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Estimates for the extinction time for the Ricci flow on certain 3manifolds and a question of Perelman
 J. Amer. Math. Soc
"... 0. introduction In this note we prove some bounds for the extinction time for the Ricci flow on certain 3–manifolds. Our interest in this comes from a question of Grisha Perelman asked to the first author at a dinner in New York City on April 25th of 2003. His question was “what happens to the Ricci ..."
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Cited by 49 (6 self)
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0. introduction In this note we prove some bounds for the extinction time for the Ricci flow on certain 3–manifolds. Our interest in this comes from a question of Grisha Perelman asked to the first author at a dinner in New York City on April 25th of 2003. His question was “what happens to the Ricci flow on the 3–sphere when one starts with an arbitrary metric? In particular does the flow become extinct in finite time? ” He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min–max argument where the minimal of all maximal slices of sweep–outs is a minimal surface; see, for instance, [CD]. The idea is then to look at how the area of this min–max surface changes under the flow. Geometrically the area measures a kind of width of the 3–manifold and as we will see for certain 3–manifolds (those that are non–aspherical like the 3–sphere) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see
Width and finite extinction time of Ricci flow
, 2007
"... This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M i ..."
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Cited by 26 (1 self)
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This is an expository article with complete proofs intended for a general nonspecialist audience. The results are twofold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2spheres. For instance, when M is a homotopy 3sphere, the width is loosely speaking the
An excursion into geometric analysis
 SURVEYS IN DIFFERENTIAL GEOMETRY
, 2003
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Onesided complete stable minimal surfaces
 J. Differential Geom
"... We prove that there are no complete onesided stable minimal surfaces in the Euclidean 3space. We classify least area surfaces in the quotient of R3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact twosided index one minimal surfaces in nonneg ..."
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Cited by 17 (3 self)
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We prove that there are no complete onesided stable minimal surfaces in the Euclidean 3space. We classify least area surfaces in the quotient of R3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact twosided index one minimal surfaces in nonnegatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface. 1.
Rigidity of minmax minimal spheres in threemanifolds. arXiv:math.DG/1105.4632
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THE EXISTENCE OF EMBEDDED MINIMAL HYPERSURFACES
, 2009
"... We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n + 1)–dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n. ..."
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Cited by 6 (0 self)
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We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth (n + 1)–dimensional Riemannian manifolds, a theorem proved first by Pitts for 2 ≤ n ≤ 5 and extended later by Schoen and Simon to any n.
Width and mean curvature flow
 Geometry and Topology
"... Given a Riemannian metric on the 2sphere, sweep the 2sphere out by a continuous oneparameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show the following usef ..."
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Cited by 5 (0 self)
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Given a Riemannian metric on the 2sphere, sweep the 2sphere out by a continuous oneparameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show the following useful property; see Theorem 1.9 below and cf. [CM1],
More about Birkhoff’s invariant and Thorne’s hoop conjecture for horizons
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Existence of good sweepouts on closed manifolds, preprint
, 2009
"... Abstract. In this note we establish estimates for the harmonic map heat flow from S 1 into a closed manifold, and we use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself ..."
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Abstract. In this note we establish estimates for the harmonic map heat flow from S 1 into a closed manifold, and we use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic. 1.