Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
CA-PCA improves manifold dimension estimation by accounting for curvature.
problem Estimating the dimension of manifolds in high-dimensional data.
method Develops CA-PCA, a local PCA method calibrated with a quadratic embedding to account for curvature.
result Improves manifold dimension estimation in various settings.
Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.
problem Understanding Kodaira dimensions on almost complex manifolds.
method Using pseudoholomorphic pluricanonical maps, defining new dimensions, and applying probabilistic combinatorics.
result Almost complex structures with top Kodaira dimension are integrable, and for compact 4-manifolds, they have elliptic fibration structures.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
The paper constructs multisections for m-spun 3-manifolds in higher dimensions.
problem Finding multisections for higher-dimensional manifolds.
method Extending the concept of spun 3-manifolds to higher dimensions, constructing multisections and diagrams.
result Infinitely many examples of non-diffeomorphic multisected manifolds in all dimensions.
New manifold type PNDP-manifold defined with Einstein warped product structure.
problem Defining manifolds with non-standard dimensions.
method Einstein warped product manifold with special base and fiber structures.
result PNDP-manifolds are Einstein warped product manifolds with specific base and fiber properties.
Study shows infinite families of manifolds with nonnegative curvature.
problem Finding nonnegatively curved metrics on manifolds.
method Exhibited infinite families of manifolds with specific properties.
result Moduli space of nonnegatively curved metrics has infinitely many components.
The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.
problem Understanding the asymptotic dimensions of manifolds and spaces with geometric decompositions.
method Analyzing the fundamental groups and using geometric decompositions to derive asymptotic dimension bounds.
result Asymptotic dimensions of certain manifolds and spaces are bounded and equal to specific values.
The paper explores Kodaira dimension on almost complex manifolds.
problem Understanding Kodaira dimension on non-integrable almost complex manifolds.
method Generalization of Kodaira dimension to almost complex manifolds and study of its behavior under deformations.
result Kodaira dimension is invariant under holomorphic deformations for smooth projective manifolds but not for non-projective manifolds.
The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.
problem Classifying Bismut Kähler-like manifolds in specific dimensions.
method Structural theorems and proving conjectures about BKL manifolds.
result Complete classifications of BKL manifolds in dimensions 4 and 5.
The study proves curvature bounds for hyperkähler manifolds.
problem Proving curvature invariants of hyperkähler manifolds.
method Analytical proof in complex dimension four, experimental proof in higher dimensions, verification for known manifolds.
result The conjectured curvature invariants are proven to be positive/negative for all known hyperkähler manifolds up to dimension eight.
We consider complex Kobayashi-hyperbolic manifolds of dimension n≥2 for which the dimension of the group of holomorphic automorphisms is equal to n2−1. We give a complete classification of such manifolds for n≥3 and discuss several examples for n=2.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
Extends Hodge theory to nearly Kähler manifolds of arbitrary dimensions.
problem Generalize Hodge-theoretic results to nearly Kähler manifolds of arbitrary dimensions.
method Apply Hodge theory to nearly Kähler manifolds of arbitrary dimensions, relating Hodge numbers to Betti numbers.
result Hodge numbers of compact nearly Kähler manifolds are related to Betti numbers in the same way as on a compact Kähler manifold.
Estimates manifold dimension using local graph structure.
problem Estimating the intrinsic dimension of manifolds from data.
method Regression on local PCA coordinates, focusing on local graph structure.
result Proposed QE and TLS estimators outperform existing methods.
Estimates manifold dimension from random samples.
problem Estimating the dimension of a manifold from random samples.
method Explicit theoretical and heuristic bounds for data set size.
result Data set needs to be sufficiently large for accurate dimension estimation.
Study the intersection form on Kähler manifolds of dimension 4 and above.
problem Understand the intersection form on higher-dimensional Kähler manifolds.
method Investigate fundamental properties and applications to birational geometry.
result Present open problems in the relationship between birational invariants and topological invariants.
Counterexample disproves HK-conjecture for flat manifolds of dimension 9 and higher.
problem Disproving the HK-conjecture for flat manifolds of various dimensions.
method Using flat manifolds of dimension 9 and higher, constructing counterexamples.
result Minimal dimension 9 is required for counterexamples, and higher dimensions are possible.
New research shows that the dimension gap between intrinsic and ambient dimensions affects adversarial vulnerability of machine learning models.
problem The mystery of adversarial attacks on machine learning models.
method Introducing two types of adversarial attacks and proving their relationship to the dimension gap.
result The dimension gap between intrinsic and ambient dimensions makes clean-trained models more vulnerable to off-manifold adversarial perturbations.
Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.
problem Understanding the Kodaira dimension of almost complex manifolds with SU(m)-structures.
method Introduced almost complex structure of splitting type and associated SU(m)-structure. Provided constructions for non-invariant almost complex structures with specific Kodaira dimensions.
result Found non-invariant almost complex structures with Kodaira dimensions 0 and -∞.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
New non-cobordant hyperbolic manifolds found in certain dimensions.
problem Identifying closed hyperbolic manifolds that are not cobordant.
method Using the cobordism class and fixed point set of an involution, combined with a geodesic embedding.
result Existence of non-cobordant closed hyperbolic manifolds in dimensions not of the form 4m+3. A diffusion model estimates data manifold dimension by tracking likelihood increases.
problem Estimating the intrinsic dimension of data manifolds.
method Trained diffusion model approximates score function, revealing manifold directionality.
result Diffusion model provides an approximation of the tangent space's dimension.
Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.
problem Understanding rolling dynamics of 2D and 3D manifolds with constraints.
method Modeling rolling as a control affine system on a fibered space Q, analyzing reachable sets.
result Identified possible dimensions of non-open rolling orbits: 2, 5, 6, 7.
Computational techniques calculate dimensions of complex structures.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
We construct a counterexamples in dimensions n>3 to Gromov's conjecture \cite{Gr1} that the macroscopic dimension of rationally essential n-dimensional manifolds equals n.
The study finds limits on dimensions of certain scales and fields for conformal manifolds.
problem Limits on dimensions of almost Einstein scales and normal conformal Killing fields for conformal manifolds.
method Analyzes the submaximal dimensions of spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds, considering different signatures and dimensions.
result Upper bounds on dimensions of almost Einstein scales and normal conformal Killing fields are determined, with examples provided for submaximal dimensions.
In this note we prove that a generic Riemannian manifold of dimension ≥3 does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analog of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symm…
Study of higher-dimensional contact manifolds and their properties.
problem Understanding contact manifolds in higher dimensions.
method Topological and symplectic methods, including open books.
result Many contact manifolds can be realized as iterated planar contact manifolds.
We show that every Sasakian manifold in dimension 2k+1 is locally generated by a free real function of 2k variables. This function is a Sasakian analogue of the Kähler potential for Kähler geometry. It is also shown that every locally Sasakian-Einstein manifold in 2k+1 dimensions is generated by a locally Kähler-…
We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension n=3, whose group of holomorphic automorphisms has dimension n2+1 and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel s…
The paper proves scalar curvature decay for uniformly contractible manifolds with finite asymptotic dimension.
problem Proving decay of scalar curvature for uniformly contractible manifolds with finite asymptotic dimension.
method Using index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control, and Lipschitz control for topological K-theory of finite dimensional simplicial complexes.
result The scalar curvature decays to zero at a rate depending only on the contractibility radius and the diameter control of the asymptotic dimension.
Proves Penrose inequality in all dimensions for specific manifolds.
problem Proving Penrose inequality in arbitrary dimensions for certain manifolds.
method Extends Bray's conformal-flow method to higher dimensions, dealing with singular outer-minimizing enclosures.
result Proves the Riemannian Penrose inequality in arbitrary dimensions.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.
problem Investigating Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds.
method Analyzing compact almost Hermitian manifolds in the Gray-Hervella class and Hermitian manifolds with nonnegative scalar curvature.
result For compact almost Hermitian manifolds with nonnegative scalar curvature, the Kodaira dimension is either -∞ or 0, with specific conditions.
This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost com…
We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension,…
The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concent…
The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 whose holomorphic automorphism group has dimension n2−3. This result complements existing classifications for automorphism group dimension n2−2 (which is in some sense critical) and greater.
New contact manifolds with many fillings found.
problem Contact manifolds with infinite fillings in odd dimensions.
method Spinal open books to construct contact manifolds.
result Contact manifolds with infinitely many different Weinstein fillings constructed.
Study shows how to effectively predict functions on manifolds using kernel methods.
problem Regression on manifolds with limited data.
method Reproducing kernel Hilbert space methods, Weyl law, effective dimension.
result Kernel regression estimator yields minimax-optimal error bounds controlled by effective dimension.
In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic (0,q)-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we e…
Paper proves rigidity for Einstein metrics in high dimensions.
problem Einstein metrics on high-dimensional manifolds.
method Liouville type rigidity result for asymptotically hyperbolic metrics.
result Established a rigidity theorem for d≥5. New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
Classifies hyperbolic manifolds with specific automorphism groups.
problem Classifying Kobayashi-hyperbolic manifolds with high-dimensional automorphism groups.
method Analyzes manifolds of dimension n≥2 with automorphism groups of dimensions n2−7 or n2−8. result Completes the classification for automorphism groups n2−7 and n2−8. The study generalizes a specific geometric correspondence to higher dimensions.
problem Understanding nondegenerate lines on holomorphic contact manifolds.
method Analyzing nondegenerate lines and corresponding distributions on higher-dimensional manifolds.
result A generalization of the (2,3,5)-distributions to higher dimensions.