1,173 research outputs found
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 be a
smooth complete solution to the Ricci flow on , with the
canonical Euclidean metric as initial data, then is trivial, i.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
with nonnegative cosmological constant is flat. When dim , 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 be a compact Riemannian manifold, be
the universal covering and be a smooth -form on with
cohomologous to zero. Suppose the fundamental group
satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we
show that there exists a smooth -form on of linear
(resp. bounded) growth such that . As applications, we
prove that on a compact Kahler manifold with
cohomologous to zero, if is or automatic (resp.
hyperbolic), then is Kahler non-elliptic (resp. Kahler hyperbolic) and the
Euler characteristic (resp.
).Comment: 22 page
Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow
Mean curvature flow evolves isometrically immersed base manifolds in the
direction of their mean curvatures in an ambient manifold . If the
base manifold 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
with negative sectional curvature is homotopic to some compact space-like
manifold , then the Euler characteristic number of satisfies
. 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.
- β¦