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Blowing up and desingularizing constant scalar curvature Kähler manifolds
"... Abstract. This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact ..."
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Cited by 33 (1 self)
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Abstract. This paper is concerned with the existence of constant scalar curvature Kähler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature Kähler metrics. We also consider the desingularization of isolated quotient singularities of compact orbifolds which already carry constant scalar curvature Kähler metrics.
Curvature and injectivity radius estimates for Einstein 4manifolds
 J. Amer. Math. Soc
"... It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2dimensional Einstein manifold, (M2, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1 ..."
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Cited by 31 (2 self)
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It is of fundamental interest to study the geometric and analytic properties of compact Einstein manifolds and their moduli. In dimension 2 these problems are well understood. A 2dimensional Einstein manifold, (M2, g), has constant curvature, which after normalization, can be taken to be −1, 0 or 1. Thus, (M2, g) is the quotient of a space form and the metric, g, is completely determined by the conformal structure. For fixed M2, the moduli space of all such g admits a natural compactification, the DeligneMumford compactification, which has played a crucial role in geometry and topology in the last two decades, e.g. in establishing GromovWitten theory in symplectic and algebraic geometry. In dimension 3, it remains true that Einstein manifolds have constant sectional curvature and hence are quotients of space forms. An essential portion of Thurston’s geometrization program can be viewed as the problem of determining which 3manifolds admit Einstein metrics. The moduli space of Einstein metrics on a 3dimensional manifold is also well understood. As a consequence of Mostow rigidity, the situation is actually simpler than in twodimensions.
Nonminimal scalarflat Kähler surfaces and parabolic stability
 Invent. Math
"... Abstract. A new construction is presented of scalarflat Kähler metrics on nonminimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank2 parabolically stable holomorphic bundles. This rather general construction is sh ..."
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Cited by 21 (5 self)
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Abstract. A new construction is presented of scalarflat Kähler metrics on nonminimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP 2 blown up at 10 suitably chosen points, admits a scalarflat Kähler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact selfdual 4manifolds. 1.
Canonical metrics on 3manifolds and 4manifolds
 Asian J. Math
"... In this paper, we discuss recent progress on the existence of canonical metrics on manifolds in dimensions 3 and 4, and the structure of moduli spaces of such metrics. The existence of a “best possible ” metric on a given closed manifold is a classical question in Riemannian geometry, ..."
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Cited by 18 (0 self)
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In this paper, we discuss recent progress on the existence of canonical metrics on manifolds in dimensions 3 and 4, and the structure of moduli spaces of such metrics. The existence of a “best possible ” metric on a given closed manifold is a classical question in Riemannian geometry,
Orbifold compactness for spaces of Riemannian metrics and applications
"... The CheegerGromov compactness theorem, cf. [C], [G], [CGv] states that the space of Riemannian nmanifolds (M n,g) satisfying the bounds R  ≤ Λ, vol ≥ v0, diam ≤ D, (1.1) is precompact in the C 1,α topology. Here R denotes the Riemann curvature tensor, vol the volume ..."
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Cited by 16 (1 self)
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The CheegerGromov compactness theorem, cf. [C], [G], [CGv] states that the space of Riemannian nmanifolds (M n,g) satisfying the bounds R  ≤ Λ, vol ≥ v0, diam ≤ D, (1.1) is precompact in the C 1,α topology. Here R denotes the Riemann curvature tensor, vol the volume
The Gradient Flow of �
, 2006
"... M Rm2 We study the gradient flow of the Riemannian functional F(g):= M Rm2. This flow corresponds to a fourthorder degenerate parabolic equation for a Riemannian metric. We prove that, as with the Ricci flow, the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we sh ..."
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Cited by 11 (5 self)
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M Rm2 We study the gradient flow of the Riemannian functional F(g):= M Rm2. This flow corresponds to a fourthorder degenerate parabolic equation for a Riemannian metric. We prove that, as with the Ricci flow, the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show shorttime existence using the DeTurck method. We prove L 2 derivative estimates of BernsteinBandoShi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.
Space of Ricci flows (I)
, 2009
"... In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control, Kähler Ricci flow and moduli space of gradient shrinking solitons. ..."
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Cited by 9 (2 self)
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In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control, Kähler Ricci flow and moduli space of gradient shrinking solitons.