aa r X i v : . [ m a t h . DG ] F e b MEAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND
JOHN LOTT
Abstract.
Following work of Ecker [3], we consider a weighted Gibbons-Hawking-Yorkfunctional on a Riemannian manifold-with-boundary. We compute its variational propertiesand its time derivative under Perelman’s modified Ricci flow. The answer has a boundaryterm which involves an extension of Hamilton’s differential Harnack expression for the meancurvature flow in Euclidean space. We also derive the evolution equations for the secondfundamental form and the mean curvature, under a mean curvature flow in a Ricci flowbackground. In the case of a gradient Ricci soliton background, we discuss mean curvaturesolitons and Huisken monotonicity. Introduction
In [3], Ecker found a surprising link between Perelman’s W -functional for Ricci flow andHamilton’s differential Harnack expression for mean curvature flow in R n . If Ω ⊂ R n is abounded domain with smooth boundary, he considered the integral over Ω of Perelman’s W -integrand [15, Proposition 9.1], the latter being defined using a positive solution u ofthe backward heat equation. With an appropriate boundary condition on u , the time-derivative of the integral has two terms. The first term is the integral over Ω of a nonnegativequantity, as in Perelman’s work. The second term is an integral over ∂ Ω. Ecker showed thatthe integrand of the second term is Hamilton’s differerential Harnack expression for meancurvature flow [6]. Hamilton had proven that this expression is nonnegative for a weaklyconvex mean curvature flow in R n .After performing diffeomorphisms generated by ∇ ln u , the boundary of Ω evolves bymean curvature flow in R n . Ecker conjectured that his W -functional for Ω is nondecreasingin time under the mean curvature flow of any compact hypersurface in R n . This conjectureis still open.In the present paper we look at analogous relations for mean curvature flow in an arbitraryRicci flow background. We begin with a version of Perelman’s F -functional [15, Section1.1] for a manifold-with-boundary M . We add a boundary term to the interior integralof F so that the result I ∞ has nicer variational properties. One can think of I ∞ as aweighted version of the Gibbons-Hawking-York functional [5, 17], where “weighted” refersto a measure e − f dV g . We compute how I ∞ changes under a variation of the Riemannianmetric g (Proposition 2). We also compute the time-derivative of I ∞ when g evolves byPerelman’s modified Ricci flow (Theorem 2). We derive the evolution equations for thesecond fundamental form of ∂M and the mean curvature of ∂M under the modified Ricciflow (Theorem 3). Date : January 31, 2012.2010
Mathematics Subject Classification.
After performing diffeomorphisms generated by −∇ f , the Riemannian metric on M evolves by the standard Ricci flow and ∂M evolves by mean curvature flow. Theorem 1. If u = e − f is a solution to the conjugate heat equation (1.1) ∂u∂t = ( −△ + R ) u on M , satisfying the boundary condition (1.2) H + e f = 0 on ∂M , then dI ∞ dt = 2 Z M | Ric + Hess( f ) | e − f dV (1.3)+ 2 Z ∂M (cid:18) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + 2 Ric( e , b ∇ f ) − e R − H Ric( e , e ) (cid:19) e − f dA. Here R is the scalar curvature of M , b ∇ is the boundarywise derivative, e is the inwardunit normal on ∂M , H is the mean curvature of ∂M and A ( · , · ) is the second fundamentalform of ∂M . Remark . If g ( t ) is flat Ricci flow on R n then the boundary integrand(1.4) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + 2 Ric( e , b ∇ f ) − e R − H Ric( e , e )becomes Hamilton’s differential Harnack expression [6, Definition 4.1](1.5) Z = ∂H∂t + 2 h V, b ∇ H i + A ( V, V )when the vector field V of (1.5) is taken to be − b ∇ f . In this flat case, Theorem 1 is the F -version of Ecker’s result. Remark . Writing (1.4) as(1.6) ∂H∂t + 2 h V, b ∇ H i − H Ric( e , e ) + (cid:28) e , ∇ V V − ∇ R − V, · ) (cid:29) (with V = − b ∇ f ) indicates a link to L -geodesics, since ∇ V V − ∇ R − V, · ) (with V = γ ′ ) enters in the Euler-Lagrange equation for the steady version of Perelman’s L -length[13, Section 4].Important examples of Ricci flow solutions come from gradient solitons. With such abackground geometry, there is a natural notion of a mean curvature soliton. We showthat (1.4) vanishes on such solitons (Proposition 7), in analogy to what happens for meancurvature flow in R n [6, Lemma 3.2]. On the other hand, in the case of convex meancurvature flow in R n , Hamilton showed that the shrinker version of (1.5) is nonnegative forall vector fields V [6, Theorem 1.1]. We cannot expect a direct analog for mean curvatureflow in an arbitrary gradient shrinking soliton background, since local convexity of thehypersurface will generally not be preserved by the flow. EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 3
Magni-Mantegazza-Tsatis [14] showed that Huisken’s monotonicity formula for mean cur-vature flow in R n [11, Theorem 3.1] extends to mean curvature flow in a gradient Ricci solitonbackground. We give two proofs of this result (Proposition 8). The first one is computationaland is essentially the same as the proof in [14]; the second one is more conceptual.In this paper we mostly deal with “steady” quantities : F -functional, gradient steadysoliton, etc. There is an evident extension to the shrinking or expanding case.The structure of the paper is as follows. Section 2 has some preliminary material. InSection 3 we define the weighted Gibbons-Hawking-York action and study its variationalproperties. Section 4 contains the derivation of the evolution equations for the secondfundamental form and the mean curvature of the boundary, when the Riemannian metricof the interior evolves by the modified Ricci flow. In Section 5 we consider mean curvatureflow in a Ricci flow background and prove Theorem 1. In the case of a gradient steady Riccisoliton background, we discuss mean curvature solitons and Huisken monotonicity.More detailed descriptions appear at the beginnings of the sections.2. Preliminaries
In this section we gather some useful formulas about the geometry of a Riemannianmanifold-with-boundary. We will use the Einstein summation convention freely.Let M be a smooth compact n -dimensional manifold-with-boundary. We denote localcoordinates for M by { x α } n − α =0 . Near ∂M , we take x to be a local defining function for ∂M .We denote the local coordinates for ∂M by { x i } ni =1 .If g is a Riemannian metric on M then we let ∇ denote the Levi-Civita connection on T M and we let b ∇ denote the Levi-Civita connection on T ∂M . Algorithmically, when takingcovariant derivatives we use Γ αβγ to act on indices in { , . . . , n − } and b Γ ijk to act on indicesin { , . . . , n − } . We let dV denote the volume density on M and we let dA denote thearea density on ∂M .Let e denote the inward-pointing unit normal field on ∂M . For calculations, we chooselocal coordinates near a point of ∂M so that ∂ (cid:12)(cid:12) ∂M = e and for all ( x , . . . , x n − ), thecurve t → ( t, x , . . . , x n − ) is a unit-speed geodesic which meets ∂M orthogonally. In thesecoordinates, we can write(2.1) g = ( dx ) + n − X i,j =1 g ij ( x , x . . . , x n − ) dx i dx j . We write A = ( A ij ) for the second fundamental form of ∂M , so A ij = g ( e , ∇ ∂ j ∂ i ), and wewrite H = g ij A ij for the mean curvature. Then on ∂M , we have(2.2) A ij = Γ ij = − Γ i j = − Γ ij = − ∂ g ij . The Codazzi-Mainardi equation(2.3) R ijk = b ∇ j A ik − b ∇ k A ij implies that(2.4) R j = b ∇ j H − b ∇ i A ij . JOHN LOTT
For later reference,(2.5) ∇ i R j = b ∇ i R j − Γ k i R kj − Γ ji R = b ∇ i R j + A ki R kj − A ij R . For any symmetric 2-tensor field v , we have(2.6) ∇ (cid:0) g ij v ij (cid:1) = g ij ∂ v ij + 2 A ij v ij = g ij ∇ v ij on ∂M . More generally, on ∂M , g ij is covariantly constant with respect to ∇ .If f ∈ C ∞ ( M ) then on ∂M , we have(2.7) ∇ i ∇ j f = b ∇ i b ∇ j f − Γ ji ∇ f = b ∇ i b ∇ j f − A ij ∇ f and(2.8) ∇ i ∇ f = b ∇ i ∇ f − Γ k i b ∇ k f = b ∇ i ∇ f + A ki b ∇ k f Variation of the weighted Gibbons-Hawking-York action
In this section we define the weighted Gibbons-Hawking-York action I ∞ and study itsvariational properties.In Subsection 3.1 we list how some geometric quantities vary under changes of the metric.As a warmup, in Subsection 3.2 we rederive the variational formula for the Gibbons-Hawkingaction. In Subsection 3.3 we derive the variational formula for the weighted Gibbons-Hawking action. In Subsection 3.4 we compute its time derivative under the modified Ricciflow.3.1. Variations.
Let δg αβ = v αβ be a variation of g . We write v = g αβ v αβ . We collect somevariational equations :(3.1) δR = ∇ α ∇ β v αβ − ∇ α ∇ α v − v αβ R αβ , (3.2) δ ( dV ) = v dV, (3.3) δ ( e ) = − v e − v k ∂ k ,δA ij = 12 ( ∇ i v j + ∇ j v i − ∇ v ij + A ij v )(3.4) = 12 (cid:16) b ∇ i v j + A ki v kj + b ∇ j v i + A kj v ki − ∇ v ij − A ij v (cid:17) , (3.5) δH = − v ij A ij + g ij δA ij = b ∇ i v i − (cid:0) g ij ∇ v ij + Hv (cid:1) , (3.6) δ ( dA ) = 12 v ii dA, EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 5
Gibbons-Hawking-York action.Definition 1.
The Gibbons-Hawking-York action [5, 17] is (3.7) I GHY ( g ) = Z M R dV + 2 Z ∂M H dA. If n = 2 then I GHY ( g ) = 4 πχ ( M ). Proposition 1. (3.8) δI GHY = − Z M v αβ (cid:18) R αβ − Rg αβ (cid:19) dV − Z ∂M v ij ( A ij − g ij H ) dA. Proof.
From (3.1) and (3.2),(3.9) δ Z M R dV = − Z M v αβ (cid:18) R αβ − Rg αβ (cid:19) dV − Z ∂M ( ∇ α v α − ∇ v ) dA. On the boundary, ∇ α v α − ∇ v = ∇ i v i − ∇ v ii (3.10) = b ∇ i v i − Γ j i v ij + Γ i i v − ∇ ( g ij v ij )= b ∇ i v i + A ij v ij − Hv − g ij ∇ v ij . From (3.5) and (3.6),(3.11) δ ( H dA ) = (cid:18) b ∇ i v i + 12 (cid:0) − g ij ∇ v ij − Hv + Hv ii (cid:1)(cid:19) dA. Combining (3.9), (3.10) and (3.11) gives δI GHY = − Z M v αβ (cid:18) R αβ − Rg αβ (cid:19) dV + Z ∂M b ∇ i v i dA − Z ∂M v ij ( A ij − g ij H ) dA (3.12) = − Z M v αβ (cid:18) R αβ − Rg αβ (cid:19) dV − Z ∂M v ij ( A ij − g ij H ) dA. This proves the proposition. (cid:3)
If the induced metric g ∂M is held fixed under the variation then v ij vanishes on ∂M and δI GHY = − R M v αβ (cid:0) R αβ − Rg αβ (cid:1) dV is an interior integral. This was the motivation forGibbons and Hawking to introduce the second term on the right-hand side of (3.7).Suppose that n >
2. We can say that with a fixed induced metric g ∂M on ∂M , the criticalpoints of I GHY are the Ricci-flat metrics on M that induce g ∂M . On the other hand, ifwe consider all variations v αβ then the critical points are the Ricci-flat metrics on M withtotally geodesic boundary. JOHN LOTT
Weighted Gibbons-Hawking-York action.
Given f ∈ C ∞ ( M ), consider the smoothmetric-measure space M = (cid:0) M, g, e − f dV (cid:1) . As is now well understood, the analog of theRicci curvature for M is the Bakry-Emery tensor(3.13) Ric ∞ = Ric + Hess( f ) . (There is also a notion of Ric N for N ∈ [1 , ∞ ] but we only consider the case N = ∞ .) Asexplained by Perelman [15, Section 1.3], the analog of the scalar curvature is(3.14) R ∞ = R + 2 △ f − |∇ f | . Considering the first variation formula for the integral of e − f over a moving hypersurface,one sees that the analog of the mean curvature is(3.15) H ∞ = H + e f. On the other hand, the analog of the second fundamental form is just A ∞ = A .If ∂M = ∅ then Perelman’s F -functional is the weighted total scalar curvature F = R M R ∞ e − f dV . Definition 2.
The weighted Gibbons-Hawking-York action is (3.16) I ∞ ( g, f ) = Z M R ∞ e − f dV + 2 Z ∂M H ∞ e − f dA. We write a variation of f as δf = h . Note that(3.17) δ (cid:0) e − f dV (cid:1) = (cid:16) v − h (cid:17) e − f dV. Proposition 2. If v − h = 0 then (3.18) δI ∞ = − Z M v αβ ( R αβ + ∇ α ∇ β f ) e − f dV − Z ∂M (cid:0) v ij A ij + v ( H + e f ) (cid:1) e − f dA. Proof.
One can check that δ ( △ f ) = △ h − (cid:0) ∇ α v αβ (cid:1) ∇ β f − v αβ ∇ α ∇ β f + 12 h∇ f, ∇ v i (3.19) = 12 △ v − (cid:0) ∇ α v αβ (cid:1) ∇ β f − v αβ ∇ α ∇ β f + 12 h∇ f, ∇ v i and(3.20) δ (cid:0) |∇ f | (cid:1) = 2 h∇ f, ∇ h i − v αβ ∇ α f ∇ β f = h∇ f, ∇ v i − v αβ ∇ α f ∇ β f. Then(3.21) δR ∞ = − v αβ ( R αβ + ∇ α ∇ β f ) e − f + ∇ β (cid:0) e − f (cid:0) ∇ α v βα − v βα ∇ α f (cid:1)(cid:1) . Hence δ (cid:18)Z M R ∞ e − f dV (cid:19) = Z M δ ( R ∞ ) e − f dV (3.22) = − Z M v αβ ( R αβ + ∇ α ∇ β f ) e − f dV − Z ∂M (cid:0) ∇ α v α − v α ∇ α f (cid:1) e − f dA. EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 7
On the boundary,(3.23) ∇ α v α − v α ∇ α f = ∇ v − v ( H + ∇ f ) + b ∇ i v i − v i b ∇ i f + v ij A ij . Next,(3.24) δ ( e f ) = − v ∇ f − v i b ∇ i f + ∇ h = − v ∇ f − v i b ∇ i f + 12 ∇ v and one finds that δ Z ∂M H ∞ e − f dA = Z ∂M δH ∞ e − f dA + Z ∂M H ∞ (cid:18) − v + 12 v ii (cid:19) e − f dA (3.25) = Z ∂M (cid:18) b ∇ i v i − v i b ∇ i f − v ( H + e f ) + 12 ∇ v (cid:19) e − f dA. Combining (3.22), (3.23) and (3.25) gives δI ∞ = − Z M v αβ ( R αβ + ∇ α ∇ β f ) e − f dV − Z ∂M (cid:0) v ij A ij + v ( H + e f ) (cid:1) e − f dA (3.26) + Z ∂M (cid:16) b ∇ i v i − v i b ∇ i f (cid:17) e − f dA = − Z M v αβ ( R αβ + ∇ α ∇ β f ) e − f dV − Z ∂M (cid:0) v ij A ij + v ( H + e f ) (cid:1) e − f dA + Z ∂M b ∇ i (cid:0) e − f v i (cid:1) dA = − Z M v αβ ( R αβ + ∇ α ∇ β f ) e − f dV − Z ∂M (cid:0) v ij A ij + v ( H + e f ) (cid:1) e − f dA. This proves the proposition. (cid:3)
Remark . If ∂M = ∅ then Proposition 2 appears in [15, Section 1.1].The variations in Proposition 2 all fix the measure e − f dV . If we also fix an inducedmetric g ∂M on ∂M then the critical points of I ∞ are gradient steady solitons on M thatsatisfy H + e f = 0 on ∂M . On the other hand, if we allow variations that do not fixthe boundary metric then the critical points are gradient steady solitons on M with totallygeodesic boundary and for which f satisfies Neumann boundary conditions.3.4. Time-derivative of the action.Assumption 1.
Hereafter we assume that H + e f = 0 on ∂M . Then on ∂M , we have(3.27) ∇ i ∇ j f = b ∇ i b ∇ j f + HA ij and(3.28) ∇ i ∇ f = − b ∇ i H + A ki b ∇ k f. JOHN LOTT
Theorem 2.
Under the assumptions (3.29) ∂g∂t = − f )) and (3.30) ∂f∂t = −△ f − R, we have dI ∞ dt = 2 Z M | Ric + Hess( f ) | e − f dV (3.31)+ 2 Z ∂M (cid:16) b △ H − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + A ij A ij H + A ij R ij + 2 R i b ∇ i f − b ∇ i R i (cid:17) e − f dA. If ( R ij + ∇ i ∇ j f ) (cid:12)(cid:12)(cid:12) ∂M = 0 and ( R i + ∇ i ∇ f ) (cid:12)(cid:12)(cid:12) ∂M = 0 then (3.32) b △ H − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + A ij A ij H + A ij R ij + 2 R i b ∇ i f − b ∇ i R i = 0 . Proof.
Equations (3.29) and (3.30) imply that e − f ( t ) dV g ( t ) is constant in t . Then Proposition2 implies that(3.33) dI ∞ dt = 2 Z M | Ric + Hess( f ) | e − f dV + 2 Z ∂M A ij ( R ij + ∇ i ∇ j f ) e − f dA. Lemma 1. On ∂M , A ij ( R ij + ∇ i ∇ j f ) e − f − b ∇ i (cid:0)(cid:0) R i + ∇ i ∇ f (cid:1) e − f (cid:1) =(3.34) (cid:16) b △ H − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + A ij A ij H + A ij R ij + 2 R i b ∇ i f − b ∇ i R i (cid:17) e − f . Proof.
As(3.35) A ij ( b ∇ i b ∇ j f ) e − f = b ∇ i (cid:16) A ij ( b ∇ j f ) e − f (cid:17) − (cid:16) b ∇ i A ij (cid:17) b ∇ j f e − f + A ( b ∇ f, b ∇ f ) e − f , we have A ij ( R ij + ∇ i ∇ j f ) e − f =(3.36) (cid:16) A ij R ij + A ij b ∇ i b ∇ j f + A ij A ij H (cid:17) e − f = (cid:16) A ij R ij − (cid:16) b ∇ i A ij (cid:17) b ∇ j f + A ( b ∇ f, b ∇ f ) + A ij A ij H (cid:17) e − f + b ∇ i (cid:16) A ij ( b ∇ j f ) e − f (cid:17) . Adding(3.37) 0 = (cid:16) b △ H − h b ∇ f, b ∇ H i (cid:17) e − f − b ∇ i (cid:16) e − f b ∇ i H (cid:17) EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 9 and using (2.4) gives A ij ( R ij + ∇ i ∇ j f ) e − f =(3.38) (cid:16) b △ H − h b ∇ f, b ∇ H i + A ij R ij − (cid:16) b ∇ i A ij (cid:17) b ∇ j f + A ( b ∇ f, b ∇ f ) + A ij A ij H (cid:17) e − f + b ∇ i (cid:16)(cid:16) A ij b ∇ j f − b ∇ i H (cid:17) e − f (cid:17) = (cid:16) b △ H − h b ∇ f, b ∇ H i + A ij R ij + R j b ∇ j f + A ( b ∇ f, b ∇ f ) + A ij A ij H (cid:17) e − f + b ∇ i (cid:16)(cid:16) A ij b ∇ j f − b ∇ i H (cid:17) e − f (cid:17) . Using (3.28), b ∇ i (cid:16)(cid:16) A ij b ∇ j f − b ∇ i H (cid:17) e − f (cid:17) = b ∇ i (cid:0)(cid:0) R i + ∇ i ∇ f (cid:1) e − f (cid:1) − b ∇ i (cid:0) R i e − f (cid:1) (3.39) = b ∇ i (cid:0)(cid:0) R i + ∇ i ∇ f (cid:1) e − f (cid:1) + (cid:16) − b ∇ i R i + R i b ∇ i f (cid:17) e − f . The lemma follows. (cid:3)
Equation (3.31) follows from (3.33), along with integrating both sides of (3.34) over ∂M .If ( R ij + ∇ i ∇ j f ) (cid:12)(cid:12)(cid:12) ∂M = 0 and ( R i + ∇ i ∇ f ) (cid:12)(cid:12)(cid:12) ∂M = 0 then (3.32) follows from (3.34). (cid:3) Evolution equations for the boundary geometry under a modified Ricciflow
In this section we consider a manifold-with-boundary whose Riemannian metric evolvesby the modified Ricci flow. We derive the evolution equations for the second fundamentalform and the mean curvature of the boundary.
Theorem 3.
Under the assumptions (4.1) ∂g∂t = − f )) and (4.2) ∂f∂t = −△ f − R, on ∂M we have (4.3) ∂g ij ∂t = − ( L b ∇ f g ) ij − R ij − HA ij ,∂A ij ∂t =( b △ A ) ij − ( L b ∇ f A ) ij − A ki R l klj − A kj R l kli + 2 A kl R kilj (4.4) − HA ik A kj + A kl A kl A ij + ∇ R i j and (4.5) ∂H∂t = b △ H − h b ∇ f, b ∇ H i + 2 A ij R ij + A ij A ij H + ∇ R . Proof.
Using (3.27), ∂g ij ∂t = − R ij − ∇ i ∇ j f (4.6) = − R ij − b ∇ i b ∇ j f − HA ij = − ( L b ∇ f g ) ij − R ij − HA ij . Next, from (3.4), ∂A ij ∂t = − ∇ i ( R j + ∇ j ∇ f ) − ∇ j ( R i + ∇ i ∇ f ) + ∇ ( R ij + ∇ i ∇ j f )(4.7) − A ij ( R + ∇ ∇ f ) . Now(4.8) ∇ ∇ i ∇ j f − ∇ j ∇ i ∇ f = ∇ ∇ j ∇ i f − ∇ j ∇ ∇ i f = − R ki j b ∇ k f − R i j ∇ f and ∇ i ∇ j ∇ f = b ∇ i ∇ j ∇ f − Γ ji ∇ ∇ f − Γ k i ∇ j ∇ k f (4.9) = b ∇ i (cid:16) b ∇ j ∇ f + A kj b ∇ k f (cid:17) − A ij ∇ ∇ f + A ki (cid:16) b ∇ j b ∇ k f + HA jk (cid:17) = − b ∇ i b ∇ j H + (cid:16) b ∇ i A kj (cid:17) b ∇ k f + A kj b ∇ i b ∇ k f − A ij ∇ ∇ f + A ki b ∇ j b ∇ k f + HA ki A jk . Then ∂A ij ∂t = − ∇ i R j − ∇ j R i + ∇ R ij − A ij ( R + ∇ ∇ f ) − R ki j b ∇ k f + HR i j (4.10) + b ∇ i b ∇ j H − (cid:16) b ∇ i A kj (cid:17) b ∇ k f − A kj b ∇ i b ∇ k f + A ij ∇ ∇ f − A ki b ∇ j b ∇ k f − HA ki A jk = b ∇ i b ∇ j H − (cid:16) b ∇ k A ij (cid:17) b ∇ k f − A kj b ∇ i b ∇ k f − A ki b ∇ j b ∇ k f − ∇ i R j − ∇ j R i + ∇ R ij − A ij R + HR i j − HA ki A jk = b ∇ i b ∇ j H − (cid:16) L b ∇ f A (cid:17) ij − ∇ i R j − ∇ j R i + ∇ R ij − A ij R + HR i j − HA ki A jk A form of Simons’ identity [16, Theorem 4.2.1] says that b ∇ i b ∇ j H =( b △ A ) ij + b ∇ i R j + b ∇ j R i − ∇ R ij (4.11) + A ki R k j + A kj R k i − A ij R + 2 A kl R kilj − HR i j − HA ki A jk + A kl A kl A ij + ∇ R i j . EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 11 (As a check, one can easily show that the contraction of both sides of (4.11) with g ij is thesame.) Then using (2.5), ∂A ij ∂t =( b △ A ) ij − (cid:16) L b ∇ f A (cid:17) ij − (cid:16) ∇ i R j − b ∇ i R j (cid:17) − (cid:16) ∇ j R i − b ∇ j R i (cid:17) (4.12) − A ij R + A ki R k j + A kj R k i + 2 A kl R kilj − HA ki A jk + A kl A kl A ij + ∇ R i j =( b △ A ) ij − ( L b ∇ f A ) ij − A ki R l klj − A kj R l kli + 2 A kl R kilj − HA ik A kj + A kl A kl A ij + ∇ R i j . This proves the evolution equation for A .Then ∂H∂t = ∂∂t (cid:0) g ij A ij (cid:1) = 2 (cid:16) R ij + b ∇ i b ∇ j f + HA ij (cid:17) A ij + g ij ∂A ij ∂t (4.13) =2 A ij R ij + b △ H − (cid:16) g ij ( L b ∇ f A ) ij − A ij b ∇ i b ∇ j f (cid:17) + A ij A ij H + g ij ∇ R i j = b △ H − h b ∇ f, b ∇ H i + 2 A ij R ij + A ij A ij H + ∇ R . This proves the theorem. (cid:3)
Proposition 3.
Under the assumptions (4.14) ∂g∂t = − f )) and (4.15) ∂f∂t = −△ f − R, we have dI ∞ dt = 2 Z M | Ric + Hess( f ) | e − f dV (4.16) + 2 Z ∂M (cid:18) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + 2 R i b ∇ i f − ∇ R − HR (cid:19) e − f dA. If ( R ij + ∇ i ∇ j f ) (cid:12)(cid:12)(cid:12) ∂M = 0 and ( R i + ∇ i ∇ f ) (cid:12)(cid:12)(cid:12) ∂M = 0 then (4.17) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + 2 R i b ∇ i f − ∇ R − HR = 0 . Proof.
From Theorem 3, b △ H − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + A ij A ij H + A ij R ij + 2 R i b ∇ i f − b ∇ i R i =(4.18) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) − A ij R ij + 2 R i b ∇ i f − b ∇ i R i − ∇ R . From the second contracted Bianchi identity,12 ∇ R = ∇ i R i + ∇ R = b ∇ i R i + A ij R ij − HR + ∇ R . (4.19)The proposition now follows from Theorem 2. (cid:3) Hypersurfaces in a Ricci flow background
In this section we consider mean curvature flow in a Ricci flow background. Mean curva-ture flow in a fixed Riemannian manifold was considered in [10].In Subsection 5.1 we translate the results of the previous sections from a fixed manifold-with-boundary, equipped with a modified Ricci flow, to an evolving hypersurface in a Ricciflow solution.Starting in Subsection 5.2, we consider mean curvature flow in a gradient Ricci solitonbackground. We define what it means for a hypersurface to be a mean curvature soliton.We show that the differential Harnack-type expression vanishes on mean curvature solitons.In Subsection 5.3 we give two proofs of the monotonicity of a Huisken-type functional.The first proof, which is calculational, is essentially the same as the one in [14]. The secondproof is noncalculational.5.1.
Mean curvature flow in a general Ricci flow background.
Let M be a smooth n -dimensional manifold and let g ( · ) satisfy the Ricci flow equation ∂g∂t = − n − { e ( · ) } be a smooth one-parameter family of immersionsof Σ in M . We write Σ t for the image of Σ under e ( t ), and consider { Σ t } to be a 1-parameter family of parametrized hypersurfaces in M . Suppose that { Σ t } evolves by themean curvature flow(5.1) dxdt = He . Proposition 4.
We have (5.2) ∂g ij ∂t = − R ij − HA ij ,∂A ij ∂t =( b △ A ) ij − A ki R l klj − A kj R l kli + 2 A kl R kilj (5.3) − HA ik A kj + A kl A kl A ij + ∇ R i j and (5.4) ∂H∂t = b △ H + 2 A ij R ij + A ij A ij H + ∇ R . Proof.
Suppose first that Σ t = ∂X t with each X t compact. Given a time interval [ a, b ], finda positive solution on S t ∈ [ a,b ] ( X t × { t } ) ⊂ M × [ a, b ] of the conjugate heat equation(5.5) ∂u∂t = ( −△ + R ) u, satisfying the boundary condition e u = Hu , by solving it backwards in time from t = b .(Choosing diffeomorphisms from { X t } to X a , we can reduce the problem of solving (5.5) toa parabolic equation on a fixed domain.) Define f by u = e − f .Let { φ t } t ∈ [ a,b ] be the one-parameter family of diffeomorphisms generated by {−∇ g ( t ) f ( t ) } t ∈ [ a,b ] ,with φ a = Id. Then φ t ( X a ) = X t for all t . Put b g ( t ) = φ ∗ t g ( t ) and b f ( t ) = φ ∗ t f ( t ). Then • b g ( t ) and b f ( t ) are defined on X a , EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 13 • ∂ b g∂t = − b g + Hess( b f )), • e b f + b H = 0 and • the measure e − b f ( t ) dV b g ( t ) is constant in t .The proposition now follows from applying ( φ ∗ t ) − to equations (4.3), (4.4) and (4.5) (thelatter three being written in terms of b g and b f ).As the result could be derived from a local calculation on Σ t , it is also valid without theassumption that Σ t bounds a compact domain. (cid:3) Example . If M = R n and g ( t ) = g flat then equations (5.2), (5.3) and (5.4) are the same as[9, Lemma 3.2, Theorem 3.4 and Corollary 3.5] Proposition 5. If u = e − f is a solution to the conjugate heat equation (5.5) then dI ∞ dt = 2 Z M | Ric + Hess( f ) | e − f dV (5.6) + 2 Z ∂M (cid:18) ∂H∂t − h b ∇ f, b ∇ H i + A ( b ∇ f, b ∇ f ) + 2 R i b ∇ i f − ∇ R − HR (cid:19) e − f dA. Proof.
This follows from Proposition 3. (cid:3)
Example . If M = R n , and g ( t ) = g flat then Proposition 5 is the same as [3, Propositions3.2 and 3.4], after making the change from the F -type functional considered in this paperto the W -type functional considered in [3].We will need the next lemma later. Lemma 2.
We have (5.7) ddt ( dA ) = − (cid:0) R ii + H (cid:1) dA. Proof.
Using (5.2),(5.8) ddt ( dA ) = 12 (cid:18) g ij ∂g ij ∂t (cid:19) dA = − (cid:0) R ii + H (cid:1) dA. This proves the lemma. (cid:3)
Using Lemma 2, we prove that a mean curvature flow of two-spheres, in a three-dimensionalimmortal Ricci flow solution, must have a finite-time singularity.
Proposition 6.
Suppose that ( M, g ( · )) is a three-dimensional Ricci flow solution that isdefined for t ∈ [0 , ∞ ) , with complete time slices and uniformly bounded curvature on compacttime intervals. If { Σ t } is a mean curvature flow of two-spheres in ( M, g ( · )) then the meancurvature flow has a finite-time singularity.Proof. We estimate the area of Σ t , along the lines of Hamilton’s area estimate for minimaldisks in a Ricci flow solution [7, Section 11], [12, Lemma 91.12]. Let A ( t ) denote the areaof Σ t . From Lemma 2,(5.9) d A dt = − Z Σ t ( R ii + H ) dA. Now(5.10) R ii = 12 (cid:0) R + R ijij (cid:1) = 12 (cid:16) R + b R − H + A ij A ij (cid:17) , where b R denotes the scalar curvature of Σ t . From a standard Ricci flow estimate [12, (B.2)],(5.11) R ( x, t ) ≥ − t . Then(5.12) d A dt = − Z Σ t (cid:16) R + b R + H + A ij A ij (cid:17) dA ≤ t A ( t ) − πχ (Σ t ) = 34 t A ( t ) − π. It follows that for any time t > A ( t ) ≤ t ≥ t (cid:16) A ( t )16 πt (cid:17) . Thus the mean curvature flow must go singular. (cid:3) Remark . The analog of Proposition 6, in one dimension lower, is no longer true, as canbeen seen by taking a closed geodesic in a flat 2-torus. This contrasts with the fact thatany compact mean curvature flow in R n has a finite-time singularity.5.2. Mean curvature solitons.
Suppose that (
M, g ( · ) , f ( · )) is a gradient soliton solutionto the Ricci flow. We recall that this means(1) ( M, g ( · )) is a Ricci flow solution,(2) At time t we have(5.13) R αβ + ∇ α ∇ β f = c t g αβ , where c = 0 in the steady case (for t ∈ R ), c = − t ∈ ( −∞ , c = 1 in the expanding case (for t ∈ (0 , ∞ )), and(3) The function f satisfies(5.14) ∂f∂t = |∇ f | . Definition 3.
At a given time t , a hypersurface Σ t is a mean curvature soliton if (5.15) H + e f = 0 . Equation (5.15) involves no choice of local orientations.When restricted to Σ t , the equations R ij + ∇ i ∇ j f = 0 and R i + ∇ i ∇ f = 0 become R ij + b ∇ i b ∇ j f + HA ij = 0 , (5.16) R i − b ∇ i H + A ki b ∇ k f = 0 . Example . If M = R n and g ( t ) = g flat , let L be a linear function on R n . Put f = L + t |∇ L | ,so that f satisfies (5.14). Then after changing f to − f , the equations in (5.16) become b ∇ i b ∇ j f = HA ij , (5.17) b ∇ i H + A ki b ∇ k f = 0 , which appear on [6, p. 219] as equations for a translating soliton. EAN CURVATURE FLOW IN A RICCI FLOW BACKGROUND 15
If (
M, g ( · ) , f ( · )) is a gradient steady soliton, let { φ t } be the one-parameter family of diffeo-morphisms generated by the time-independent vector field −∇ g ( t ) f ( t ), with φ = Id. If Σ isa mean curvature soliton at time zero then its ensuing mean curvature flow { Σ t } consists ofmean curvature solitons, and { Σ t } differs from { φ t (Σ ) } by hypersurface diffeomorphisms.There is a similar description of the mean curvature flow of a mean curvature soliton if( M, g ( · ) , f ( · )) is a gradient shrinking soliton or a gradient expanding soliton. Proposition 7. If ( M, g ( · ) , f ( · )) is a gradient steady soliton and { Σ t } is the mean curvatureflow of a mean curvature soliton then (5.18) ∂H∂t − h b ∇ f , b ∇ H i + A ( b ∇ f , b ∇ f ) + 2 R i b ∇ i f − ∇ R − HR = 0 . Proof.
We clearly have (cid:0) R ij + ∇ i ∇ j f (cid:1) (cid:12)(cid:12)(cid:12) Σ t = 0 and (cid:0) R i + ∇ i ∇ f (cid:1) (cid:12)(cid:12)(cid:12) Σ t = 0. The propositionnow follows from Proposition 3, along the lines of the proof of Proposition 4. (cid:3) Example . Suppose that M = R n , g ( t ) = g flat , L is a linear function on R n and f = L + t |∇ L | . After putting V ( t ) = − b ∇ f , Proposition 7 is the same as [6, Lemma 3.2].5.3. Huisken monotonicity.Proposition 8. [14] If { Σ t } is a mean curvature flow of compact hypersurfaces in a gradientsteady Ricci soliton ( M, g ( · ) , f ( · )) then R Σ t e − f ( t ) dA is nonincreasing in t . It is constant in t if and only if { Σ t } are mean curvature solitons.Proof.
1. Using the mean curvature flow to relate nearly Σ t ’s, Lemma 2 gives ddt Z Σ t e − f ( t ) dA = − Z Σ t (cid:18) dfdt + R ii + H (cid:19) e − f ( t ) dA (5.19) = − Z Σ t (cid:18) ∂f∂t + He f + R ii + H (cid:19) e − f ( t ) dA = − Z Σ t (cid:0) |∇ f | + He f + R ii + H (cid:1) e − f ( t ) dA. From the soliton equation,(5.20) 0 = R ii + ∇ i ∇ i f = R ii + b ∇ i b ∇ i f + Γ i i ∇ f = R ii + b △ f − He f . Then ddt Z Σ t e − f ( t ) dA = − Z Σ t (cid:16) − b △ f + | b ∇ f | + | e f | + 2 He f + H (cid:17) e − f ( t ) dA (5.21) = − Z Σ t (cid:0) H + e f (cid:1) e − f ( t ) dA. The proposition follows. (cid:3)
Proof.
2. Let { φ t } be the one-parameter family of diffeomorphisms considered after Example3. Put b g ( t ) = φ ∗ t g ( t ) and b f ( t ) = φ ∗ t f ( t ). Then for all t , we have b g ( t ) = g (0) and b f ( t ) = f (0).Put b Σ t = φ − t (Σ t ). In terms of g (0) and f (0), the surfaces b Σ t satisfy the flow(5.22) dxdt = He + ∇ f (0) = ( H + e f (0)) e + b ∇ f (0) , which differs from the flow(5.23) dxdt = ( H + e f (0)) e by diffeomorphisms of the hypersurfaces. The flow (5.23) is the negative gradient flow of thefunctional b Σ → R b Σ e − f (0) dA . Hence R b Σ t e − f (0) dA is nonincreasing in t , and more precisely,(5.24) ddt Z b Σ t e − f (0) dA = − Z b Σ t (cid:0) H + e f (0) (cid:1) e − f (0) dA. The proposition follows. (cid:3)
Remark . There are evident analogs of Proposition 8, and its proofs, for mean curvatureflows in gradient shrinking Ricci solitons and gradient expanding Ricci solitons. For theshrinking case, where t ∈ ( −∞ , τ = − t . Then τ − ( n − / R Σ t e − f dA is nonincreasingin t . When M = R n , g ( τ ) = g flat and f ( x, τ ) = | x | τ , we recover Huisken monotonicity [11,Theorem 3.1]. Remark . With reference to the second proof of Proposition 8, the second variation ofthe functional Σ → R Σ e − f dA was derived in [1]; see [4, 8] for consequences. The secondvariation formula also plays a role in [2, Section 4]. Remark . As a consequence of the monotonicity statement in Remark 5, we can saythe following about singularity models; compare with [11, Theorem 3.5]. Suppose that(
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Department of Mathematics, University of California - Berkeley, Berkeley, CA 94720-3840, USA
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