arXiv research

Survey on two non-Kähler geometry conjectures.

Proves section conjecture for curves and surface bundles over various fields.

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.

Proves a conjecture for Calabi-Yau manifolds.

Researchers confirm conjecture for complex nilmanifolds in higher dimensions.

This paper generalizes biharmonic Riemannian submersions to higher dimensions.

Verify conjecture for special Hermitian manifolds.

The Strominger conjecture is confirmed for compact Hermitian manifolds in 2D and special higher dimensions.

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

The article confirms a complex geometry conjecture for a specific type of manifold.

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…

Shows Anosov flows with genus one sections, supporting a conjecture.

Researchers prove a complex geometric conjecture about certain manifolds.

The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.

The survey is devoted to Toponogov's conjecture, that {\it if a complete simply connected Riemannian manifold with sectional curvature $\le 4$ and injectivity radius $\ge π/2$ has extremal diameter $π/2$, then it is isometric to CROSS}. In Section 1 the relations of problem with geodesic foliations of a round sphere ar…

Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.

Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.

A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry.

Proves Stolz conjecture for specific types of manifolds.

The paper confirms a conjecture for Bismut torsion parallel metrics.

Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.

The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.

The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.

Proves stability condition for Lagrangian sections in toric weak Fano manifolds.

The note confirms a conjecture for specific Lie groups.

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with …

Minimal hypersurfaces are the only $H$-tensional in 4D space forms.

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

In this paper we define the bi-orthogonal sectional curvature and we present two modified Yamabe invariants for compact 4-dimensional manifolds. In particular we obtained a relationship between one of these invariants and a Hopf conjecture.

Falsehood of Pólya's conjecture for spheres shown.

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's eq…

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.

We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.

In this paper we review the development and recent results of the Siu-Yang conjecture which is that every Kähler-Einstein compact complex manifold of complex dimension two with negative sectional curvature is biholomorphic to a compact quotient of the complex 2-ball.

We investigate a new property for compact Kahler manifolds. Let X be a Kahler manifold of dimension n and let H^{1,1} denote the (1,1) part of its real second cohomology. On this space, we have an degree n form given by cup product. Let K denote the open cone of Kahler classes in H^{1,1}, and K_1 the level set consisti…

The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature

Introduces arithmetic analogues of Orr invariants and spaces for absolute Galois groups.

In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold m…

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on $S^2 \times S^2$ D.Gromoll and K.Tapp in [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on $S^2 \times R^3$): "The boundary $S^2\times S^2…

The article confirms a conjecture for solvmanifolds with complex commutator.

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section) which meets every leaf orthogonally. In addition the set of regular points is open and dense in ea…

A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to …

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection be…

Proves a 1930s Hopf conjecture about positive curvature manifolds.

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.