequalities.
Establishes a Penrose-type inequality for axisymmetric initial data with angular momentum and charge.
problem Establishing a Penrose-type inequality for axisymmetric initial data with angular momentum and charge.
method Maximal, axisymmetric initial data for the Einstein-Maxwell equations satisfying the weak energy condition. Rigidity statement proven.
result Reduces to the conjectured Penrose inequality with angular momentum and charge under certain conditions.
This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.
problem Classifying geodesics of projectively flat sprays and determining sprays based on geodesics.
method Introduction of a geodesic method to determine an n-dimensional spray based on a family of curves with 2(n-1) free parameters as geodesics.
result Classification of geodesics of projectively flat sprays and determination of sprays based on geodesics.
Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.
problem Addressing the rigidity-flexibility dichotomy for wrinkled patterns in thin elastic sheets.
method Develops hodograph transformation and parametrix-corrector decomposition to handle corner singularities and prove well-posedness.
result Proves existence and uniqueness of hodograph weak solutions and derives energy estimates for stability.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. The study describes quasiflats in 2D Artin groups and their properties.
problem Understanding the structure and properties of quasiflats in 2D Artin groups.
method Metric systolicity and combinatorial analysis of tilings.
result Precise description of building blocks (atomic sectors) for quasiflats in 2D Artin groups.
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…
The paper proves rigidity theorems for hypersurfaces in spherical space forms.
problem Proving rigidity of hypersurfaces in spherical space forms under certain curvature conditions.
method Topological and homotopical methods, including diffeomorphisms and weak homotopy equivalences.
result The universal cover of the hypersurface is diffeomorphic to the n-sphere and fundamental group bounds.
Study of cubulations in hyperbolic groups and their invariant cross ratios.
problem Understanding geometric structures in hyperbolic groups.
method Analysis of cubulations and cross ratios in Gromov hyperbolic groups.
result Essential cubulations of hyperbolic groups are length-spectrum rigid.
We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike.…
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.
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.
Brezis' open problem on harmonic maps resolved
problem Existence of explicit solutions to the harmonic map equation with Dirichlet boundary condition
method Boundary rigidity argument
result Proving the uniqueness of the explicit maps as weak harmonic maps
Let X be a compact connected CR manifold of dimension 2n−1,n≥2. We assume that there is a transversal CR locally free S1 action on X. Let Lk be the k-th power of a rigid CR line bundle L over X. Without any assumption on the Levi-form of X, we obtain a scaling upper-bound for the partial Szegő …
Maps close to isometries on Riemannian manifolds are close to isometries.
problem Understanding the rigidity of maps on Riemannian manifolds.
method Optimal linear estimate using Sobolev maps, weak Riemannian Piola identity, harmonic map heat flow, and linearization.
result Optimal rigidity estimate for maps of a compact Riemannian manifold to itself.
Minimal submanifolds either fill space or are confined with geometric restrictions.
problem Understanding the behavior of minimal submanifolds in space.
method A dichotomy principle and volume doubling theorem.
result Quantitative restrictions on confined minimal submanifolds, including volume growth and optimal density rates.
Study of spacetimes in cosmology without symmetry assumptions.
problem Proving timelike incompleteness and properties of time functions.
method Rigorous mathematical proofs and analysis of spacetime properties.
result Timelike incompleteness for specific spacetimes with mean curvature constraints.
We present and study a family of metrics on the space of compact subsets of RN (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our s…
The paper proves the regularity of solutions to a specific differential equation describing physical phenomena.
problem Understanding the regularity of solutions to a curvature-dependent differential equation.
method Weaves together analysis and geometry to prove the optimal regularity of solutions.
result The second derivative of solutions is continuous only in very rigid situations.
The paper splits local rigidity into vertical and horizontal types.
problem Local rigidity of Clifford-Klein forms in homogeneous spaces.
method Introducing a splitting of local rigidity into vertical and horizontal rigidity.
result Refined results and a new approach to Baklouti's conjecture.