The Entropy Formula for the Ricci Flow and Its Geometric Applications Grigori Perelman

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 The Entropy Formula for the Ricci Flow and Its Geometric Applications 2002 arXiv preprint math/0211159 perelman2002-ricci-flow perelman2002-ricci-flow.txt, the project's staged copy 21812f75822eadd3306317df86d61b066ca67ad6 arxiv-open mathematics

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 0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. mathematical-proof Quotation withheld because this record is not cleared for redistribution. THE SAME OBJECT HAS A NAME IN ANOTHER FIELD, AND THE PAPER SUPPLIES THE TRANSLATION: the functional's first variation formula CAN BE FOUND IN THE LITERATURE ON THE STRING THEORY, WHERE IT DESCRIBES THE LOW ENERGY EFFECTIVE ACTION - and THE FUNCTION f IS CALLED DILATON FIELD there. One mathematical object, two literatures, two names, and a citation offered so the reader can cross between them about this quote: The span begins mid-sentence, at "and its first variation formula", because everything before it in that sentence is interleaved mathematics - "can be found in [H 4, § psi = (delta m)2 psi ... M gijRm R and its first variation formula". So the quote does not name WHICH functional, and the tag does not claim to. The translation it records - the same object called 'dilaton field' in the string theory literature - is intact. Quotation withheld because this record is not cleared for redistribution. A MATHEMATICAL FREEDOM IS EXPLAINED BY A PHYSICS TERM RATHER THAN A MATHEMATICAL ONE: THE REMARKABLE FACT HERE IS THAT DIFFERENT CHOICES lead to THE SAME FLOW, up to a diffeomorphism - THAT IS, THE CHOICE OF m IS ANALOGOUS TO THE CHOICE OF GAUGE. The invariance is named by importing a word from another discipline, and the import is what makes it recognisable Quotation withheld because this record is not cleared for redistribution. THE ANALOGY'S STATUS IS MARKED AS HEURISTIC IN ITS OWN NUMBERED REMARK, AND THE DIFFERENCE IS SPELLED OUT: HEURISTICALLY, this statistical analogy IS RELATED TO the description of the renormalization group flow - IN THE LATTER one obtains quantities BY AVERAGING OVER HIGHER ENERGY STATES, WHEREAS IN THE FORMER THOSE STATES ARE SUPPRESSED by the exponential factor. The resemblance is asserted and the point of divergence given in the same breath about this quote: CONTEXT THIS QUOTE CANNOT CARRY, recorded because it is the paper's central terminological hedge. The sentence that states the analogy is SCRAMBLED in the staged text and cannot be quoted: "In this section we show that the functional a sense analogous to minus entropy. W , introduced in section 3, is in" - the clean reading is "the functional W, introduced in section 3, is in a sense analogous to minus entropy". Damage shape 5. The TITLE of the paper says 'the entropy formula'; section 5 says 'in a sense analogous to MINUS entropy' - qualified twice, by the hedge and by the sign - and that sentence is the one the extraction destroyed. Quotation withheld because this record is not cleared for redistribution. A PROPERTY IS DISCOVERED AND IMMEDIATELY GIVEN A NAME BUILT FROM TWO OTHER NAMES: a REMARKABLE PROPERTY of solutions with nonnegative curvature operator IN ARBITRARY DIMENSION is CALLED A DIFFERENTIAL HARNACK INEQUALITY, and what it ALLOWS is stated in the same sentence - comparing the curvatures at different points and different times. The name records neither the property nor its use but its lineage Quotation withheld because this record is not cleared for redistribution. A RESULT IS VALUED BY WHAT IT UNBLOCKS IN SOMEONE ELSE'S PROGRAMME: the theorem is said to REMOVE THE MAJOR STUMBLING BLOCK IN HAMILTON'S APPROACH TO GEOMETRIZATION. The contribution is located relative to another person's plan rather than stated on its own terms Quotation withheld because this record is not cleared for redistribution. A QUESTION IS PROPOSED AND CALLED NATURAL, WHICH IS THE ONLY WARRANT OFFERED FOR IT: Quotation withheld The word NATURAL carries the whole justification for why this is the question to ask next, and no other reason is given Quotation withheld because this record is not cleared for redistribution. A TERM IS ATTRIBUTED TO A PERSON RATHER THAN DEFINED: a solution defined on a time interval of a certain form is ANCIENT, IN HAMILTON'S TERMINOLOGY - and the attribution is the definition's provenance, given in place of a criterion Quotation withheld because this record is not cleared for redistribution. what this limits: the gradient-flow construction for general m: it MAY NOT EXIST EVEN FOR SHORT TIME Quotation withheld because this record is not cleared for redistribution. what this limits: the completeness of the programme as presented here: THE MORE TECHNICALLY COMPLICATED ARGUMENTS, RELATED TO THE SURGERY, WILL BE DISCUSSED ELSEWHERE Quotation withheld because this record is not cleared for redistribution. what this limits: the eternal-solutions claim: WE HAVE NOT BEEN ABLE TO CONFIRM HAMILTON'S HOPE THAT THE SOLUTION THAT EXISTS FOR ALL TIME NECESSARILY HAS BOUNDED NORMALIZED CURVATURE Quotation withheld because this record is not cleared for redistribution. what this limits: the canonical-flow-through-singularities claim: IT IS LIKELY THAT ... ONE WOULD GET A CANONICALLY DEFINED RICCI FLOW THROUGH SINGULARITIES, BUT AT THE MOMENT I DON'T HAVE A PROOF OF THAT Quotation withheld because this record is not cleared for redistribution. what this limits: the rigour of the shrinking-case argument as given: THIS ARGUMENT IS NOT HARD TO MAKE RIGOROUS; THE DETAILS ARE LEFT TO THE READER - the support is not in the paper Quotation withheld because this record is not cleared for redistribution. what this limits: the physical analogy only: UNFORTUNATELY, THIS SUBJECT IS BEYOND MY UNDERSTANDING AT THE MOMENT - the Black Hole Thermodynamics parallel is offered without being vouched for Quotation withheld because this record is not cleared for redistribution. what this limits: the geometrization result attributed to this paper: what it contains at that point is described by the author as a brief sketch of the proof Quotation withheld because this record is not cleared for redistribution. what this limits: the nonnegative-curvature property of blow-up limits used here: it rests on results of Ivey and Hamilton proved in dimension three, where large scalar curvature makes the negative part of the curvature tensor comparatively small Quotation withheld because this record is not cleared for redistribution. what this limits: any comparison of this reduced-volume formula with Huisken's monotonicity formula for mean curvature flow: the author states the two resemble each other but that here the monotonicity runs in the opposite direction Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Ricci flow Quotation withheld because this record is not cleared for redistribution. geometrization conjecture Quotation withheld because this record is not cleared for redistribution. Hamilton program Quotation withheld because this record is not cleared for redistribution. reduced volume Quotation withheld because this record is not cleared for redistribution. monotonicity formula Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Hamilton program geometrization conjecture strength of the relation: 0.9 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span says the program WOULD IMPLY the conjecture; a logical implication typed as cause. Quotation withheld because this record is not cleared for redistribution. reduced volume monotonicity formula strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. Quotation withheld because this record is not cleared for redistribution. Ricci soliton about this quote: QUOTE REPAIRED: the extractor split a hyphen-space and the quote was recorded faithfully with the damage in it, which made it unanchorable. The repaired string is an EXACT SUBSTRING of the staged text - that is why it was accepted, not because a rule produced it. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. surgery Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://arxiv.org/abs/math/0211159.