Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons

Feldman, Mikhail; Ilmanen, Tom; Knopf, Daniel F

doi:10.4310/jdg/1090511686

Cited by 239 publications

(322 citation statements)

References 28 publications

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“…Noncompact expanding solitons were also constructed by the first author [13]. More recently, Feldman, Ilmanen and Knopf [46] constructed new examples of noncompact shrinking and expanding Kähler-Ricci solitons.…”

Section: 14)mentioning

confidence: 99%

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

Cao¹,

Zhu²

2006

Asian Journal of Mathematics

305

365

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Section: 14)mentioning

confidence: 99%

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

Cao¹,

Zhu²

2006

Asian Journal of Mathematics

305

365

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“…In particular, when the initial metric is smooth except for a conic singularity x 0 , as may arise by evolution from a smooth metric defined for negative times [10], then θ + (t) will not be useful in proving that a blowup about the vertex (x 0 , 0) for small t > 0 yields an expander.…”

Section: Proposition 23 the Forward Reduced Volume Is Bounded Below Bymentioning

confidence: 99%

“…2 This is unfortunate, because non compact expanders are needed for modeling the relaxation out of a previously formed singularity of conic type; see, for example, [2,18,10] and remarks after Proposition 2.3.…”

mentioning

confidence: 99%

Entropy and reduced distance for Ricci expanders

Feldman

Ilmanen

2005

J Geom Anal

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ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expandersSmall, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities corresponding to the first two of these. The functional F is monotone on the Ricci flow and constant precisely on steadies [25, Section 1]. The shrinking, or localizing, entropy W is monotone in general and constant precisely on shrinking solitons [25, Section 3]. Closely related is the notion of the (backward) reduced volume [25, Section 7]. The latter two show that developing singularities and ancient histories are modeled on shrinkers, when an appropriate blowup, or blowdown is taken [25,27].In this note, by tweaking some signs, we observe similar monotonicity formulae for the expanding case. In Section 1, analogous to [25, Section 1], we define an expanding, or delocalizing entropy W + , which is monotone in general and constant precisely on expanders. It follows from Math Subject Classifications. Primary: 58G11.

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“…Those for v σ are of course the same as for u σ . For v, we have 14) in which C is again a constant depending only on F * and n.…”

Section: Pointwise Estimates For U and Vmentioning

confidence: 99%

“…Here the manifold is a noncompact warped product R × f S n , and a supersolution of the Ricci flow PDE is used to prove that f vanishes in finite time on a proper subset of R. A second class of examples is constructed in [14]. Here the manifold is a complex line bundle C → L n −k → CP n−1 with 1 ≤ k ≤ n − 1.…”

Section: Antecedentsmentioning

confidence: 99%