In the mathematical fields of differential geometry and geometric analysis, the Ricci flow (/ˈriːtʃi/REE-chee, Italian:[ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen.
Following Shing-Tung Yau's suggestion[citation needed] that the singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture, Hamilton produced a number of results in the 1990s which were directed towards the conjecture's resolution. In 2002 and 2003, Grigori Perelman presented a number of fundamental new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's program. Perelman's work is now widely regarded as forming the proof of the Thurston conjecture and the Poincaré conjecture, regarded as a special case of the former. It should be emphasized that the Poincare conjecture has been a well-known open problem in the field of geometric topology since 1904. These results by Hamilton and Perelman are considered as a milestone in the fields of geometry and topology.
geometric analysis, the Ricciflow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricciflow, is a certain partial differential...
years. In 2002 and 2003, he developed new techniques in the analysis of Ricciflow, and proved the Poincaré conjecture and Thurston's geometrization conjecture...
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian...
In combination with their results, Grigori Perelman's construction of Ricciflow with surgery in 2003 provided a complete characterization of these topologies...
announced a proof of the full geometrization conjecture in 2003 using Ricciflow with surgery in two papers posted at the arxiv.org preprint server. Perelman's...
This represents one sense in which the Kähler-Ricciflow is significantly simpler than the usual Ricciflow, where there is no (known) computation of the...
g 0 ) {\displaystyle (M,g_{0})} yields a self-similar solution to the Ricciflow equation ∂ t g t = − 2 Ric ( g t ) . {\displaystyle \partial _{t}g_{t}=-2\operatorname...
the mean curvature flow and Ricciflow, solving a question concerning the uniqueness of self-similar solutions to the Ricciflow which arose in the context...
twenty years. Hamilton and Perelman's work revolved around Hamilton's Ricciflow, which is a complicated system of partial differential equations defined...
entropy formula for the Ricciflow and its geometric applications". arXiv:math.DG/0211159. Perelman, Grisha (10 March 2003). "Ricciflow with surgery on three-manifolds"...
S. Hamilton showed that the normalized Ricciflow on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric)...
mean curvature flow of hypersurfaces. In 1984, he adapted Hamilton's seminal work on the Ricciflow to the setting of mean curvature flow, proving that...
introduced by Richard Hamilton in his work on the Ricciflow. It has since been applied to other geometric flows as well as to other systems such as the Navier–Stokes...
Hamilton, which started after Yau learned of the latter's work on the Ricciflow, is also mentioned. Subsequently, the article describes Yau in relation...
James; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei (2010). The Ricciflow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical...
mean curvature flow Intrinsic geometric flows are flows on the Riemannian metric, independent of any embedding or immersion. Ricciflow, as in the solution...
to use the Ricciflow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricciflow, called Ricciflow with surgery...
conjecture. His work contained a number of notable new results on the Ricciflow, although many proofs were only sketched and a number of details were...
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