Upper half-plane

Several stages of Ricci flow on a 2D manifold.

In differential geometry, the Ricci flow (Template:IPAc-en) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.

The Ricci flow, named after Gregorio Ricci-Curbastro, was first introduced by Richard Hamilton in 1981 and is also referred to as the Ricci–Hamilton flow. It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the differentiable sphere theorem by Brendle and Schoen.

Mathematical definition

Given a Riemannian manifold with metric tensor $g_{ij}$, we can compute the Ricci tensor $R_{ij}$, which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called "time" (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation

$\partial _{t}g_{ij}=-2R_{ij}.$

The normalized Ricci flow makes sense for compact manifolds and is given by the equation

$\partial _{t}g_{ij}=-2R_{ij}+{\frac {2}{n}}R_{\mathrm {avg} }g_{ij}$

where $R_{\mathrm {avg} }$ is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and $n$ is the dimension of the manifold. This normalized equation preserves the volume of the metric.

The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.)

Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

Examples

• If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is Ricci-flat.
• If the manifold is a sphere (with the usual metric) then Ricci flow collapses the manifold to a point in finite time. If the sphere has radius 1 in n dimensions, then after time $t$ the metric will be multiplied by $(1-2t(n-1))$, so the manifold will collapse after time $1/2(n-1)$. More generally, if the manifold is an Einstein manifold (Ricci = constant × metric), then Ricci flow will collapse it to a point if it has positive curvature, leave it invariant if it has zero curvature, and expand it if it has negative curvature.
• For a compact Einstein manifold, the metric is unchanged under normalized Ricci flow. Conversely, any metric unchanged by normalized Ricci flow is Einstein.

In particular, this shows that in general the Ricci flow cannot be continued for all time, but will produce singularities. For 3 dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

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• A significant 2-dimensional example is the cigar soliton solution, which is given by the metric (dx² + dy²)/(e^4t + x² + y²) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons. An example of a 3-dimensional steady Ricci soliton is the "Bryant soliton", which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations.

Relationship to uniformization and geometrization

The Ricci flow (named after Gregorio Ricci-Curbastro) was introduced by Richard Hamilton in 1981 in order to gain insight into the geometrization conjecture of William Thurston, which concerns the topological classification of three-dimensional smooth manifolds. Hamilton's idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston; the possibilities, called Thurston model geometries, include the three-sphere S³, three-dimensional Euclidean space E³, three-dimensional hyperbolic space H³, which are homogeneous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes.) Hamilton's idea was that these special metrics should behave like fixed points of the Ricci flow, and that if, for a given manifold, globally only one Thurston geometry was admissible, this might even act like an attractor under the flow.

Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time.) This doesn't prove the full geometrization conjecture because the most difficult case turns out to concern manifolds with negative Ricci curvature and more specifically those with negative sectional curvature. (A strange and interesting fact is that all closed three-manifolds admit metrics with negative Ricci curvatures! This was proved by L. Zhiyong Gao and Shing-Tung Yau in 1986.) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

Note that the term "uniformization" correctly suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" correctly suggests placing a geometry on a smooth manifold. Geometry is being used here in a precise manner akin to Klein's notion of geometry (see Geometrization conjecture for further details). In particular, the result of geometrization may be a geometry that is not isotropic. In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

The Ricci flow does not preserve volume, so to be more careful in applying the Ricci flow to uniformization and geometrization one needs to normalize the Ricci flow to obtain a flow which preserves volume. If one fail to do this, the problem is that (for example) instead of evolving a given three-dimensional manifold into one of Thurston's canonical forms, we might just shrink its size.

It is possible to construct a kind of moduli space of n-dimensional Riemannian manifolds, and then the Ricci flow really does give a geometric flow (in the intuitive sense of particles flowing along flowlines) in this moduli space.

Relation to diffusion

To see why the evolution equation defining the Ricci flow is indeed a kind of nonlinear diffusion equation, we can consider the special case of (real) two-manifolds in more detail. Any metric tensor on a two-manifold can be written with respect to an exponential isothermal coordinate chart in the form

$ds^{2}=\exp(2\,p(x,y))\,\left(dx^{2}+dy^{2}\right).$

(These coordinates provide an example of a conformal coordinate chart, because angles, but not distances, are correctly represented.)

The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan. Take the coframe field

$\sigma ^{1}=\exp(p)\,dx,\;\;\sigma ^{2}=\exp(p)\,dy$

so that metric tensor becomes

$\sigma ^{1}\otimes \sigma ^{1}+\sigma ^{2}\otimes \sigma ^{2}=\exp(2p)\,\left(dx\otimes dx+dy\otimes dy\right).$

Next, given an arbitrary smooth function $h(x,y)$, compute the exterior derivative

$dh=h_{x}dx+h_{y}dy=\exp(-p)h_{x}\,\sigma ^{1}+\exp(-p)h_{y}\,\sigma ^{2}.$

Take the Hodge dual

$\star dh=-\exp(-p)h_{y}\,\sigma ^{1}+\exp(-p)h_{x}\,\sigma ^{2}=-h_{y}\,dx+h_{x}\,dy.$

Take another exterior derivative

$d\star dh=-h_{yy}\,dy\wedge dx+h_{xx}\,dx\wedge dy=\left(h_{xx}+h_{yy}\right)\,dx\wedge dy$

(where we used the anti-commutative property of the exterior product). That is,

$d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)\,\sigma ^{1}\wedge \sigma ^{2}.$

Taking another Hodge dual gives

$\Delta h=\star d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)$

which gives the desired expression for the Laplace/Beltrami operator

$\Delta =\exp(-2\,p(x,y))\left(D_{x}^{2}+D_{y}^{2}\right).$

To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe:

$d\sigma ^{1}=p_{y}\exp(p)dy\wedge dx=-\left(p_{y}dx\right)\wedge \sigma ^{2}=-{\omega ^{1}}_{2}\wedge \sigma ^{2}$

$d\sigma ^{2}=p_{x}\exp(p)dx\wedge dy=-\left(p_{x}dy\right)\wedge \sigma ^{1}=-{\omega ^{2}}_{1}\wedge \sigma ^{1}.$

From these expressions, we can read off the only independent connection one-form

${\omega ^{1}}_{2}=p_{y}dx-p_{x}dy.$

Take another exterior derivative

$d{\omega ^{1}}_{2}=p_{yy}dy\wedge dx-p_{xx}dx\wedge dy=-\left(p_{xx}+p_{yy}\right)\,dx\wedge dy.$

This gives the curvature two-form

${\Omega ^{1}}_{2}=-\exp(-2p)\left(p_{xx}+p_{yy}\right)\,\sigma ^{1}\wedge \sigma ^{2}=-\Delta p\,\sigma ^{1}\wedge \sigma ^{2}$

from which we can read off the only linearly independent component of the Riemann tensor using

${\Omega ^{1}}_{2}={R^{1}}_{212}\,\sigma ^{1}\wedge \sigma ^{2}.$

Namely

${R^{1}}_{212}=-\Delta p$

from which the only nonzero components of the Ricci tensor are

$R_{22}=R_{11}=-\Delta p.$

From this, we find components with respect to the coordinate cobasis, namely

$R_{xx}=R_{yy}=-\left(p_{xx}+p_{yy}\right).$

But the metric tensor is also diagonal, with

$g_{xx}=g_{yy}=\exp(2p)$

and after some elementary manipulation, we obtain an elegant expression for the Ricci flow:

${\frac {\partial p}{\partial t}}=\Delta p.$

This is manifestly analogous to the best known of all diffusion equations, the heat equation

${\frac {\partial u}{\partial t}}=\Delta u$

where now $\Delta =D_{x}^{2}+D_{y}^{2}$ is the usual Laplacian on the Euclidean plane. The reader may object that the heat equation is of course a linear partial differential equation—where is the promised nonlinearity in the p.d.e. defining the Ricci flow?

The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking $p(x,y)=0$. So if $p$ is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation. This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate. But if our hot plate is finite in size, and has no boundary where heat can be carried off, we can expect to homogenize the temperature, but clearly we cannot expect to reduce it to zero. In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.

Recent developments

The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher-dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving n-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time $t_{0}$. In certain cases, such neckpinches will produce manifolds called Ricci solitons.

There are many related geometric flows, some of which (such as the Yamabe flow and the Calabi flow) have properties similar to the Ricci flow.

See also

Applications

• Uniformization theorem
• Geometrization conjecture
• Solution of the Poincaré conjecture
• Differentiable sphere theorem

General context

• Ricci curvature
• Calculus of variations
• Geometric flow

References

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• Template:Cite arxiv
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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Erratum.
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• Template:Cite arxiv
• Anderson, Michael T. Geometrization of 3-manifolds via the Ricci flow, Notices AMS 51 (2004) 184–193.
• John Milnor, Towards the Poincaré Conjecture and the classification of 3-manifolds, Notices AMS. 50 (2003) 1226–1233.
• John Morgan, Recent progress on the Poincaré conjecture and the classification of 3-manifolds, Bull. AMS 42 (2005) 57–78.
• Notes from the Clay math institute Summer School Program 2005 on Ricci flow.
• Richard Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom 17 (1982), 255–306.
• Collected Papers on Ricci Flow ISBN 1-57146-110-8.
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. A popular book that explains the background for the Thurston classification programme.
• Ricci flow Theme on arxiv.org