 
Summary: HitchinThorpe Inequality for Noncompact Einstein
4Manifolds
Xianzhe Dai
Guofang Wei
November 15, 2006
Abstract
We prove a HitchinThorpe inequality for noncompact Einstein 4manifolds 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.
1 Introduction
Einstein manifolds are important both in mathematics and physics. They are good candidates for
canonical metrics on general Riemannian manifolds and they are the vacuum solutions of Einstein's
field equation (with cosmological constant) in general relativity. As a result, they are extensively
studied (Cf. [8], [25]).
Besides space forms and irreducible symmetric spaces, a large class of compact Einstein manifolds
is given by the solution of Calabi conjecture. Namely, a compact Kšahler manifold with a nonpositive
first Chern class admits a KšahlerEinstein metric [36], [6]. In the case of positive first Chern class,
the work of [30, 29] says that CP2
