Proves almost flat manifolds with mixed curvature bounds.
problem Finding structures with mixed curvature bounds.
method Mixed curvature analogue of Gromov's almost flat manifolds theorem.
result Proves upper and lower curvature bounds for almost flat manifolds.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
The paper explores mixed curvature for Hermitian manifolds and its implications.
problem Investigating the properties of mixed curvature for Hermitian manifolds.
method Analyzing convex combinations of first Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant mixed curvature are Kähler unless specific conditions are met.
The paper studies constant kth-mixed curvature on Hermitian manifolds and finds self-duality and Kähler conditions.
problem Investigating constant kth-mixed curvature on Hermitian manifolds. method Analyzing Hermitian manifolds with convex combinations of Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant kth-mixed curvature are self-dual, and if k=2, the metric is Kähler. Study on Kähler manifolds with non-positive mixed curvature and its implications.
problem Characterizing properties of Kähler manifolds with non-positive mixed curvature.
method Analyzing the canonical line bundle properties based on curvature types.
result Canonical line bundle is nef, big, and ample under certain curvature conditions.
The paper proves conditions for projectivity and rational connectedness of complex manifolds with quasi-positive mixed curvature.
problem Conditions for projectivity and rational connectedness of complex manifolds with quasi-positive mixed curvature.
method Convex combination of Ricci curvature and holomorphic sectional curvature, proving projectivity and rational connectedness under specific curvature conditions.
result Compact complex manifolds with quasi-positive mixed curvature are projective and rationally connected under certain conditions.
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
Develops a new VAE for mixed-curvature latent spaces.
problem Lack of a generic approach for VAEs in curved latent spaces.
method Mixed-curvature Variational Autoencoder (MVAE) that trains on a product of constant curvature Riemannian manifolds.
result Generalizes Euclidean VAE to curved latent spaces and recovers Euclidean VAE when curvatures are zero.
We establish the cancellation of the first |2j-q| terms in the diagonal asymptotic expansion of the restriction to the (0, 2j)-forms of the Bergman kernel associated to the modified spin^c Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a (non necessarily holomorphic) complex vecto…
Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
Closed Riemannian manifolds with positive mixed sectional curvature
problem Constructing closed Riemannian manifolds with positive mixed sectional curvature
method Explicit construction using totally geodesic foliations
result Positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds
In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold (M,g) equipped with a vector field X. We define several functions (qth Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The n…
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
The paper improves L2-estimates for Dirac-Dolbeault operators on complex manifolds.
problem Improving L2-estimates for Dirac-Dolbeault operators on complex manifolds. method Generalized classical method to handle mixed curvature cases and provided bounds on error terms.
result Full asymptotic expansion for Bergman kernel obtained.
We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical (ρ,η)-system with respect to a (ρ,η)-spray, the Berwald (ρ,η)-…
Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension at least three that are not of Berwald type. The exis…
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
The paper introduces and studies a new type of submersion in Riemannian geometry.
problem Exploring new submersions in Riemannian geometry.
method Defining and studying Clairaut Riemannian warped product submersions.
result Established conditions for a Riemannian warped product submersion to satisfy the Clairaut condition.