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A VARIATIONAL APPROACH FOR COMPACT HOMOGENEOUS EINSTEIN MANIFOLDS
"... Einstein metrics of volume 1 on a closed manifold can be characterized variationally as the critical points of the Hilbert action [Hi], which associates to each Riemannian metric of volume 1 the integral of its scalar curvature. As is wellknown, the gradient vector of the Hilbert action with respec ..."
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Einstein metrics of volume 1 on a closed manifold can be characterized variationally as the critical points of the Hilbert action [Hi], which associates to each Riemannian metric of volume 1 the integral of its scalar curvature. As is wellknown, the gradient vector of the Hilbert action with respect to the natural L2 metric is precisely the negative of the traceless Ricci tensor. Consequently, if G is a compact group of diffeomorphisms of the manifold, then unit volume Ginvariant Einstein metrics are the critical points of the restriction of the Hilbert action to the subspace of unit volume Ginvariant metrics. In this paper we shall exploit this principle to study homogeneous Einstein metrics from a variational viewpoint. Suppose G is a compact Lie group and H is a closed subgroup so that M = G/H is connected. The restriction of the Hilbert action to the finitedimensional space MG 1 of unit volume Ginvariant metrics is then just the scalar curvature function S: MG 1 → R that associates to each invariant metric its scalar curvature (a constant), and Ginvariant Einstein metrics are precisely the critical points of S. In the homogeneous case, the Einstein equations are algebraic, and many authors (see, e.g., references in [Bes], [Wa]) have made substantial progress in solving these equations, often explicitly.
Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates
, 2003
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Convergence of KählerRicci flow on Fano manifolds
 II, J. Reine Angew. Math
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MANIFOLDS WITH QUADRATIC CURVATURE DECAY AND FAST VOLUME GROWTH
, 2002
"... Abstract. We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms. 1. ..."
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Abstract. We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms. 1.
SMOOTH YAMABE INVARIANT AND SURGERY
, 2008
"... We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}. ..."
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We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}.
Injectivity Radius of Lorentzian Manifolds
 COMMUN. MATH. PHYS.
, 2007
"... Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geome ..."
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Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) –where p is a point of the manifold and T is a futureoriented timelike unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity, our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.
Finiteness theorems for nonnegatively curved vector bundles
 Duke Math. J
"... We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of “bounded geometry.” 1. ..."
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We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of “bounded geometry.” 1.
Desingularization of Coassociative 4folds with Conical Singularities
, 2006
"... Given a coassociative 4fold N with a conical singularity in a ϕclosed 7manifold M (a manifold endowed with a distinguished closed 3form ϕ), we construct a smooth family, {N ′ (t) : t ∈ (0, τ)} for some τ> 0, of (smooth, nonsingular,) compact coassociative 4folds in M which converge to N in t ..."
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Given a coassociative 4fold N with a conical singularity in a ϕclosed 7manifold M (a manifold endowed with a distinguished closed 3form ϕ), we construct a smooth family, {N ′ (t) : t ∈ (0, τ)} for some τ> 0, of (smooth, nonsingular,) compact coassociative 4folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4fold in R 7, dilated by t, then deforming the resulting compact submanifold of M to the required coassociative 4fold. 1
Overview of the proof of the Bounded L2 Curvature Conjecture
"... This memoir contains an overview of the proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einsteinvacuum equations depends only on the L2norm of the curvature and a lower bound on the volume radius of the corresponding in ..."
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This memoir contains an overview of the proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einsteinvacuum equations depends only on the L2norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to another, more subtle, scaling tied to its causal geometry. Indeed, L2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 + 1 (based on Strichartz estimates) were obtained in [2], [3], [49], [50], [19] and optimized in [20], [36], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. The entire proof is obtained in the following sequence of papers S. Klainerman, I. Rodnianski, J. Szeftel, The bounded L2 curvature conjecture. arXiv:1204.1767,