Nonnegative Sectional Curvature Research Articles

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

A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

On the topology of moduli spaces of non-negatively curved Riemannian metrics

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

On pull-backs of the universal connection

AbstractNarasihman and Ramanan proved in [Amer. J. Math. 83(1961), 563–572] that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tensor, and (3) induces a connection metric with nonnegative sectional curvature on the vector bundle (or positive sectional curvature on the sphere bundle).

Harnack inequality for nonlocal operators on manifolds with nonnegative curvature

We establish the Krylov–Safonov Harnack inequalities and Hölder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity estimates for these operators which recover the classical estimates for second order local operators as limits.

Odd-Dimensional GKM-Manifolds of Non-Negative Curvature

Abstract Let $M$ be a closed, odd GKM$_3$ manifold of non-negative sectional curvature. We show that in this situation one can associate an ordinary abstract GKM$_3$ graph to $M$ and prove that if this graph is orientable, then both the equivariant and the ordinary rational cohomology of $M$ split off the cohomology of an odd-dimensional sphere.

Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85.

Correction to “Moduli spaces of nonnegative sectional curvature and non-unique souls”, J. Diff. Geom. 89 (2011), no. 1, 49–85.

On the entropy of parabolic Allen–Cahn equation

We define a local (mean curvature flow) entropy for Radon measures in \mathbb{R}^n or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen–Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen–Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen–Cahn equation has unit density for all future time.

Remarks on compact quasi-Einstein manifolds with boundary

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $\Bbb{S}^n_+,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.

A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and infty -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Liouville-type theorems on manifolds with nonnegative curvature and strictly convex boundary

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.

Fake $13$-projective spaces with cohomogeneity one actions

We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, that are fake real $13$-dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard $\mathbb{R}\mathrm{P}^{13}$. As observed by F. Wilhelm and the second named author in [RW], the Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on Shimada's exotic $15$-spheres descend to the cohomogeneity one actions on the $P^{13}$s. We prove that the $P^{13}$s are diffeomorphic to well-known $\mathbb{Z}_2$ quotients of certain Brieskorn varieties, and that the Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on the $P^{13}$s are equivariantly diffeomorphic to well-known actions on these Brieskorn quotients. The $P^{13}$s are octonionic analogues of the Hirsch-Milnor fake $5$-dimensional projective spaces, $P^{5}$s. K. Grove and W. Ziller showed that the $P^{5}$s admit metrics of non-negative curvature that are invariant with respect to the Davis $\mathsf{SO}(2)\times \mathsf{SO}(3)$-cohomogeneity one actions. In contrast, we show that the $P^{13}$s do not support $\mathsf{SO}(2)\times \mathsf{G}_2$-invariant metrics with non-negative sectional curvature.

Complete linear Weingarten spacelike submanifolds immersed in the anti-de Sitter space

We deal with $n$-dimensional complete linear Weingarten spacelike submanifolds having nonnegative sectional curvature and immersed in the anti-de Sitter space $\mathbb H_p^{n+p}$ of index $p$ with parallel normalized mean curvature vector field. We recall

Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy R P 5 \mathbb {R} P^5 has infinitely many path components. We also show that in each dimension 4 k + 1 4k+1 there are at least 2 2 k 2^{2k} homotopy R P 4 k + 1 \mathbb {R} P^{4k+1} ’s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4 k + 3 ≥ 7 4k+3\geq 7 .

Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature.