Hamilton’s Harnack inequalities for Ricci flow
E890839
Harnack inequality
curvature estimate
differential inequality
geometric analysis result
tool in Ricci flow theory
Hamilton’s Harnack inequalities for Ricci flow are fundamental differential inequalities that provide monotonicity and curvature control along solutions to the Ricci flow, playing a key role in the analysis of geometric evolution and singularity formation.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Harnack inequality
ⓘ
curvature estimate ⓘ differential inequality ⓘ geometric analysis result ⓘ tool in Ricci flow theory ⓘ |
| appliesTo |
Ricci flow
NERFINISHED
ⓘ
Riemannian manifolds evolving by Ricci flow ⓘ solutions of the Ricci flow equation ⓘ |
| assumes |
complete solutions to Ricci flow in many statements
ⓘ
nonnegative Ricci curvature in some versions ⓘ nonnegative curvature operator in some versions ⓘ |
| concerns |
comparison of curvature at different space–time points
ⓘ
space–time differential inequalities for curvature ⓘ |
| field |
Riemannian geometry
ⓘ
geometric analysis ⓘ partial differential equations ⓘ |
| generalizationOf | Li–Yau Harnack inequality for the heat equation NERFINISHED ⓘ |
| governs |
evolution of Ricci curvature under Ricci flow
ⓘ
evolution of full curvature tensor under Ricci flow ⓘ evolution of scalar curvature under Ricci flow ⓘ |
| hasAuthor | Richard S. Hamilton NERFINISHED ⓘ |
| hasType |
matrix Harnack inequality
ⓘ
trace Harnack inequality ⓘ |
| implies |
backward and forward Harnack estimates for curvature
ⓘ
lower bounds on scalar curvature at later times from earlier times ⓘ monotonicity of certain curvature combinations along space–time paths ⓘ |
| inspired |
Perelman’s differential Harnack inequalities for the conjugate heat equation
ⓘ
subsequent Harnack inequalities for other geometric flows ⓘ |
| introducedBy | Richard S. Hamilton NERFINISHED ⓘ |
| mathematicalDomain |
differential geometry
ⓘ
geometric evolution equations ⓘ |
| playsRoleIn |
Hamilton’s program for using Ricci flow to understand 3-manifolds
ⓘ
Perelman’s proof of the Poincaré conjecture ⓘ |
| provides |
differential inequalities for curvature quantities
ⓘ
gradient estimates for curvature under Ricci flow ⓘ monotonicity formulas along Ricci flow ⓘ space–time curvature control ⓘ |
| publishedIn | Communications in Analysis and Geometry NERFINISHED ⓘ |
| relatedTo |
Perelman’s entropy functionals
NERFINISHED
ⓘ
ancient solutions of Ricci flow ⓘ gradient shrinking Ricci solitons ⓘ |
| relatedWork | Hamilton’s paper "The Harnack estimate for the Ricci flow" NERFINISHED ⓘ |
| usedFor |
classifying singularity models
ⓘ
curvature control along Ricci flow ⓘ deriving Li–Yau type estimates in geometric flows ⓘ deriving a priori estimates for Ricci flow ⓘ proving noncollapsing and nondegeneracy properties ⓘ understanding singularity formation in Ricci flow ⓘ |
| yearProposed | 1993 ⓘ |
Referenced by (1)
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