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165
Ricci Flow with Surgery on ThreeManifolds
"... This is a technical paper, which is a continuation of [I]. Here we verify most of the assertions, made in [I, §13]; the exceptions are (1) the statement that a 3manifold which collapses with local lower bound for sectional curvature is a graph manifold this is deferred to a separate paper, as the ..."
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Cited by 448 (2 self)
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This is a technical paper, which is a continuation of [I]. Here we verify most of the assertions, made in [I, §13]; the exceptions are (1) the statement that a 3manifold which collapses with local lower bound for sectional curvature is a graph manifold this is deferred to a separate paper, as the proof has nothing to do with the Ricci flow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustified, and, on the other hand, irrelevant for the other conclusions. The Ricci flow with surgery was considered by Hamilton [H 5,§4,5]; unfortunately, his argument, as written, contains an unjustified statement (RMAX = Γ, on page 62, lines 710 from the bottom), which I was unable to fix. Our approach is somewhat different, and is aimed at eventually constructing a canonical Ricci flow, defined on a largest possible subset of spacetime, a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius h, which is the radius of the necks, where the surgeries are performed, and the much larger radius r, such that the solution on the scales less than r has standard geometry. The point is to make h arbitrarily small while keeping r bounded away from zero. Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P(x, t, r, △t) denotes a parabolic neighborhood, that is the set of all points (x ′ , t ′ ) with x ′ ∈ B(x, t, r) and t ′ ∈ [t, t + △t] or t ′ ∈ [t + △t, t], depending on the sign of △t. A ball B(x, t, ǫ −1 r) is called an ǫneck, if, after scaling the metric with factor r −2, it is ǫclose to the standard neck S 2 × I, with the product metric, where S 2 has constant scalar curvature one, and I has length 2ǫ −1; here ǫclose refers to C N topology, with N> ǫ −1. A parabolic neighborhood P(x, t, ǫ −1 r, r 2) is called a strong ǫneck, if, after scaling with factor r −2, it is ǫclose to the evolving standard neck, which at each
Filling Riemannian manifolds
 J. of Differential Geometry
, 1983
"... We want to discuss here several unsolved problems concerning metric invariants of a Riemannian manifold V = (V, g) which mediate between the curvature and topology of V. ..."
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Cited by 323 (6 self)
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We want to discuss here several unsolved problems concerning metric invariants of a Riemannian manifold V = (V, g) which mediate between the curvature and topology of V.
The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality
 J. DIFFERENTIAL GEOM
, 1998
"... In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announc ..."
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Cited by 201 (0 self)
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In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian version of the Penrose inequality for the total mass of an asymptotically flat 3manifold of nonnegative scalar curvature, announced in [HI1]. Let M be a smooth Riemannian manifold of dimension n 2 with metric g = (g ij ). A classical solution of the inverse mean curvature flow is a smooth family x : N \Theta [0; T ] !M
Monopoles and contact structures
 Invent. Math
, 1997
"... (i) Fourmanifolds with contact boundary The monopole invariants, or SeibergWitten invariants, introduced by Witten [27] are invariants of a smooth, closed, oriented 4manifold X. When bC(X) is greater than 1, they can be regarded as defining a map ..."
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Cited by 92 (4 self)
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(i) Fourmanifolds with contact boundary The monopole invariants, or SeibergWitten invariants, introduced by Witten [27] are invariants of a smooth, closed, oriented 4manifold X. When bC(X) is greater than 1, they can be regarded as defining a map
Spectral Theory Of Elliptic Operators On NonCompact Manifolds
, 1992
"... preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs ..."
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Cited by 80 (8 self)
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preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...
L.F.Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature
 J. Differential Geom
"... Abstract. In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of SchoenYau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative sca ..."
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Cited by 80 (11 self)
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Abstract. In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary. Using a general version of Positive Mass Theorem of SchoenYau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is spin and its boundary can be isometrically embedded into Euclidean space as a strictly convex hypersurface, then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in Euclidean space. Moreover, equality holds if and only if the manifold is isometric with a domain in the Euclidean space. Conversely, under the assumption that the theorem is true, then one can prove the ADM mass of an asymptotically flat manifold is nonnegative, which is part of the Positive Mass Theorem. §0 Introduction. The structure of a manifold with positive or nonnegative scalar curvature has been studied extensively. There are many beautiful results for compact manifolds without boundary, see [L, SY12, GW13]. For example, in [L], Lichnerowicz found that some compact manifolds admit no Riemannian metric with positive scalar curvature. In [SY12] Schoen and
Hardy spaces of differential forms on Riemannian manifolds
, 2006
"... Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transfo ..."
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Cited by 55 (11 self)
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Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H pboundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.
Metrics of Positive Scalar Curvature and Connections With Surgery
 Annals of Math. Studies
, 2001
"... this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be ne ..."
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Cited by 45 (1 self)
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this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be necessary to consider manifolds with boundary