arXiv research

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

Holomorphic Euler number vanishes for certain Kähler manifolds.

For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.

Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.

The paper shows that certain manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.

We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E …

Let $M^n$ be a closed manifold of almost nonnegative sectional curvature and nonzero first de Rham cohomology group. For any $[θ] \in H^1_{dR}(M^n), [θ] \neq 0$, we show that the Morse- Novikov cohomology group $H^p(M^n, θ)$ vanishes for any $p$. A similar result holds for a closed manifold of almost nonnegative Ricci …

The paper proves fibration theorems for manifolds with almost nonnegative Ricci curvature.

We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold.

We prove that a fundamental group of codimension one nonnegative Ricci curvature C2-foliation of a closed Riemannian manifold is finitely generated and almost abelian, i.e. it contains abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of codimension one non…

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 ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The n…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

Study on extremizers for Sobolev inequality on curved manifolds.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

The paper constructs solutions to the Kähler-Ricci flow and studies complex structures.

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+εI \in C…

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…

The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.

For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group $π_1(M^n)$ can be generated by $C(n)$ generators. Inspired by their work, we give a quantitative proof of the above theorem and show that $C(n)\…

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.

The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

Survey on open manifolds with nonnegative Ricci curvature and open questions.

In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-Émery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).

Ancient Ricci flows with nonnegative curvature operator have bounded entropy.

The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.

Study classifies special 4D shapes with certain curvature.

Paper studies fundamental groups of certain Ricci solitons.

Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.

We study manifolds with almost nonnegative curvature operator (ANCO) and provide first examples of closed simply connected ANCO mannifolds that do not admit nonnegative curvature operator.

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $π:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $føπ$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume est…

Complete Ricci flow from singular 3D manifold with pseudolocality.

The study shows black hole horizons at low temperatures have limited topology.

Proves inequalities on curved spaces with positive curvature.

Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.

Flat open manifolds with full first Betti number have zero curvature.

The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.

Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to fo…

The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.

New topological restrictions found for spaces with nonnegative Ricci curvature.

Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.

The paper proves a theorem about splitting manifolds with specific curvature properties.