The Harnack estimate for the Ricci flow

Hamilton, Richard S.

doi:10.4310/jdg/1214453430

Cited by 281 publications

(297 citation statements)

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“…Since g ij has weakly positive curvature operator, by the trace Harnack inequality for the Ricci flow proved by the second author in [Ham93a], ∂ ∂t R + R t + 2∇R · ∇u + 2R ij u i u j ≥ 0. It follows from (2.3) and maximum principle that…”

Section: Proof (Proof Of Theorem 11) It Is Easy To See That For T Smentioning

confidence: 99%

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Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials

Cao

Hamilton

2009

Geom. Funct. Anal.

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Section: Proof (Proof Of Theorem 11) It Is Easy To See That For T Smentioning

confidence: 99%

“…In [Ham93a], the second author proved a Harnack estimate for the Ricci flow on Riemannian manifolds with weakly positive curvature operator, its trace version, (1.1) ∂R ∂t + R t + 2∇R · V + 2Rc(V, V ) ≥ 0 [Ni06] or [CCG + 07, Chapter 16] for a detailed proof).…”

Section: Introductionmentioning

confidence: 99%

Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials

Cao

Hamilton

2009

Geom. Funct. Anal.

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“…where H (X) := ∂R/∂t + 2 ∇R, X + 2Rc(X, X) + R/t is the exactly twice the traced Li-YauHamilton differential Harnack expression in [11]. (Notice that the H in [25,Section 7] also equals the LYH expression, but evaluated at a negative time t = −τ .)…”

Section: Forward Reduced Distance and Reduced Volumementioning

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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“…Hamilton's Ricci flow [32] on Riemannian manifolds (M, g) is given by the evolution equation ∂g ∂t = −2Ric(g). The Ricci flow became a fundamental tool in the study of the geometry and topology of manifolds, starting with the seminal results of Hamilton [32], [35], [36], [37], and culminating more recently with Perelman's proof of the Poincaré conjecture [49], [50], [51], see also [1], [47] and [10], [42], [44].…”

Section: Introductionmentioning

confidence: 99%