Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(n\ge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>\sigma _{n}K_{\max }$ , where $\sigma _n\in (\frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(\ge {\!\!4})$ -dimensional compact Einstein manifold satisfies $K_{\min }\ge \eta _n R_0,$ where $\eta _n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $\eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(\ge {\!\!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $c\ge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.     

Einstein 4-manifolds and nonpositive isotropic curvature

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space.     

Geography of 4-manifolds with positive scalar curvature

We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov’s observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.     

Noncompact 4-manifolds with uniformly positive isotropic curvature

In this paper, we study Ricci flow on noncompact 4-manifolds with uniformly positive isotropic curvature and with no essential imcompressible space form. That means there is positive lower bound of isotropic curvature and bounded geometry. Then by Perelman's technique, we can analyze the structures of such manifolds.     

Finite isometry groups of 4-manifolds with positive sectional curvature

Let M be an oriented compact Riemannian 4-manifold with positive sectional curvature. Let G be a finite subgroup of the isometry group of M. We prove that, if G is a finite group of order , then

Rigidity for closed manifolds with positive curvature

Let M be an n-dimensional complete connected Riemannian manifold with sectional curvature sec(M) ≥ 1 and radius rad(M) > π/2. In this article, we show that M is isometric to a round n-sphere if for any x ∈ M, the first conjugate locus of x is a single point and if M contains a geodesic loop of length 2 · rad(M). We also show that the same conclusion is true if the conjugate value function at any point of M is a constant function. This work was done while the author was visiting MPI for Mathematics in Leipzig, Germany. The author is very grateful to MPI for Mathematics in Leipzig for its hospitality and CAPES.     

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ?secM?1. We prove that M is homeomorphic to a spherical space form if one of the following conditions holds:

(i)

The center of the fundamental group has index ?w(δ), a constant depending on δ;

(ii)

and the fundamental group is a non-cyclic group of order ?C, a constant;

(iii)

The volume is less than ?(δ) and the fundamental group is either isomorphic to a spherical 5-space group or has an odd order, and it has a center of index ?w, a constant.

     

Rigidity of Riemannian manifolds with positive scalar curvature

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the \(L^{\frac{n}{2}}\)-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

Examples of 4-manifolds with almost nonpositive curvature

We construct an infinite family of non-homeomorphic 4-manifolds with almost nonpositive sectional curvature whose universal covering space is not contractible. As a consequence, these manifolds do not support metrics with nonpositive sectional curvature. To achieve this, we use a generalization of Bavard's surgery construction, combined with an open book decomposition and knot theory.     

Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with specified asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

Simply connected open 3-manifolds with slow decay of positive scalar curvature

The goal of this paper is to investigate the topological structure of open simply connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is diffeomorphic to ${\mathbb{R}}^3$. Furthermore, using this result, we prove that any open simply connected 3-manifold M with ${\pi }_2(M)=\mathbb{Z}$ and a complete metric as above is diffeomorphic to ${\mathbb{S}}^2\times \mathbb{R}$.     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Almost Hermitian 4-manifolds of pointwise constant anti-holomorphic sectional curvature

We show that a 4-dimensional almost Hermitian manifold (M, J, g) is of pointwise constant anti-holomorphic sectional curvature if and only if (M, J, g) is self-dual with J-invariant Ricci tensor and K₁₂₁₂ = 0, where K is the complexification of the Riemannian curvature tensor.     

TheL 2 structure of moduli spaces of Einstein metrics on 4-manifolds

Partially supported by NSF Grants DMS 87-01137 and 89-01303