arXiv research

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature $K<c<0$ and Ricci curvature $Ric > d$, where $c$ and $d$ are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe …

The paper extends Gray's result to quaternion-Kähler manifolds.

New results on tori restrict sectional curvature when Ricci curvature is negative and bounded.

New compact K-E manifolds with negative curvature found.

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

The study proves the non-existence of certain Kähler metrics with specific curvature properties.

The paper proves properties of manifolds with negative holomorphic sectional curvature.

Directly proves Wu's theorem on negative curvature metrics.

Negative curvature implies ample canonical bundle on manifolds.

The cross curvature flow preserves negative sectional curvature in 3-manifolds for all time.

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

Ricci flow deforms metrics with positive curvature to include negative curvature.

A new proof connects curvature negativity to bundle positivity using Kähler-Ricci flow.

Study shows Kähler-Ricci flow on manifolds with negative curvature converges to a Kähler-Einstein metric.

Locally symmetric metrics on 4-manifolds with non-negative curvature.

Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.

Compact complex manifolds with specific curvature properties have positive canonical bundles.

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

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently smal…

Motivated to study the geometry of the exotic spheres constructed in [5], we derive a necessary condition for non-negative sectional curvature in certain total spaces of Riemannian submersions with totally geodesic fibers. In particular, we prove that the bundles in [5] and [1] have sections of negative curvature.

Generalizes DeTurck's theorem to non-compact manifolds.

Two remarks on curvature properties of Kähler manifolds.

The sectional curvature of the Weil-Petersson metric on Teichmuller space is known to be negative. We show that this Weil-Petersson sectional curvature is not pinched from above by any negative constants, i.e., there is no negative upper bound.

In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…

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

Study Killing forms on negatively curved manifolds, introducing generalized vector cross products.

We give a geometric obstruction to the non-negativity of the sectional curvature in the total spaces of certain Riemannian submersions with totally geodesic fibers; applications of this obstruction to several examples are given.

Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

The study examines symmetries in spaces with positive or non-negative curvature.

New rigidity theorem for manifolds with specific negative curvature.

The paper proves conditions for Kähler manifolds with negative curvature.

Study on moduli spaces of non-negative curvature metrics on manifolds.

We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.

We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas…

Anosov geodesic flow proven in non-compact manifolds with negative curvature.

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal sectional curvatures are locally equivalent to canal hypersurfaces in Euclidean spa…

The study shows that certain curved solitons are flat.

We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.

In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms…

Odd GKM-manifolds with non-negative curvature split cohomology.

Conditions ensure constant curvature in negatively curved manifolds.

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the self-dual part of Weyl tensor, then it will be shown that there is no metric $g$ on $S^2 \times S^2$ with both (i) $K > 0$ and (ii) ${1/6} s - W_+ \ge 0$. We also investigate o…

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

New findings on stable minimal hypersurfaces in curved 4-manifolds.

The paper constructs metrics with negative curvature on complex manifolds.