The Entropy Formula for the Ricci Flow and Its Geometric Applications

Grigori Perelman · arXiv preprint · 2002 · mathematics · record perelman2002-ricci-flow
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Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. The PDF extraction badly scrambles all displayed mathematics (formulas, indices and section symbols are interleaved with the prose), so every quote here is taken from intact prose sentences only; no formula or theorem statement was quoted. Claim type was set to 'method' because the paper's own framing is that it supplies the technical tools (entropy/monotonicity formula, no-local-collapsing) that carry out an existing program, though several of its results (no nontrivial breathers) read as null-results in shape. One further explicit relation was deliberately NOT recorded as a claim_relation because the paper denies any relation: '5.3* An entropy formula for the Ricci flow in dimension two was found by Chow [C]; there seems to be no relation between his formula and ours.' 2026-08-14 and 2026-08-17 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node. Update, 15 September 2026: EXCEPTION, measured with the merge gate's own predicate (Paper.locate): 2 quote(s) on this record are not found as one exact string in the current staged text - content_tags.definitions[0].evidence, content_tags.findings[0].evidence. The sentence above does not hold for them as written. Where a row records why - the quote is grounded in another copy of the paper, joins two passages with an ellipsis, or differs from the staged text by extraction noise - that note stays on the row. Update, 16 September 2026: EXCEPTION TO THE DATED PASS ABOVE, on this record only: this record carries 2 composite quote fields - a quote assembled from two or more passages joined by an ellipsis. A COMPOSITE IS NOT AN EXACT CONTIGUOUS SUBSTRING of the staged text and is deliberately not anchored: each of its parts is verified separately at the offsets in evidence composite spans, and a part occurring more than once carries span null rather than a guessed offset. So the 'every quote verified as an exact substring' clause of the dated pass above DOES NOT HOLD for those fields, and never did - the parts were verified, the whole was not. Nothing is deleted, re-grounded or withdrawn: this writes down an exception that was always true of composites and had not been recorded, so a reader was told more than the pass had checked.

The quality of the text these quotes were cut from. symbol-font-not-extractable: 2 '(cid:NNN)' glyph-index markers remain in a staged text of 81,142 characters, where the PDF's embedded font had no ToUnicode map. THE QUOTES ON THIS RECORD ARE THE PAPER'S AND THE STAGED COPY IS THE DAMAGED ONE - which is the opposite of the usual case and is why it went unnoticed: 2 of its 46 live quotes are not exact substrings of this text, and 1 of those 2 were located in it by aligning the quote's WORD SEQUENCE under exactly this damage - the words are there, the characters inside them are not. Not repaired in place: editing the staged text moves every offset after the edit and re-hashes every anchor on the record. Measured at 113aae60fefc3eb1, re-measured unchanged at 54b63482778ddf56.

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Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 8 concept tags on this paper, 5 are grounded in a quoted sentence.

Ricci flow Poincaré conjecture geometrization entropy functional geometrization conjecture Hamilton program reduced volume monotonicity formula 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 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 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 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 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 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 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 curvature metric manifold entropy monotonicity surgery soliton Ricci soliton mathematical-proof

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relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
concept edge	`concept_edges`	an asserted relation between two concepts, with the sentence asserting it
concept	`concepts`	an idea the paper works with, quoted where the paper uses it
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
sense	`senses`	a distinct use of a term, quoted where the paper fixes it
transfer	`transfers`	a method or concept carried in from another field

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://arxiv.org/abs/math/0211159. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 6 · claim_type 1 · concept_edges 2 · concepts 5 · content_tags 3 · limitations 9 · mechanisms 7 · method 1 · phenomenon 1 · senses 3 · transfers 1

Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 2 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.

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facet	tag	field	conf.	whose words
claim_type	method Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`		the paper's words
phenomenon	singularity formation in the Ricci flow Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`		the paper's words
method	mathematical-proof Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mathematical-proof	`method`		the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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. also in: 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	`mechanisms[0]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[1]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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. also in: 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	`mechanisms[2]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[3]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[4]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[5]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[6]`	0.95	the tagger's words
limitations	for general m the gradient flow may not exist even for short time Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the gradient-flow construction for general m: it MAY NOT EXIST EVEN FOR SHORT TIME	`limitations[0]`		the tagger's words
limitations	The surgery arguments, which the program needs, are deferred to another paper. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the completeness of the programme as presented here: THE MORE TECHNICALLY COMPLICATED ARGUMENTS, RELATED TO THE SURGERY, WILL BE DISCUSSED ELSEWHERE	`limitations[1]`		the tagger's words
limitations	The author could not establish Hamilton's expected boundedness of normalized curvature for eternal solutions. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[2]`		the tagger's words
limitations	No proof is given that the limiting construction yields a canonically defined Ricci flow through singularities. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[3]`		the tagger's words
limitations	Details of the rigorous version of the shrinking-case argument are not supplied. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[4]`		the tagger's words
limitations	The author states his own understanding is insufficient for the physics analogy invoked (Black Hole Thermodynamics). Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[5]`		the tagger's words
limitations	The result the paper is famous for is presented as a brief sketch. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[6]`	1.0	the tagger's words
limitations	The curvature fact the blow-up analysis rests on is imported from earlier work and holds in dimension three. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[7]`	1.0	the tagger's words
limitations	A formula of the paper's resembles a known one and moves in the opposite direction. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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	`limitations[8]`	1.0	the tagger's words
transfers	importing entropy and monotonicity ideas from statistical physics to build monotonic functionals for the Ricci flow, used to resolve geometric and topological questions Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`transfers[0]`		the tagger's words
concepts	Ricci flow Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Ricci flow	`concepts[0]`		the paper's words
concepts	geometrization conjecture Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: geometrization conjecture	`concepts[4]`		the paper's words
concepts	Hamilton program Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Hamilton program	`concepts[5]`		the paper's words
concepts	reduced volume Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: reduced volume	`concepts[6]`		the paper's words
concepts	monotonicity formula Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: monotonicity formula	`concepts[7]`		the paper's words
senses	the conjecture this paper is known for - NEVER NAMED IN IT: 'Poincare' occurs nowhere; what the paper addresses is the geometrization conjecture, from which the more famous result follows, so the absence is not a later coinage but a stronger claim standing in for a weaker one Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[0]`	1.0	the tagger's words
senses	monotonicity - RESEMBLING A KNOWN FORMULA AND RUNNING THE OTHER WAY: the reduced-volume expression looks like Huisken's for mean curvature flow, and the author states the direction of monotonicity is opposite, so matching the two on their form would join quantities that disagree in sign Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[1]`	1.0	the tagger's words
senses	entropy - BY STATISTICAL ANALOGY, AND HEURISTIC: the monotonicity formula is interpreted in terms of the entropy of a canonical ensemble, and the author calls the connection to renormalization group flow heuristic rather than derived Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[2]`	1.0	the tagger's words
claim_relations	builds-on → Hamilton Ricci flow Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`		the tagger's words
claim_relations	builds-on → statistical physics / thermodynamics (entropy concepts) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[1]`		the tagger's words
claim_relations	builds-on → Hamilton's program for the Ricci flow and geometrization Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[2]`		the tagger's words
claim_relations	builds-on → Hamilton's program (implication for the geometrization conjecture) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[3]`		the tagger's words
claim_relations	extends → Ivey [I], no breathers theorem in dimension three Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[4]`		the tagger's words
claim_relations	builds-on → renormalization group flow picture from quantum field theory Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[5]`		the tagger's words
concept_edges	Hamilton program —causes→ geometrization conjecture Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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. also in: Hamilton program · geometrization conjecture	`concept_edges[0]`		the tagger's words
concept_edges	reduced volume —contrasted-with→ monotonicity formula Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: reduced volume · monotonicity formula	`concept_edges[1]`		the tagger's words
content_tags	Where singularities (necks) form, one performs SURGERY — cutting the neck and gluing caps — then continues the Ricci flow. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[1]`		the tagger's words
content_tags	Ricci soliton Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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. also in: Ricci soliton	`content_tags.definitions[1]`		the tagger's words
content_tags	surgery Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: surgery	`content_tags.definitions[2]`		the tagger's words

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