The study explores stable diffeomorphism groups in 4-manifolds using localisation and invariants.
problem Understanding stable diffeomorphism groups in 4-manifolds.
method Localisation of n-manifolds, inverting connected sum construction, using Bauer--Furuta invariants.
result K3-stable Bauer--Furuta invariants determine S^2xS^2-stable invariants.
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
Constructs operations on stable moduli spaces to compare manifold cohomology.
problem Comparing cohomology of moduli spaces of closed manifolds.
method Constructs operations on stable moduli spaces and uses them to compare cohomology.
result Obtains isomorphisms in a stable range for all primes not invertible in coefficients.
Researchers study the geometric properties of a specific type of stable processes.
problem Understanding the information geometry of tempered stable processes.
method Derivation of α-divergence, Fisher information matrices, and α-connections.
result Obtained Fisher information matrices and α-connections for statistical manifolds.
Characterizes hyperbolic links with stable maps to the plane.
problem Understanding hyperbolic links through stable maps.
method Characterization of hyperbolic links via stable maps to the plane.
result Complete characterization of hyperbolic links with specific stable maps.
We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simp…
In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian manifold which admits a stable solution. Furthermore, under the non-negativity of …
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
Generalized complex (GC) geometry interpolates between ordinary symplectic and complex geometry. Stable generalized complex manifolds (first introduced by Cavalcanti, Gualtieri in 2015) carry a Poisson structure which is generically symplectic, but degenerates on a (real) codimension-2 submanifold. Up to gauge equivale…
Study on stable Hamiltonian topology finds non-density of certain structures.
problem Non-density of stable hypersurfaces and Hamiltonian structures.
method Proving non-density results for stable hypersurfaces and Hamiltonian structures in various dimensions.
result Non-density of stable hypersurfaces and Hamiltonian structures in specific isotopy and homotopy classes.
This paper shows stable mappings are never dense on non-compact manifolds.
problem Density of stable mappings on non-compact manifolds.
method Complementing Mather's theory, proving stability results for non-compact manifolds.
result The set of stable mappings is never dense on non-compact manifolds.
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
In this note we give a direct method to classify all stable forms on Rn as well as to determine their automorphism groups. We show that in dimension 6,7,8 stable forms coincide with non-degnerate forms. We present necessary conditions and sufficient conditions for a manifold to admit a stable form. We also discuss …
Non-compact manifolds prevent C0-stable mappings from being dense.
problem Density of C0-stable mappings on non-compact manifolds. method Using topologically critical points to show non-density.
result The set of C0-stable mappings is never dense on non-compact manifolds. The study restricts stable minimal immersions in product spaces to specific configurations.
problem Prohibiting stable minimal immersions in certain product spaces.
method Analyzing stable minimal immersions in products of complex, quaternionic, and octonionic projective spaces.
result The only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold.
Stable approach solves equivariant Hopf theorem for G-manifolds.
problem Describe homotopy classes of G-equivariant maps into a G-sphere.
method Equivariant stable homotopy theory with semi-free G-universe.
result Degrees of maps are characterized by congruences.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
New classification for some unorientable 4-manifolds using modified surgery theory.
problem Classifying stable diffeomorphism classes of unorientable 4-manifolds.
method Modified surgery theory applied to unorientable 4-manifolds with specific fundamental groups.
result Found nine stable diffeomorphism classes for pin+ manifolds, one for pin−, and four for neither, under certain conditions. New stable minimal hypersurfaces found in 4-manifolds, proving topology results.
problem Finding stable minimal hypersurfaces with specific topologies in 4-manifolds.
method Geometric measure theory and 4-manifold topology techniques.
result Existence of stable minimal hypersurfaces diffeomorphic to S3 or S2imesS1. Homotopy theory for (2n+1)-dimensional manifold triads with fixed boundary.
problem Classifying stable moduli spaces of (2n+1)-dimensional manifold triads. method Homotopy-theoretic description of stable moduli spaces, stabilization by boundary connected sum with SnimesDn+1. result Established homology of stable moduli spaces for (2n+1)-dimensional manifold triads. We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.
We show that closed 3-manifolds with high Heegaard distance and bounded subsurface Heegaard distance are primitive stable when they are regarded as representations from the free group corresponding to the handlebody. This implies that any point on the boundary of Schottky space can be approximated by primitive stable r…
In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manif…
The study examines stable regions in weighted manifolds with boundary properties.
problem Studying stable regions in weighted manifolds with boundary properties.
method Using deformations constructed from parallel vector fields tangent to the boundary, the study deduces rigidity properties for stable sets.
result The classification of stable sets in some Riemannian cylinders and uniqueness results for minimizers.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M) with stable commutator length less than d is represented by a geodesic with leng…
Paper introduces a method to generate stable shapes using Grassmann manifolds.
problem Generating stable shapes with minimal extraneous transformations.
method Continuous normalization flows on Grassmann manifolds to eliminate extraneous transformations.
result The method significantly outperforms state-of-the-art methods in generating high-quality samples.
In this article, we systematically investigate the stability properties of certain warped product Einstein manifolds. We characterize stability of these metrics in terms of an eigenvalue condition of the Einstein operator on the base manifold. In particular, we prove that all complete manifolds carrying imaginary Killi…
This paper extends homological stability results for configuration spaces of manifolds.
problem Homological stability of configuration spaces of manifolds.
method Analyzing the cohomology of configuration spaces of manifolds, focusing on stability in odd and even degrees.
result The stable range for homology groups of configuration spaces depends on the dimension of the manifold and the number of configuration points.
It follows from a theorem of Gromov that the stable systolic category of a closed manifold is bounded from below by the rational cup-length of the manifold. In the paper we study the inequality in the opposite direction. In particular, combining our results with Gromov's theorem, we prove the equality of stable systoli…
Polystable bundles on Kaehler manifolds are stable if their classes are stable.
problem Stability of holomorphic vector bundles on Kaehler manifolds.
method Quasi-linear Hodge theory and geometric invariant theory.
result Stability of vector bundles is equivalent to stability of their classes in geometric invariant theory.
New method proves h-principles for stable forms on manifolds.
problem Proving h-principles for stable forms on manifolds. method Convex integration applied to stable forms.
result Proved h-principles for 4 classes of stable forms. The paper proves vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
problem Vanishing theorems for harmonic forms and spinors on stable minimal hypersurfaces.
method Positive curvature assumptions on the ambient manifold.
result Vanishing of L2-harmonic forms and spinors on stable minimal hypersurfaces. For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then th…
Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.
problem Classifying stable diffeomorphism of spin 4-manifolds with given fundamental groups.
method Formulated conjectural relationships between algebraic invariants and obstructions, proved for specific groups.
result Proved conjectures for specific fundamental groups, providing complete algebraic stable classification.
New bounds set for stable 2-systole in specific geometric spaces.
problem Bounding stable 2-systole in metrics with positive scalar curvature.
method Proving uniform bounds on specific manifolds.
result Stable 2-systole is uniformly bounded for certain manifolds.
The study proves stable minimal immersions in positively curved manifolds are totally geodesic.
problem Proving stable minimal immersions in positively curved manifolds are totally geodesic.
method Formulating stable Bernstein type theorems in certain positively curved ambient manifolds.
result Proves stable minimal immersions in positively curved manifolds are totally geodesic.
Extends Einstein-Hilbert functional definition for stable manifolds.
problem Stability of Einstein manifolds on Riemannian manifolds.
method Second variation of generalized Einstein-Hilbert functional.
result Properties of stable Einstein manifolds presented.
In this paper, we prove that any polarized K-stable manifold is CM-stable. This extends what I did for Fano manifolds in my 2012 paper.
We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.
We will define a version of Seiberg-Witten-Floer stable homotopy types for a closed, oriented 3-manifold Y with b1(Y)>0 and a spin-c structure c on Y with c1(c) torsion under an assumption on Y. Using the Seiberg-Witten-Floer stable homotopy type, we will construct a gluing formula…
The paper constructs multiple manifolds with similar properties.
problem Finding multiple distinct manifolds with stable diffeomorphism.
method Constructing n pairwise homotopically inequivalent manifolds. result Each constructed manifold is stably diffeomorphic to one another.
The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.
problem Stable ergodicity of group actions on smooth manifolds restricted to one-dimensional cases.
method Geometric method using quasi-conformal blender for constructing stable local dynamics.
result Every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C1+α. Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.
problem Understanding scarring behavior of minimal hypersurfaces along stable ones.
method Analyzing a generic metric on a manifold to show scarring of minimal hypersurfaces.
result Closed, embedded minimal hypersurfaces scarring along stable ones, with diverging area and Morse index.
Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h\in H_k(X,R) then the stable norm of h is the infimum of the Riemannian k-volumes of re…
The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.
problem Understanding criticality and splitting theorems for manifolds with spectral Ricci bounds.
method Proving criticality criteria and spectral splitting theorems for manifolds with more than one end and spectral Ricci bounds.
result New insights into Li-Wang's theory and applications to stable and δ-stable minimal hypersurfaces.