arXiv research

Ricci flow preserves positive sectional curvature on homogeneous spheres

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…

Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.

In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional c…

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, an…

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.

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

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

New theorem links quaternionic-Kähler manifolds to symmetric spaces.

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

No conformal product structures on compact manifolds with constant curvature.

Ricci flow can change metrics with positive curvature to those without.

The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.

The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.

Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.

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

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

Extends Perelman's theorem to positive intermediate curvature conditions.

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7…

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

We show that the indices of certain twisted Dirac operators vanish on a $Spin$-manifold $M$ of positive sectional curvature if the symmetry rank of $M$ is $\geq 2$ or if the symmetry rank is one and $M$ is two connected. We also give examples of simply connected manifolds of positive Ricci curvature which do not admit …

Proves existence of infinite order elements in curvature spaces.

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

Curvature tensors can always be matched to a metric tensor under certain conditions.

We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.

We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.

Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature.

The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…

New manifolds found with almost everywhere positive curvature.

Positive curvature forces foliation leaf spaces to have boundaries.

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomor…

In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming …

Introduces positivity for $(1,1)$ classes in foliated manifolds.

We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterization for a dense open subset of the space of operators in dimension $4$. We also briefly examine higher-dimentional curvature operators.

Sharp lower bound for curvature in Kähler manifolds.

New findings on curvature and null spaces of Laplacians.

The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

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…

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…

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

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional cu…