Nonnegative Sectional Curvature Research Articles

Ancient Solutions of Ricci Flow with Type I Curvature Growth

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a simply connected, noncompact, κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution with nonnegative sectional curvature necessarily splits a Euclidean factor. It follows that a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}-noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.

Some results on four-manifolds with nonnegative biorthogonal curvature

Let (M,g) be a 4-dimensional compact oriented Riemannian manifold with nonnegative biorthogonal curvature and W− be the anti-self-dual component of the Weyl curvature tensor W. If M has constant scalar curvature and |W−| of M is constant, or if M has harmonic Weyl tensor and detW− or |W−| of M is constant, then we give two classification theorems for M. As two applications, a 4-dimensional compact oriented Einstein manifold with nonnegative sectional curvature whose detW− or |W−| is constant is isometric to one of S4, CP2, S2×S2 or a flat manifold; a 4-dimensional compact oriented self-dual Riemannian manifold with positive sectional curvature and constant scalar curvature either is conformal to S4, CP2, or is diffeomorphic to CP2♯CP2, CP2♯CP2♯CP2.

The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature

The logarithmic Sobolev inequality for a submanifold in manifolds with asymptotically nonnegative sectional curvature

Diameter Estimate in Geometric Flows

We prove the upper and lower bounds of the diameter of a compact manifold (M,g(t)) with dimRM=n(n≥3) and a family of Riemannian metrics g(t) satisfying some geometric flows. Except for Ricci flow, these flows include List–Ricci flow, harmonic-Ricci flow, and Lorentzian mean curvature flow on an ambient Lorentzian manifold with non-negative sectional curvature.

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.

Parallel Codazzi tensors with submanifold applications

AbstractA decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

Index theory and deformations of open nonnegatively curved manifolds

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.

The Gauss–Bonnet inequality beyond aspherical conjecture

Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group \(H_k\) for all \(k\ge 3\), either it is flat or it has Gauss–Bonnet quantity (defined by (1.3)) no greater than \(8\pi \). In the second case, the equality for Gauss–Bonnet quantity yields that the universal covering splits as the Riemannian product of a 2-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical 2-systole estimate unifies the results from Bray et al. (Comm Anal Geom 18(4):821–830, 2010) and Zhu (Proc Am Math Soc 148(8):3479–3489, 2020).

The moduli space of nonnegatively curved metrics on quotients of S2 × S3 by involutions

We show that for an orientable non-spin manifold with fundamental group Z2 and universal cover S2×S3, the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on S3×S3 or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative η invariant of the spinc Dirac operator computed by means of a Lefschetz fixed point theorem.

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of S^7-bundles over S^8 which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional curvature on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Unbounded Pontryagin numbers on nonnegatively curved spin manifolds

We prove that any rational linear combination of Pontryagin numbers that does not factor through the universal elliptic genus is unbounded on connected closed spin manifolds of nonnegative sectional curvature.

Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics

We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.

Vector bundles of non-negative curvature over cohomogeneity one manifolds

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization.Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.

Sobolev Inequalities in Manifolds with Nonnegative Curvature

AbstractWe prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael‐Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.

Moduli space of nonnegatively curved metrics on manifolds of dimension 4k + 1

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

Local symmetry rank bound for positive intermediate Ricci curvatures

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

Minimal Legendrian submanifolds of $\mathbb S^{9}$ with nonnegative sectional curvature

Minimal Legendrian submanifolds of $\mathbb S^{9}$ with nonnegative sectional curvature