Strong Uniqueness of the Ricci Flow

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.Comment: 21 page

On stationary solutions to the vacuum Einstein field equations

We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation $Ric(g_{M})=\lambda g_{M}$ with nonnegative cosmological constant $\lambda\geq 0$ is flat. When dim $\geq 5$, if the spacetime is assumed to be static additionally, we prove that its universal cover splits isometrically as a product of a Ricci flat Riemannian manifold and a real line.Comment: 24 page

On stationary solutions to the non-vacuum Einstein field equations

We derive a local curvature estimate for four-dimensional stationary solutions to the inheriting Einstein-Maxwell-Klein-Gordon equations. In particular, it implies that any such stationary geodesically complete solution with vanishing Poynting vector and proper coupling constants (like dark energy) is flat. We also generalize the result to higher dimensions.Comment: 20 page

On Euler characteristic and fundamental groups of compact manifolds

Let $M$ be a compact Riemannian manifold, $\pi:\widetilde{M}\rightarrow M$ be the universal covering and $\omega$ be a smooth $2$-form on $M$ with $\pi^*\omega$ cohomologous to zero. Suppose the fundamental group $\pi_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $1$-form $\eta$ on $\widetilde M$ of linear (resp. bounded) growth such that $\pi^*\omega=d \eta$. As applications, we prove that on a compact Kahler manifold $(M,\omega)$ with $\pi^*\omega$ cohomologous to zero, if $\pi_1(M)$ is $\mathrm{CAT}(0)$ or automatic (resp. hyperbolic), then $M$ is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic $(-1)^{\frac{\dim_\mathbb{R} M}{2}}\chi(M)\geq 0$ (resp. $>0$).Comment: 22 page

Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.Comment: 40 page

Euler characteristic numbers of space-like manifolds

In this note, we prove that if a compact even dimensional manifold $M^{n}$ with negative sectional curvature is homotopic to some compact space-like manifold $N^{n}$, then the Euler characteristic number of $M^{n}$ satisfies $(-1)^{\frac{n}{2}}\chi(M^{n})>0$. We also show that the minimal volume conjecture of Gromov is true for all compact even dimensional space-like manifolds

Volume Growth and Curvature Decay of Positively Curved K\"{a}hler manifolds

In this paper we obtain three results concerning the geometry of complete noncompact positively curved K\"{a}hler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact K\"{a}hler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompact K\"{a}hler manifold with positive bisectional curvature decays at least linearly in the average sense. The third result is concerned with the relation between the volume growth and the curvature decay. We prove that the curvature decay of a complete noncompact K\"{a}hler manifold with nonnegative curvature operator and with the maximal volume growth is precisely quadratic in certain average sense.Comment: 44 page

A Conformally Invariant Classification Theorem in Four Dimensions

In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.Comment: 18 pages. we supplement the reducible case b) in rigidity theorem 1.6 and add more references in the new versio

Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.Comment: 25 page

Uniqueness of the Ricci Flow on Complete Noncompact Manifolds

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.Comment: 33 pages (Previous version has some typing errors, the present one is correct.