佩雷尔曼关于庞加莱猜想的论文0303109
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A metric on B3 or RP3 \ B¯3, such that each point outside some compact subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively.
1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature.
A ball B(x, t, ǫ−1r) is called an ǫ-neck, if, after scaling the metric with factor பைடு நூலகம்−2, it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1; here ǫ-close refers to CN topology, with N > ǫ−1.
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 7-10 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 space-time, - 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.
Proof. A gradient shrinking soliton gij(t), −∞ < t < 0, satisfies the equation
A parabolic neighborhood P (x, t, ǫ−1r, r2) is called a strong ǫ-neck, if, after scaling with factor r−2, it is ǫ-close to the evolving standard neck, which at each
We denote by ǫ a fixed small positive constant. In contrast, δ denotes a positive quantity, which is supposed to be as small as needed in each particular argument.
arXiv:math.DG/0303109 v1 10 Mar 2003
Ricci flow with surgery on three-manifolds
Grisha Perelman∗
August 21, 2006
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 3-manifold 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.
∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelman@pdmi.ras.ru or perelman@math.sunysb.edu
1
time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′)−1. A metric on S2 × I, such that each point is contained in some ǫ-neck, is
called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively.
1 Ancient solutions with bounded entropy
1.1 In this section we review some of the results, proved or quoted in [I,§11], correcting a few inaccuracies. We consider smooth solutions gij(t) to the Ricci flow on oriented 3-manifold M , defined for −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded nonnegative sectional curvature, κ-noncollapsed on all scales for some fixed κ > 0; such solutions will be called ancient κ-solutions for short. By Theorem I.11.7, the set of all such solutions with fixed κ is compact modulo scaling, that is from any sequence of such solutions (M α, giαj(t)) and points (xα, 0) with R(xα, 0) = 1, we can extract a smoothly (pointed) convergent subsequence, and the limit (M, gij(t)) belongs to the same class of solutions. (The assumption in I.11.7. that M α be noncompact was clearly redundant, as it was not used in the proof. Note also that M need not have the same topology as M α.) Moreover, according to Proposition I.11.2, the scalings of any ancient κ-solution gij(t) with factors (−t)−1 about appropriate points converge along a subsequence of t → −∞ to a non-flat gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamilton’s strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infinite cylinder, or its Z2 quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by [H 1]. The noncompact case is ruled out below.
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.
1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature.
A ball B(x, t, ǫ−1r) is called an ǫ-neck, if, after scaling the metric with factor பைடு நூலகம்−2, it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1; here ǫ-close refers to CN topology, with N > ǫ−1.
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 7-10 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 space-time, - 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.
Proof. A gradient shrinking soliton gij(t), −∞ < t < 0, satisfies the equation
A parabolic neighborhood P (x, t, ǫ−1r, r2) is called a strong ǫ-neck, if, after scaling with factor r−2, it is ǫ-close to the evolving standard neck, which at each
We denote by ǫ a fixed small positive constant. In contrast, δ denotes a positive quantity, which is supposed to be as small as needed in each particular argument.
arXiv:math.DG/0303109 v1 10 Mar 2003
Ricci flow with surgery on three-manifolds
Grisha Perelman∗
August 21, 2006
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 3-manifold 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.
∗St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelman@pdmi.ras.ru or perelman@math.sunysb.edu
1
time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′)−1. A metric on S2 × I, such that each point is contained in some ǫ-neck, is
called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively.
1 Ancient solutions with bounded entropy
1.1 In this section we review some of the results, proved or quoted in [I,§11], correcting a few inaccuracies. We consider smooth solutions gij(t) to the Ricci flow on oriented 3-manifold M , defined for −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded nonnegative sectional curvature, κ-noncollapsed on all scales for some fixed κ > 0; such solutions will be called ancient κ-solutions for short. By Theorem I.11.7, the set of all such solutions with fixed κ is compact modulo scaling, that is from any sequence of such solutions (M α, giαj(t)) and points (xα, 0) with R(xα, 0) = 1, we can extract a smoothly (pointed) convergent subsequence, and the limit (M, gij(t)) belongs to the same class of solutions. (The assumption in I.11.7. that M α be noncompact was clearly redundant, as it was not used in the proof. Note also that M need not have the same topology as M α.) Moreover, according to Proposition I.11.2, the scalings of any ancient κ-solution gij(t) with factors (−t)−1 about appropriate points converge along a subsequence of t → −∞ to a non-flat gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamilton’s strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infinite cylinder, or its Z2 quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by [H 1]. The noncompact case is ruled out below.
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.