Manifolds Of Positive Ricci Curvature Research Articles

Generalized surgery on Riemannian manifolds of positive Ricci curvature

The surgery theorem of Wraith states that positive Ricci curvature is preserved under surgery if certain metric and dimensional conditions are satisfied. We generalize this theorem as follows: instead of attaching a product of a sphere and a disc, we glue a sphere bundle over a manifold with a so-called core metric, a type of metric which was recently introduced by Burdick to construct metrics of positive Ricci curvature on connected sums. As applications we extend a result of Burdick on the existence of core metrics on certain sphere bundles and obtain new examples of 6-manifolds with metrics of positive Ricci curvature.

A new explicit complete Kähler metric on ℂ n

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on [Klembeck P. A complete Kähler metric of positive curvature on . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature.

A note on some remarkable differential equations on a Riemannian manifold

The Fischer-Marsden conjecture asserts that an n-dimensional compact manifold admitting a nontrivial solution of the so-called Fischer-Marsden differential equation is necessarily an Einstein space. If this were true, then a classical theorem of Obata would imply that the underlying manifold is either a standard sphere or a Ricci flat space. Although counterexamples to this conjecture have been found by Kobayashi and Lafontaine, it has recently been proved by Cernea and Guan that the Fischer-Marsden conjecture holds, provided that the space of nonconstant solutions of the Fischer-Marsden equation is of dimension at least n, the authors actually proving that in this case (M,g) is nothing but a standard sphere. The main aim of this article is to show that any compact Riemannian manifold of positive Ricci curvature that admits a nontrivial concircular vector field with the potential function satisfying the Fischer-Marsden equation must be isometric to a standard sphere and the converse is also valid. Moreover, we prove that the existence of a nontrivial solution to another remarkable differential equation on Riemannian manifolds, namely the stationary Schrödinger equation, it also leads to a characterization of the sphere, provided that some pinching conditions are satisfied.

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers

Manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type

Manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type

Local stability of Einstein metrics under the Ricci iteration

We provide a sufficient condition for the local stability of closed Einstein manifolds of positive Ricci curvature under the Ricci iteration in terms of the spectrum of the Lichnerowicz Laplacian acting on divergence-free tensor fields. We use this result to consider the stability of several Einstein manifolds under the Ricci iteration, including symmetric spaces of compact type.

On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of k -positive Ricci curvature and prove that for a projective n -dimensional Kähler manifold of k -positive Ricci curvature the MRC fibration has generic fibers of dimension at least n-k+1 . We also establish an analogous result for projective Kähler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.

A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.

Reductions of minimal Lagrangian submanifolds with symmetries

Let M be a Fano manifold equipped with a Kahler form $$\omega \in 2\pi c_1(M)$$ and K a connected compact Lie group acting on M as holomorphic isometries. In this paper, we show the minimality of a K-invariant Lagrangian submanifold L in M with respect to a globally conformal Kahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $$L_0=L/K$$ in a Kahler quotient $$M_0$$ with respect to the Hsiang–Lawson metric. Furthermore, we give some examples of Kahler reductions by using a circle action obtained from a cohomogenenity one action on a Kahler–Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.

Min–max hypersurface in manifold of positive Ricci curvature

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques-Neves in their proof of the Willmore conjecture \cite{MN12}, is a bridge to connect sweepouts appearing naturally in geometry to sweepouts used in the min-max theory. Our result removes a critical assumption of \cite{MN12}, called the no mass concentration condition, and hence confirms a conjecture by Marques-Neves in \cite{MN12}.

Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

In this paper we exhibit deformations of the hemisphere [Formula: see text], [Formula: see text], for which the ambient Ricci curvature lower bound [Formula: see text] and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.

On the Yamabe constants of [formula omitted] and [formula omitted

We compare the isoperimetric profiles of S2×R3 and of S3×R2 with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of S2×R3 and S3×R2. Explicitly we show that Y(S3×R2,[g03+dx2])>(3/4)Y(S5) and Y(S2×R3,[g02+dx2])>0.63Y(S5). We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.

On the moment-angle manifolds of positive Ricci curvature

We construct Riemannian metrics of positive Ricci curvature on some moment-angle manifolds. In particular, we construct a nonformal moment-angle Riemannian manifold of positive Ricci curvature.

Harmonic maps from compact Kähler manifolds with positive scalar curvature to Kähler manifolds of strongly seminegative curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. If one reduces the assumption on the Ricci curvature to one on th

Kähler Manifolds Positive Ricci Curvature

The aim of this paper is to give an overview of some results obtained in the field of Kähler manifolds of positive Ricci curvature.

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.

Schouten tensor and some topological properties

The notion of positive curvature plays an important role in differential geometry. The existence of such a metric often implies some topological properties of the underlying manifold. A typical example is the Bochner vanishing theorem on manifolds of positive Ricci curvature. In this paper, we consider Riemannian metrics with certain type of positivity on the Schouten tensor. This notion of curvature was introduced by Viaclovsky [18] which extends the notion of scalar curvature. Let (M,g) be an oriented, compact and manifold of dimension n > 2. And let Sg denote the Schouten tensor of the metric g, i.e.,

SCALAR CURVATURE, KILLING VECTOR FIELDS AND HARMONIC ONE-FORMS ON COMPACT RIEMANNIAN MANIFOLDS

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

Metrics of positive Ricci curvature on bundles

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 E × ℝ p admits a complete metric of positive Ricci curvature for all large p.

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given m and C there exists a positive constant e = e(m, C) > 0 such that any m-dimensional complete Riemannian manifold with Ricci curvature Ricc > m-1, sectional curvature K π-e is Lipschitz close and diffeomorphic to the standard unit m-sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.