Flow by mean curvature inside a moving ambient space

FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE ANNIBALE MAGNI, CARLO MANTEGAZZA, AND EFSTRATIOS TSATIS arXiv:0911.5130v1 [math.DG] 27 Nov 2009 A BSTRACT. We show some computations related in particular to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the analogous of Huisken’s monotonicity formula and its connection with the validity of some Li–Yau–Hamilton Harnack–type inequalities in a moving manifold. C ONTENTS 1. Static Ambient Space 2. Moving Ambient Space 3. Ricci and Back–Ricci Flow 3.1. Ricci Flow Case 3.2. Back–Ricci Flow Case 4. Li–Yau–Hamilton Harnack Inequalities and Ricci Flow 4.1. Computation I: Ricci Flow 4.2. Computation II: Back–Ricci Flow 4.3. Dimension 2 4.4. A Very Special Case References 1 2 3 3 4 4 5 7 8 9 9 1. S TATIC A MBIENT S PACE First we show the extension of Huisken’s monotonicity formula by Hamilton [3]. Let u be a positive solution of the backward heat equation on a Riemannian manifold (M, g), ut = −∆M u . Let us assume we have a smooth, compact, immersed submanifold N with dim N = n evolving by the mean curvature flow in the ambient space M with dim M = m, the metric on N is the induced metric and we let µ to be the associated measure. We denote the normal indices with α, β, γ, . . . and the tangent ones with i, j, k, . . . , then, ∆M u = ∆N u + g αβ ∇α ∇β u − Hα ∇α u . Now we compute Z Z Z d u dµ = ut + Hα ∇α u − H2 u dµ = −∆M u + Hα ∇α u − H2 u dµ . dt N N N Using (1.1) and integrating by parts we obtain Z Z d −g αβ ∇α ∇β u + 2Hα ∇α u − H2 u dµ . u dµ = dt N N α u we get Adding and subtracting the quantity ∇α u∇ u  Z  Z Z d ∇α u∇α u ∇α u∇α u 2 α − ∇α ∇α u − dµ . u dµ = − H u − 2H ∇α u + dt N u u N N Date: November 27, 2009. 1 (1.1) FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 2 This becomes d dt Z N u dµ = − Z N H− ∇⊥ u u 2 u dµ − Z N ∇α ∇α u − ∇α u∇α u dµ . u Finally, setting τ = T − t for some constant T ∈ R one obtains, for every t < T ,   Z Z 2 m−n m−n ∇⊥ u d u dµ = − τ 2 H− τ 2 u dµ dt u N N Z m−n ∇α u∇α u u ∇α ∇α u − + (m − n) dµ −τ 2 u 2τ N Z 2 m−n ∇⊥ u 2 H− u dµ = −τ u N ! Z ∇2αβ u ∇α u∇β u gαβ m−n −τ 2 g αβ u dµ + − 2 u u 2τ N Z m−n 2 H + ∇⊥ f e−f dµ = −τ 2 ZN  m−n gαβ  αβ −f ∇2αβ f − g e dµ , +τ 2 2τ N (1.2) where in the last passage we substituted u = e−f , as u > 0. Notice that ft = −∆M f + |∇f |2 . This is Hamilton’s result in [3]. 2. M OVING A MBIENT S PACE Let us assume now that the metric of the ambient space evolves by the rule gt = −2Q (if Q = Ric we have the Ricci flow) and the backward heat equation is modified to ut = −∆M u + Ku for some function K. If now we repeat the previous computation we have two extra terms, the first is coming from the modification to the equation for u, the second from the time derivative of the measure on N . Indeed, the associated metric on N is affected not only by the motion of the submanifold but also by the evolution of the ambient metric on M . After some computation we have d µ = (−H2 − g ij Qij )µ = (−H2 − tr Q + g αβ Qαβ )µ . dt Therefore we get   Z Z 2 m−n m−n d ∇⊥ u τ 2 u dµ u dµ = − τ 2 H− dt u N N ! Z ∇2αβ u ∇α u∇β u gαβ m−n −τ 2 g αβ u dµ + − u u2 2τ N Z m−n 2 (K − tr Q + g αβ Qαβ )u dµ +τ N Z m−n 2 = −τ 2 H + ∇⊥ f e−f dµ ZN  m−n gαβ  αβ −f ∇2αβ f + Qαβ − g e dµ +τ 2 2τ ZN m−n +τ 2 (K − tr Q)e−f dµ , N (2.1) FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 3 where we substituted u = e−f , hence, ft = −∆M f + |∇f |2 − K. This computation suggests that a good choice is K = tr Q as the last term vanishes and we get   Z Z m−n m−n d 2 τ 2 u dµ = − τ 2 H + ∇⊥ f e−f dµ dt N ZN  m−n gαβ  αβ −f ∇2αβ f + Qαβ − +τ 2 g e dµ . (2.2) 2τ N Moreover, notice that with the choice K = tr Q, we have Z Z Z d −∆M u = 0 , ut − tr Qu = u= dt M M M R R at least when the ambient manifold M is compact, hence the integral M u = M e−f is constant during the flow. 3. R ICCI AND B ACK –R ICCI F LOW 3.1. Ricci Flow Case. We choose now Q = Ric, that is, the metric g on M evolves by the Ricci flow in some time interval (a, b) ⊂ R, and we set K = R to be the scalar curvature. By the previous computation we get   Z Z m−n m−n d 2 2 2 τ u dµ = − τ H + ∇⊥ f e−f dµ dt N ZN  m−n gαβ  αβ −f ∇2αβ f + Rαβ − g e dµ , (3.1) +τ 2 2τ N for a positive solution of the conjugate backward heat equation and f = − log u. Hence, m−n 2 ut = −∆u + Ru (3.2) ft = −∆f + |∇f |2 − R . (3.3) R Monotonicity u dµ  is so related to the nonpositivity of the Li–Yau–Hamilton type N  of τ gαβ 2 expression ∇αβ f + Rαβ − 2τ g αβ . Notice that the same conclusion holds also if ut ≤ −∆u + Ru. If (M, g(t)) is a gradient soliton of Ricci flow and f its “potential” function, it is well known that u = e−f satisfies the conjugate heat equation (3.2) and we have • Expanding Solitons: flow defined on (Tmin , +∞) and ∇2 f + Ric = g/2(Tmin − t) • Steady Solitons: eternal flow and ∇2 f + Ric = 0 • Shrinking Solitons: flow defined on (−∞, Tmax ) and ∇2 f + Ric = g/2(Tmax − t) Substituting, in the three cases, the above expression becomes   1 1 • Expanding Soliton: m−n 2 Tmin −t − T −t which is always negative as t ∈ (Tmin , T ).   − T 1−t which is always negative as t ∈ (−∞, T ). • Steady Soliton: m−n 2   1 1 − • Shrinking Soliton: m−n 2 Tmax −t T −t which is nonpositive if T ≤ Tmax as t ∈ (−∞, min{T, Tmax }). Proposition 3.1. If (M, g(t)) is a steady or expanding gradient soliton and f is its potential function, then monotonicity holds for every T ≥ Tmin. If (M, g(t)) is a shrinking gradient soliton on (−∞, Tmax ) and f is its potential function, then monotonicity holds for every T ≤ Tmax . FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 4 3.2. Back–Ricci Flow Case. If we choose Q = −Ric, that is, the metric g evolves by back–Ricci flow in some time interval (a, b) ⊂ R, and we set K = R to be the scalar curvature. By the previous computation we get   Z Z m−n m−n d 2 2 2 u dµ = − τ H + ∇⊥ f e−f dµ τ dt N ZN  m−n gαβ  αβ −f ∇2αβ f − Rαβ − +τ 2 g e dµ , (3.4) 2τ N for a positive solution of the conjugate backward heat equation ut = −∆u − Ru and f = − log u. Hence, (3.5) ft = −∆f + |∇f |2 + R m−n R 2 Monotonicity is so related to the nonpositivity of the Li–Yau–Hamilton type N u dµ   of τ gαβ 2 expression ∇αβ f − Rαβ − 2τ g αβ . Notice that the same conclusion holds also if ut ≤ −∆u − Ru. 4. L I –YAU –H AMILTON H ARNACK I NEQUALITIES AND R ICCI F LOW • We denote with fij = ∇2ij f the second covariant derivative of f , then ∇2ij f = ∂2f ∂f − Γkij . ∂xi ∂xj ∂xk • Let ωi a 1–form, then we have the following formula for interchanging of covariant derivatives ∇pq ωi − ∇qp ωi = Rpqis ωs . Let ωij a 2–form, then ∇pq ωij − ∇qp ωij = Rpqis ωsj + Rpqj s ωis . • II Bianchi Identity: contracted, that is, ∇s Rijkl + ∇l Rijsk + ∇k Rijls = 0 g js ∇s Rijkl − ∇l Ricik + ∇k Ricil = 0 div Riemikl = ∇k Ricil − ∇l Ricik contracted again (Schur Lemma), div Rick = ∇k R − div Rick that is, div Ric = ∇R/2 . • Evolution equations for Ricci tensor and scalar curvature under Ricci flow: ∂t Ricij = ∆Ricij + 2Ricpq Ripjq − 2g pq Ricip Ricqj ∂t R = ∆R + 2|Ric|2 . • Evolution equations for Christoffel symbols under Ricci flow: ∂t Γkij = −g kl (∇i Ricjl + ∇j Ricil − ∇l Ricij ) . FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 5 • Interchange of Laplacian and second derivatives: ∇2ij ∆f = ∇i ∇j ∇k ∇k f = ∇i (Rjkkp ∇p f ) + ∇i ∇k ∇j ∇k f = − ∇i (Ricjp ∇p f ) + ∇i ∇k ∇k ∇j f = − ∇i Ricjp ∇p f − Ricjp fip + ∇i ∇k ∇k ∇j f = − ∇i Ricjp ∇p f − Ricjp fip + Rikkp fpj + Rikjp fkp + ∇k ∇i ∇k ∇j f = − ∇i Ricjp ∇p f − Ricjp fip − Ricip fpj − Rikpj fkp + ∇k (Rikjp ∇p f ) + ∇k ∇k ∇i ∇j f = − ∇i Ricjp ∇p f − ∇k Rikpj ∇p f − Ricjp fip − Ricip fpj − Rikpj fkp − Rikpj fkp + ∆∇i ∇j f = − (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f − Ricjp fip − Ricip fpj − 2Rikpj fkp + ∆∇i ∇j f where in the last passage we used the II Bianchi identity. Hence, ∇2ij ∆f − ∆∇i ∇j f = −(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f − Ricjp fip − Ricip fpj − 2Rikpj fkp . 4.1. Computation I: Ricci Flow. Suppose that ut = −∆u + Ru and u > 0, we want to show the nonpositivity of the term ∇2ij f + Rij − gij 2τ for f = − log u which satisfies ft = −∆f + |∇f |2 − R . Equivalently, if we had chosen f = log u, we can show the positivity of ∇2ij f − Rij + gij 2τ for f = log u which satisfies ft = −∆f − |∇f |2 + R . We set τ = T − t, Lij = fij − Ricij , Hij = τ Lij + gij /2 = τ [fij − Ricij ] + gij /2. FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 6 (∂t + ∆)Hij = − Lij − Ricij + τ [∆fij + ∇2ij ft + (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [∂t Ricij + ∆Ricij ] = − Lij − Ricij + τ [∆fij − ∇2ij ∆f − ∇2ij |∇f |2 + (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] = − Lij − Ricij + τ [(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f + Ricjp fip + Ricip fpj + 2Rikpj fkp − ∇2ij |∇f |2 + (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] = − Lij − Ricij + τ [Ricjp fip + Ricip fpj + 2Rikpj fkp − 2fip fjp − 2∇3ijk f ∇k f + 2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] = − Lij − Ricij + τ [Ricjp fip + Ricip fpj + 2Rikpj fkp − 2fip fjp − ∇3ijk f ∇k f − ∇3jik f ∇k f + 2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] = − Lij − Ricij + τ [Ricjp fip + Ricip fpj + 2Rikpj fkp − 2fip fjp − 2∇3kij f ∇k f − 2Rikjp ∇p f ∇k f + 2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] = − Lij − Ricij + τ [Ricjp fip + Ricip fpj − 2fip fjp − 2∇3kij f ∇k f ] − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2 R] + τ [2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f − 2Rikjp fpk − 2Rikjp ∇p f ∇k f ] substituting, Lij = [Hij − gij /2]/τ and fij = [Hij − gij /2]/τ + Ricij , we get FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 7 (∂t + ∆)Hij = − Hij /τ + gij /2τ − Ricij + τ [Ricjp Ricip + Ricip Ricpj − 2∇k Ricij ∇k f ] 2 − 2τ [Hij /τ 2 − Hij /τ 2 + gij /4τ 2 + Ricik Rickj + Ricik Hjk /τ + Ricjk Hik /τ − Ricij /τ ] + [Ricjp Hip + Ricip Hpj − 2∇k Hij ∇k f ] − Ricij − τ [2∆Ricij + 2Ricpq Ripjq − 2Ricip Ricpj − ∇2ij R] + τ [2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − 2τ Rikjp Ricpk − 2Rikjp Hpk + Ricij − 2τ Rikjp ∇p f ∇k f 2 = [Hij − 2Hij ]/τ + Ricij + τ [Ricjp Ricip + Ricip Ricpj − 2∇k Ricij ∇k f ] − 2τ [Ricik Rickj + Ricik Hjk /τ + Ricjk Hik /τ ] + [Ricjp Hip + Ricip Hpj − 2∇k Hij ∇k f ] − τ [2∆Ricij + 4Ricpq Ripjq − 2Ricip Ricpj − ∇2ij R] + τ [2(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − 2Rikjp Hpk − 2τ Rikjp ∇p f ∇k f 2 = [Hij − 2Hij ]/τ − 2∇k Hij ∇k f − [Ricik Hjk + Ricjk Hik + 2Rikjp Hpk ] − τ [2∆Ricij − 2Ricjp Ricip + 4Ricpq Ripjq − ∇2ij R − Ricij /τ ] + τ [2(∇i Ricjk + ∇j Ricik − 2∇k Ricij )∇k f ] − 2τ Rikjp ∇p f ∇k f so finally, we get 2 (∂t + ∆)Hij = [Hij − 2Hij ]/τ − 2∇k Hij ∇k f − Ricki Hkj − Rickj Hki − 2Ripjq H pq − τ [2∆Ricij − 2g pq Ricip Ricjq + 4Ricpq Ripjq − ∇2ij R − Ricij /τ ] + 2τ (∇i Ricjk + ∇j Ricik − 2∇k Ricij )∇k f − 2τ Ripjq ∇p f ∇q f . Notice that the second and third lines gives the Hamilton’s Harnack quadratic with a wrong term −Ricij /τ . 4.2. Computation II: Back–Ricci Flow. Suppose that ut = −∆u − Ru and u > 0, we want to show the nonpositivity of the term gij ∇2ij f − Rij − 2τ for f = − log u which satisfies ft = −∆f + |∇f |2 + R . Equivalently, if we had chosen f = log u we can show the positivity of gij ∇2ij f + Rij + 2τ for f = log u which satisfies ft = −∆f − |∇f |2 − R . • Evolution equations for Ricci tensor and scalar curvature under back–Ricci flow: ∂t Ricij = −(∆Ricij + 2Ricpq Ripjq − 2g pq Ricip Ricqj ) ∂t R = −(∆R + 2|Ric|2 ) . FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 8 • Evolution equations for Christoffel symbols under back–Ricci flow: ∂t Γkij = g kl (∇i Ricjl + ∇j Ricil − ∇l Ricij ) . We set fi = ∇i f , fij = ∇2ij f and Lij = fij + Ricij , Hij = τ Lij + gij /2 = τ [fij + Ricij ] + gij /2, (∂t + ∆)Hij = − Lij + Ricij + τ [∆fij + ∇2ij ft − (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] + τ [∂t Ricij + ∆Ricij ] = − fij + τ [∆fij − ∇2ij ∆f − ∇2ij |∇f |2 − (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] = − fij + τ [(∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f + g pq Ricjp fiq + g pq Ricip fqj − 2Ripjq f pq ] + τ [−∇2ij |∇f |2 − (∇i Ricjk + ∇j Ricik − ∇k Ricij )∇k f ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] = − fij + τ [g pq Ricjp fiq + g pq Ricip fqj − 2Ripjq f pq − ∇2ij |∇f |2 ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] = − fij + τ [g pq Ricjp fiq + g pq Ricip fqj − 2Ripjq f pq ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] − τ [2fip fjp + 2∇3ijk f ∇k f ] = − fij + τ [g pq Ricjp fiq + g pq Ricip fqj − 2Ripjq f pq ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] − τ [2fip fjp + 2∇3kij f ∇k f + 2Ripjq ∇p f ∇q f ] . Suppose now that at time t > 0, the tensor Hij (which goes +∞ as t → T − ) get its “last” zero eigenvalue at some point (p, t) in space and time, with V i unit zero eigenvector. We extend V i in space such that ∇V (p) = ∇2 V (p) = 0 and constant in time. Then if Z = Hij V i V j we have that Z has a global minimum on M × [t, T ] at (p, t). At such point we have Z = 0, ∇Z = 0 and ∆Z ≥ 0, hence, fij V i V j = −Ricij V i V j − 1/2τ , and as ∇Z = 0, ∇k fij V i V j = −∇k Ricij V i V j . Then 0 ≤ ∂t Z + ∆Z = (∂t Hij + ∆Hij )V i V j = {−fij + τ [g pq Ricjp fiq + g pq Ricip fqj − 2Ripjq f pq ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] − τ [2fip fjp + 2∇3kij f ∇k f + 2Ripjq ∇p f ∇q f ]}V i V j = {Ricij + gij /2τ + τ [−2Ric2ij − Ricij /τ + 2Ricpq Ripjq + Ricij /τ ] − τ [2Ricpq Ripjq − 2g pq Ricip Ricqj + ∇2ij R] − τ [2Ric2ij + 2Ricij /τ + gij /2τ 2 − 2∇k Ricij ∇k f + 2Ripjq ∇p f ∇q f ]}V i V j = {−Ricij − τ [2Ric2ij + ∇2ij R − 2∇k Ricij ∇k f + 2Ripjq ∇p f ∇q f ]}V i V j = − τ {∇2ij R + 2Ric2ij + Ricij /τ − 2∇k Ricij ∇k f + 2Ripjq ∇p f ∇q f ]}V i V j . By this computation, it follows that we would get a contradiction by maximum principle, if the following Hamilton–Harnack type inequality is true. ∇2ij R + 2Ric2ij + Ricij /τ − 2∇k Ricij U k + 2Ripjq U p U q ≥ 0 . See [4] and also [2]. 4.3. Dimension 2. In the special two–dimensional case of a surface with bounded and positive scalar curvature this inequality holds, see [1, Chapter 15, Section 3]. If a positive function u satisfies ut = −∆u − Ru FLOW BY MEAN CURVATURE INSIDE A MOVING AMBIENT SPACE 9 for a closed curve moving by its curvature k inside a surface evolving by gt = 2Ric = Rg, we have   Z Z √ d √ 2 u ds ≤ − τ τ k − ∇⊥ log u u ds , dt γ γ where ν is the unit normal to the curve γ. 4.4. A Very Special Case. In dimension 2, for a surface with positive and bounded scalar curvature, we consider the scalar curvature function u = R > 0. It satisfies ut = −∆u − Ru as, under the back–Ricci flow, we have ∂t R = −∆R − R2 . In this case we can get directly the monotonicity formula   Z Z √ d √ 2 R ds ≤ − τ τ k − ∇⊥ log R R ds , dt γ γ as the Li–Yau quadratic in this case, that is, 1 , 2τ is nonnegative being exactly the “special” form of Hamilton–Harnack inequality for surfaces with positive scalar curvature (see [1]). This inequality becomes an equality (for every curve) iff M is a gradient expanding Ricci soliton with R > 0 and k = ∇⊥ log R. ∇2νν log R + R/2 + R EFERENCES 1. 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Mantegazza: [email protected] (Efstratios Tsatis) D EPARTMENT OF P HYSICS , U NIVERSITY OF PATRAS , G REECE , GR-26500 E-mail address, Efstratios Tsatis: [email protected] View publication stats