Ricci flow
E48279
geometric evolution equation
method in Riemannian geometry
parabolic partial differential equation
partial differential equation
tool in geometric analysis
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
Aliases (2)
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
geometric evolution equation
→
method in Riemannian geometry → parabolic partial differential equation → partial differential equation → tool in geometric analysis → |
| actsOn |
Riemannian metric
→
|
| aimsToProduce |
canonical geometric structures on manifolds
→
|
| analogy |
heat equation
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|
| appliedIn |
3-manifold topology
→
Kähler geometry → study of Einstein metrics → |
| canDevelop |
finite-time singularities
→
|
| centralToWorkOf |
Grigori Perelman
→
|
| definedOn |
Riemannian manifold
→
|
| dimension |
applicable in any dimension
→
|
| drivingTensor |
Ricci curvature
→
|
| evolves |
Riemannian metric g(t)
→
|
| field |
differential geometry
→
geometric analysis → global Riemannian geometry → |
| generalizationOf |
curve shortening flow on 1-manifolds
→
|
| governingEquation |
∂g_ij/∂t = -2 Ric_ij
→
|
| hasVariant |
Kähler–Ricci flow
→
Ricci flow with surgery → normalized Ricci flow → |
| introducedBy |
Richard S. Hamilton
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|
| invariantUnder |
diffeomorphisms
→
pullback by diffeomorphisms → |
| isGeometric |
true
→
|
| isLocal |
true
→
|
| relatedConcept |
Hamilton’s compactness theorem
→
Hamilton’s maximum principle → Perelman’s entropy functionals → Ricci curvature → reduced volume → scalar curvature → sectional curvature → surgery in Ricci flow → κ-solutions → |
| singularityAnalysisUses |
blow-up techniques
→
rescaling arguments → |
| specialCaseOf |
geometric heat flow
→
|
| tendsTo |
even out curvature
→
smooth out irregularities in the metric → |
| type |
nonlinear PDE
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|
| usedInProofOf |
Poincaré conjecture
→
geometrization conjecture → |
| wellPosedness |
short-time existence for smooth initial metrics
→
|
| yearIntroduced |
1982
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|
Referenced by (4)
| Subject (surface form when different) | Predicate |
|---|---|
|
Ricci curvature tensor
→
Ricci curvature tensor ("Ricci solitons") → |
usedIn |
|
Ricci flow
("Ricci flow with surgery")
→
|
hasVariant |
|
Ricci scalar
→
|
relatedTo |