APPEARED IN BULLETIN OF THE

arXiv:math/9301218v1 [math.AP] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 90-94A NEW RESULT FOR THE POROUS MEDIUM EQUATIONDERIVED FROM THE RICCI FLOWLang-Fang WuAbstract. Given R2, with a “good” complete metric, we show that the uniquesolution of the Ricci flow approaches a soliton at time infinity.

Solitons are solutionsof the Ricci flow, which move only by diffeomorphism. The Ricci flow on R2 is thelimiting case of the porous medium equation when m is zero.

The results in the Ricciflow may therefore be interpreted as sufficient conditions on the initial data, whichguarantee that the corresponding unique solution for the porous medium equationon the entire plane asymptotically behaves like a “soliton-solution”.On R2, any metric can be expressed as ds2 = eu(dx2 + dy2), where {x, y} arerectangular coordinates on R2. Let R be the scalar curvature, then the so-calledRicci flow on R2 is(*0)∂∂tds2 = −Rds2,which may also be expressed as(*1)∂∂tu = e−u∆u,where ∆= ∂2x + ∂2y.The porous medium equation is defined as(*2)∂∂τ v = ∆vm,where 0 < m < ∞, and v is a function on R2.

If we let τ = t/m, then (*2) can beexpressed as∂∂tv = ∆(vm −1)m.The limiting case of the porous medium equation as m →0 [PME (m = 0)] istherefore(*3)∂∂tv = ∆ln v.In dimension 1, the PME (m = 0) with initial data v /∈L1 has been studied by[ERV, H1, H2, V].We are grateful to Sigurd B. Angenent for pointing out the following observationreflected in [W2, Appendix].1991 Mathematics Subject Classification. Primary 58D25 35K05.Received by the editors September 24, 1991 and, in revised form, on November 13, 1991 andMay 1, 1992.

Presented on July 2 at the 1992 Regional Geometry Institute at Utahc⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2L.-F. WUProposition [Angenent]. On R2, if we let v = eu, then the limiting case of theporous medium equation as m →0 (∗3) is equivalent to the Ricci flow (∗1).On a complete (R2, ds2) we may define:(a) the circumference at infinity to beC∞(ds2) = supKinfD2{L(∂D2)|∀compact set K ⊂R2, ∀open set D2 ⊃K}where L(∂D2) is the length of ∂D2 with respect to the metric ds2;(b) the aperture to beA(ds2) = 12π limr→∞L(∂Br)r,where Br is a geodesic ball at any given point on R2 with radius r.A gradient soliton is a solution of the Ricci flow, which moves only by diffeomor-phism and there exists a function f such that∂∂tgij = L▽fgij, where L▽f is theLie derivative in the direction of the gradient f. There are two types of gradientsolitons on R2.

Namely, the flat soliton (C∞= ∞, A = 1) and the cigar soliton(C∞< ∞, A = 0). The flat soliton is the standard flat metric on R2.

The cigarsoliton is a metric, which can be expressed as ds2 = (du2 + dv2)/(1 + u2 + v2),where {u, v} are rectangular coordinates on R2. The cigar solitons are the so-calledBarenblatt solutions in the field of the porous medium equations.The Ricci flow and the soliton phenomenon gave a new proof for the uniformiza-tion theorem on compact surfaces and orbifolds without boundary ([C, Ha1, W2,CW]).

Understanding the solitons may provide insights toward studying the Ricciflow on higher-dimensional K¨ahler manifolds. For an announcement of related work,see [Shi].

As a step towards further studying the soliton phenomenon on higher-dimensional K¨ahler manifolds, Richard Hamilton raised the following question:Question. On what manifolds do solutions to the Ricci flow asymptotically ap-proach nontrivial solitons?One of the simplest spaces is R2 with complete metric.

We say that the Ricci flowon R2 has weak modified convergence at time infinity if there exists a 1-parameterfamily of diffeomorphisms {φt}t∈[0,∞) on R2 such that for any sequence of timesgoing to infinity there is a subsequence of times {tj}∞j=0 and the modified metricds2(φtj( · ), tj) converges uniformly on every compact set as j →∞. We will nowstate our results concerning the Ricci flow.Main Theorem (Ricci flow [W2]).

Given a complete (R2, ds2(0)) with |R| ≤Cand |Du| ≤C at t = 0, the Ricci flow has weak modified convergence at time infinityto a limiting metric. In the case when R > 0 at t = 0, the limiting metric is a cigarsoliton if C∞(ds2(0)) < ∞, or a flat metric if A(ds2(0)) > 0.In the process of proving the main theorem we have the following lemmas.

LetR denote the scalar curvature and R−= max{−R, 0}.Lemma 1 (Ricci flow) (Long time existence). Given a complete (R2, ds2) with|R| ≤C and |Du| ≤C at t = 0, under the Ricci flow (∗1), the solution of the flowexists for infinite time.

A NEW RESULT FOR THE POROUS MEDIUM EQUATION3Lemma 2 (Ricci flow). Given a complete (R2, ds2) with |R| ≤C,RR2 R−dµ < ∞,|Du| ≤C, and C∞> 0 at t = 0, under the Ricci flow (∗1), we have the following :1.

(Uniqueness) The solution of the flow is unique.2. (Geometric Properties) C∞, A(gij), andRR2 Rdµ < ∞are constants underthe flow.Lemma 3 (Ricci flow).

Given a complete (R2, ds2) with 0 < R ≤C and |Du| ≤Cat t = 0. Then, under the Ricci flow (∗1), limt→∞eu(x,y,t) converges uniformly onevery compact set and limt→∞eu(x,y,t) is either identically zero or positive every-where.

If limt→∞eu(x,y,t) > 0 then limt→∞eu(x,y,t)(dx2 + dy2) induces a metric onR2 with curvature identically zero.Nevertheless, it is possible to choose a 1-parameter family of diffeomorphisms toget weak modified convergence. To see why modifying the solution by diffeomor-phism is needed, we will illustrate the following example.Example 2.2.

Given a cigar soliton (or one of the Barenblatt solutions) ds2(0) =(dx2 + dy2)/(1 + x2 + y2) on R2, it is easy to compute that the solution of theRicci flow with initial data ds2(0) is ds2(t) = (dx2 + dy2)/(e4t + x2 + y2). Theneu(x,y,t) = 1/(e4t + x2 + y2) goes to zero as time approaches infinity; therefore, wecannot claim that limt→∞eu(x,y,t) yields a metric on R2.Nevertheless, if we let diffeomorphism φt(A, B) = (e2tA, e2tB) = (x, y), thends2(x, y, t) = ds2(φt(A, B), t) = dA2 + dB21 + A2 + B2 .Let cds2(A, B, t) = ds2(φt(A, B), t) and ebu = 1/(1 + A2 + B2).

Then ebu is stationaryin time.Note that the Ricci flow on other complete noncompact surfaces is also discussedin [W2].Now we will list the corresponding relations between the function v and thegeometric properties. If {r, θ} are polar coordinates on R2, then ds2 = v(dr2 +r2dθ2) and(R2, ds2) is complete =⇒v > 0 andZ ∞r=0v1/2(θ, r) dr = ∞(*4)∀0 ≤θ ≤2π,(*5)|R| ≤C ⇐⇒∆ln vv ≤C,(∗5′)0 < R ≤C ⇐⇒0 < −∆ln vv≤C,(*6)|Du| ≤C ⇐⇒|(v2x + v2y)/v3| ≤C2;

4L.-F. WU(*7)ZR−dµ ≤C ⇐⇒Zmax{−∆ln v, 0} dx dy ≤C,(*8)C∞= limr→∞Zrv1/2dθ > 0 =⇒v /∈L1,(*9)A(ds2) = limr→∞Rrv1/2dθRv1/2dr > 0.The relations (*5), (*6), and (*8) follow fromR = −e−u∆u = −∆ln vv,(**5)u = ln vand|Du|2 = 1v< vx, vy >v· < vx, vy >v,(**6)Zv dx dy =Zeu dx dy = ∞. (**8)We say that the PME (m = 0) on R2 has weak modified convergence at timeinfinity if there exists a 1-parameter family of reparametrizations {φt}t∈[0,∞) on R2such that for any sequence of times going to infinity there is a subsequence of times{tj}∞j=0 and the modified solution v(φtj ( · ), tj) converges uniformly to a positivefunction on every compact set as j →∞.

Then we haveMain Theorem* [PME (m = 0)]. On R2, let the positive function v satisfy(∗4), (∗5), and (∗6) at t = 0.

Then, under the PME (m = 0), the solution v(x, t) ofPME (m = 0) with v(x, 0) = v(x) has weak modified convergence at time infinityto a limiting positive function v∞satisfying (∗4). In the case when (∗5′) also holdsat t = 0, v∞is one of the Barenblatt solutions if C∞(v( · , 0)) < ∞, or a constantif A(v( · , 0)) > 0.Lemma *1 [PME (m = 0)] (Long time existence).

On R2, if the positive functionv satisfies (∗4), (∗5), and (∗6) at t = 0, then, under the PME (m = 0), the solutionv(x, t) of PME (m = 0) with v(x, 0) = v(x) exists for infinite time.Lemma *2 (PME (m = 0)). On R2, if the positive function v satisfies (∗4), (∗5),(∗6), (∗7), and (∗8) at t = 0, then, under the PME (m = 0), we have the following :1.

(Uniqueness) The solution of the flow is unique.2. (Geometric Properties) C∞, A(ds2), andRR2 R dµ < ∞are constants underthe flow.Lemma *3 (PME (m = 0)).

On R2, if the positive function v satisfies (∗4), (∗5′),and (∗6) at t = 0, then, under the PME (m = 0), limt→∞v(x, y, t) convergesuniformly on every compact set and limt→∞v(x, y, t) is either identically zero orpositive everywhere. If limt→∞v(x, y, t) > 0 then limt→∞v(x, y, t)(dx2 + dy2) in-duces a metric on R2 with curvature identically zero, in particular, limt→∞v(x, y, t)is a constant.Note that there is still a large class of Riemannian structures with C∞= ∞andA = 0, which our method fails to classify the limit.

A NEW RESULT FOR THE POROUS MEDIUM EQUATION5Sketch of the proof. The evolution equation of h = R + |Du|2 provides the infinitetime existence and uniform bounds for |Du|, |Dku|, R, and |DkR| for all k ≥1after a short time.

Finite total curvature and C∞> 0 imply that the curvaturedecays to zero at distance infinity. This yields that C∞, A(ds2), andRR dµ arepreserved under the flow.

Furthermore, the solution of the flow is unique and theinjectivity radius i(M) decays at most exponentially.The positivity of the curvature of an initial metric provides pointwise convergenceof the function eu at time infinity. The uniform bounds on |Dmu| imply limt→∞euis a smooth function and is either identically zero or positive everywhere.

In thecase, when limt→∞eu > 0, the limiting solution is a flat metric.We also may choose a 1-parameter family of diffeomorphisms φt : R2 →R2 suchthat there exists a sequence of times {tj}∞j=0 and limt→∞ds2(φtj( · ), tj) convergesuniformly on every compact set. If R > 0 at t = 0 and C∞(ds2(0)) < ∞, then someintegral bound classifies limt→∞ds2(φtj( · ), tj) as a cigar soliton with circumferenceno bigger than C∞(ds2(0)) < ∞.

If R > 0 at t = 0 and A(ds2(0)) > 0, then theHarnack’s inequality classifies limt→∞ds2(φtj( · ), tj) as a flat metric.References[A]D. G. Aronson, The porous medium equations, Some Problems in Nonlinear Diffusion (A.Fasano and M. Primicerio, eds. ), Lecture Notes in Maths., vol.

1224, Springer, New York,1986.[CW]B. Chow and L. Wu, The Ricci flow on compact 2-orbifolds with curvature negative some-where, Comm.

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XLIV, Wiley, New York, 1991, pp. 275–286.

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Analysis 103 (1988), 985–1039.[Ha1]R. Hamilton, The Ricci flow on surfaces, Contemp.

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237–262. [Ha2], Notes on Harnack’s inequality, preprint.[H1]M.

A. Herrero, A limiting case in nonlinear diffusion, Nonlinear Anal. 13 (1989), 611–628.

[H2], Singular diffusion on the line (to appear).[Shi]W. X. Shi, Complete noncompact K¨ahler manifolds with positive holomorphic bisectionalcurvature, Bull.

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Soc. (N.S.) 23 (1990), 437–440.[V]J.

L. Valazquez, Two nonlinear diffusion equations with finite speed of propagation, Pro-ceedings of the conference in honor of Jack Hale on the occasion of his 60th birthday,preprint.[W1]L. Wu, The Ricci flow on 2-orbifolds with positive curvature, J. Differential Geom 33(1991), 575–596.

[W2], The Ricci flow on complete R2 (The limiting case of the porous medium equationsas m →0), submitted.Center for Math Analysis, Australian National University, Canberra, Act 2601AustraliaE-mail address: lang@gauss.anu.edu.auCurrent address: Princeton University, Mathematics Department, Princeton, NJ08544E-mail address: lfwu@math.princeton.edu

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