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The Geometry of Differential Harnack Estimates 3 Sebastian Helmensdorfer, Peter Topping 1 0 2 January 9, 2013∗ n a J 8 Abstract In this short note, we hope to give a rapid induction for non-experts ] G into the world of Differential Harnack inequalities, which have been so D influential in geometric analysis and probability theory overthe past few decades. Atthecoarsestlevel,theseareoftenmysterious-lookinginequal- . h ities that holdfor ‘positive’ solutions of someparabolic PDE,and canbe t verified quicklyby grinding out a computation and applyinga maximum a m principle. In this note we emphasise the geometry behind the Harnack inequalities, which typically turn out to be assertions of the convexity of [ some natural object. As an application, we explain how Hamilton’s Dif- 1 ferential Harnack inequality for mean curvature flow of a n-dimensional v submanifold of Rn+1 can be viewed as following directly from the well- 3 knownpreservationofconvexityundermeancurvatureflow,butthistime 4 of a(n+1)-dimensional submanifold ofRn+2. Wealso brieflysurveythe 5 earlier work that led us to these observations. 1 . 1 1 Introduction 0 3 1 Perhaps the simplest situation in which to explain Differential Harnack Esti- : v mates is that of the ordinary, scalar, linear heat equation on Euclidean space. i Let u:Rn (0,T] (0, ) be a positive bounded solution of X × → ∞ r ∂u a = u. (1) ∂t △ R. Hamilton’s matrix Harnack estimate [HA2], restricted to this special case, says that the solution u satisfies the following differential inequality: Theorem 1.1. For each t (0,T], any positive, bounded solution u to the heat ∈ equation satisfies I Hess(logu)+ 0 (2) 2t ≥ where I denotes the identity matrix. ∗PaperinvitedbyActesduséminaireThéoriespectraleetgéométrie,UniversitédeGreno- ble. 1 Before we try to understand the geometry behind this inequality, let us try to understand why it is so useful. Taking the trace of (2) yields the following special case of the seminal inequalities of Li-Yau [LY], which predate the work of Hamilton: n (logu)+ 0, △ 2t ≥ and by rewriting the heat equation (1) as ∂ logu= (logu)+ (logu)2, ∂t △ |∇ | we find: Corollary 1.2. For each t (0,T], any positive, bounded solution u to the heat ∈ equation satisfies ∂ n logu (logu)2+ 0. (3) ∂t −|∇ | 2t ≥ In particular, this implies that ∂ logu n, i.e. that u cannot decrease ∂t ≥ −2t too fast. Note that we have managed to get rid of all spatial derivatives of u. To extractthe greatestpossibleamountofinformationfromCorollary1.2,pick two times 0 < t < t T, and consider a smooth path γ : [t ,t ] Rn from 1 2 1 2 x Rn to x Rn. W≤e may then compute, also using Young’s ineq→uality, 1 2 ∈ ∈ d ∂logu logu(γ(t),t)= + (logu),γ˙ dt ∂t h∇ i n 1 (logu)2 (logu)2+ γ˙ 2 (4) ≥ |∇ | − 2t − |∇ | 4| | (cid:16) (cid:17) (cid:18) (cid:19) n 1 = γ˙ 2, −2t − 4| | and then integrate to find that n t 1 t2 logu(x ,t ) logu(x ,t ) log 2 γ˙(t)2dt. 2 2 1 1 − ≥−2 t − 4 | | (cid:18) 1(cid:19) Zt1 Because there is no longer any mention of γ on the left-hand side, we may now optimise this inequality by taking γ to be the minimising geodesic from x to 1 x , i.e. a straight line, to obtain 2 n t x x 2 2 2 1 logu(x ,t ) logu(x ,t ) log | − | , 2 2 1 1 − ≥−2 t − 4(t t ) (cid:18) 1(cid:19) 2− 1 and we have proved a classical Harnack estimate: Corollary 1.3. For 0 < t < t T and x ,x Rn, any positive, bounded 1 2 1 2 solution u:Rn (0,T] (0, ) t≤o the heat equati∈on satisfies: × → ∞ t n2 x x 2 1 2 1 u(x ,t ) u(x ,t ) exp | − | . 2 2 1 1 ≥ (cid:18)t2(cid:19) −4(t2−t1)! 2 This beautiful estimate tells us that positive solutions cannot decrease too quicklyastimeadvances,evenifwemovealittleinspace. Moreoveritissharp, as can be seen by considering the fundamental solution of the heat equation 1 x2 ρ(x,t)= exp | | . (5) (4πt)n/2 − 4t (cid:18) (cid:19) In particular, this sharpness is manifested in the fact that ρ achieves equality in (2): I Hess(logρ)+ 0. (6) 2t ≡ If we now subtract (6) from (2), we obtain the following simple geometric rephrasing of Theorem 1.1, which is entirely in the spirit of this note. Corollary 1.4. For t (0,T], the function log u is convex. ∈ ρ (cid:16) (cid:17) In fact, with this formulation one can reduce the proof of the full matrix Harnackinequality in this case to the fact that the sum of log-convexfunctions is again log-convex: Proof. (Corollary 1.4.) By translating time by an arbitrarily small amount, we may assume that our solution is smooth on the whole of Rn [0,T], andu(,0) × · is a positive function. Fix t>0, and write u(x,t)= u(y,0)ρ(x y,t)dy, Rn − Z or alternatively u(x,t) F(x):= = u(y,0)G(x,y)dy, ρ(x,t) Rn Z where G(x,y) := ρ(x−y,t). We must therefore prove that F is log-convex, i.e. ρ(x,t) that for all x,z Rn and α (0,1), we have ∈ ∈ F(αx+(1 α)z) F(x)αF(z)1−α. − ≤ But for all y Rn, the function G(,y) is log-convex (even log-affine) and thus ∈ · F(αx+(1 α)z)= u(y,0)G(αx+(1 α)z,y)dy − Rn − Z u(y,0)G(x,y)αG(z,y)1−αdy ≤ Rn (7) Z α 1−α u(y,0)G(x,y)dy u(y,0)G(z,y)dy ≤ Rn Rn (cid:18)Z (cid:19) (cid:18)Z (cid:19) =F(x)αF(z)1−α by Ho¨lder. 3 2 The Differential Harnack estimate for mean curvature flow Now we consider solutions of another heat equation, namely of the mean curvature flow (see [EC]). Let (M ) Rn+1, M = F (Mn) be a family of t t∈[0,T] ⊂ t t smoothly immersed hypersurfaces, satisfying the nonlinear PDE ∂ F =H~ = Hν (8) t ∂t − where H~ is the mean curvature vector, ν is a choice of unit normal and H the corresponding mean curvature of M . We are especially interested in convex t initial data M . Convexity is preserved along the flow and compact convex 0 solutions shrink to a point at a finite time T (see [HU]). max We are also particularly interested in so-called self-expanders of the mean curvature flow, which are hypersurfaces M of Euclidean space that solve the 1 self-expander equation x,ν H + h i =0 (9) 2 on M , where again, ν is the unit normal such that H~ = Hν. The family of 1 − hypersurfacesM =√tM , t>0, then providesa self-expanding solutionof (8) t 1 up to tangential diffeomorphisms (see [EC]). Underthemeancurvatureflow,themeancurvatureitselfsatisfiesaparabolic equation, and one might hope to prove a Harnack inequality for H under some sortofpositivitycondition. R.Hamilton[HA3]showedthatthecorrectpositiv- ity hypothesis is to ask for the solutions to be convex, and proved an estimate that we state here only for compact submanifolds. Theorem 2.1 (Hamilton [HA3]). Let (M ) be a compact solution of (8) t t∈[0,T] such that M is convex. Then all M are convex and satisfy for t (0,T] 0 t ∈ ∂H H Z(V,V):= +2 H,V +h(V,V)+ 0 (10) ∂t h∇ i 2t ≥ for any tangent vector V, where , h and , denote the induced connection ∇ h· ·i on Mn, the second fundamental form of M and the Euclidean inner product t respectively. Again one can integrate this estimate along extremal curves to obtain a Harnackinequalityforthemeancurvatureintheclassicalsense(seealso[HA3]). Corollary 2.2. Under the assumptions of Theorem 2.1 we have for 0 < t < 1 t T, x M and x M : 2 ≤ 1 ∈ t1 2 ∈ t2 t 1 −∆ H(x2,t2) H(x1,t1) e 4 ≥ t r 2 where t2 dγ 2 ∆=inf dt γ dt Zt1 (cid:12) (cid:12)Mt (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 and the infimum is taken over all C1-paths γ : [t ,t ] Rn, which remain on 1 2 → the surface, i.e. γ(t) M , and with γ(t ) = x and γ(t ) = x . Here dγ ∈ t 1 1 2 2 dt Mt denotes the length of the component of the velocity vector of γ that is t(cid:12)ang(cid:12)ent (cid:12) (cid:12) to M . (cid:12) (cid:12) t Harnack inequalities for geometric flows have numerous applications, e.g. Hamilton’sHarnackestimatefor meancurvatureflow canbe appliedto classify convex eternal solutions of the mean curvature flow, if the mean curvature as- sumesitsspace-timemaximum,astranslatingsolitons(see[HA3]). Theoriginal proof of Theorem 2.1 directly uses a tensor maximum principle type argument. Asisthecaseforotherequations,thefirststeptoobtainTheorem2.1istofind therightquantityfrom(10),andthiswasoriginallydonebylookingforexpres- sions that vanish on self-expanding solutions of (8) and then trying to combine these in an appropriate way. While this method has provedto be extremely ef- fective, it does not give much geometric insight into Harnackexpressions. Such an insight can be gained by considering appropriate space-time constructions for geometric flows, as we now roughly describe. Thepioneeringworkinthedirectionofspace-timeconstructionsfortheRicci flow (see [TO]) and for the mean curvature flow was done by B. Chow and S. Chu(see[CC1,CC2]). TheymanagedtoshowthatZ(V,V) H –anexpression −2t that is constant along translating solitons – approximately corresponds to the second fundamental form of an extreme stretching by a factor N in the time direction of the so-calledspace-time track of the mean curvature flow. One can take the limit as N of this second fundamental form, yielding exactly → ∞ Z(V,V) H. − 2t In the context of Ricci flow, B. Chow and D. Knopf developed this idea further by considering a form of rescaled Ricci flow in order to obtain a precise correspondence between the relevant Harnack quantity and the curvature of a degenerate space-time construction (see [CK] for further details). E. Cabezas- Rivas and P. Topping extended these ideas in [CT] by constructing a non- degenerate expanding space-time approximate Ricci soliton, the limit of whose curvatures gave the existing, and new, Harnack quantities. Thus Harnack inequalities correspond to the preservation of certain curvature conditions (positive curvature operator, positive complex sectional curvature etc.) under Ricci flow. In this note we will see the mean curvature flow analogue of these ideas, wherethecorrespondencebetweenpreservationof‘positivecurvature’andHar- nack inequalities turns out to be particularly clean and precise. B.Kotschwar[KO]hasrecentlyconsideredvariantsoftheseideas,recovering theHarnackquantitiesforalargeclassofcurvatureflows(theHarnackestimates aredue to B.Andrews andK. Smoczyk,see [AN, SM]), as the limit as N →∞ of the second fundamental form of certain stretched variants of the space-time track. MoreoverhegaveanalternativeproofofHamilton’sHarnackestimatefor themeancurvatureflowbyshowingconvexityofthesespace-timetrackvariants. This was achieved by proving a generalised tensor maximum principle, applied on slices of the degenerate limit of the space-time track variants. In the remainder of this paper, we give a rigorous, purely geometric proof 5 of Hamilton’s Harnack estimate (10) which does not rely on any form of the maximum principle other than the classical preservation of convexity in solutions of mean curvature flow. The mean curvature flow to which we apply this convexity-preservation is not the original flow (M ); instead we take the flow t starting at a cone over the original initial surface M . In particular, we never 0 haveto applyanymaximumprinciple toanydegenerateobjects,andthe whole proof is phrased purely in terms of convexity. The remainder of the paper is organised as follows. In the next section we describehowtoflowgraphicalspace-timeconesoverhypersurfacesbytheirmean curvature. In Section 4 we state the relationship between so-called canonical self-expanders and the Harnack quantity for the mean curvature flow. Finally, in Section 5, we put things together and show how preservation of convexity along the mean curvature flow directly yields the Harnack inequality, Theorem 2.1. 3 Mean curvature flow of space-time cones Let (M ) be a compact convex solution of (8). By translating the flow t t∈[0,Tmax) within the ambient space, we may assume that M encloses the originin Rn+1. 0 For N 1, we define the cone to be the (n+1)-dimensionalsubmanifold of N Rn+2 g≥iven by C = t(x,N): x M ֒ Rn+1,t [0, ) . N 0 C ∈ → ∈ ∞ Wecanview asthe(cid:8)entiregraphofaLipschitzfunctionf(cid:9):Rn+1 [0, ). N N C → ∞ The cone becomes steeper as N becomes larger and the Lipschitz constant N C of f is bounded by C(M )N. N 0 K. Ecker and G. Huisken [EH1] developed a theory of mean curvature flow of entire graphs, that applies to the cone . The essential properties of N N here are that it is the graph of a LipschitzCfunction f on the whole of Rn+C1, N andthat it is ‘straightatinfinity’, whichis guaranteedin particularby the fact that z,νCN =0, h i for all z . The Ecker-Huisken theory then implies the existence of a self- N expander∈Σ˜C = graph(v˜ ), where v˜ : Rn+1 [0, ) has Lipschitz constant N N N → ∞ no greater than that of f (which in turn is bounded by C(M )N, as we have N 0 observed) such that √tΣ˜ N defines (for t > 0) a mean curvature flow starting at that is smooth away N fromtheinitialconepoint. (Inparticular,Σ˜ isasympCtoticto .) See Figure N N C 1. We can also say something useful about the infimum of v˜ ; more precisely N we can argue that minv˜ C(M ,n)N. (11) N 0 ≤ 6 Figure 1: The graphical self-expander Σ˜ N M ΣÑ CN 0 To see this, let d(M )>0 be defined to be half the radius of the largest sphere 0 centred at (0,1) Rn+1 R that does not intersect the region ∈ × t(x,y): x M ֒ Rn+1,t [0, ),y <1 0 ∈ → ∈ ∞ belowtheconeC(cid:8). ThenforallN 1,weseethatthe(n+1)(cid:9)-spherecentredat 1 (0,N) Rn+1 Rofradiusd(M )w≥illnotintersectthecone ,orequivalently, 0 N thatfor∈allh>×0,thespherecentredat(0,h) Rn+1 RofrCadius hd(M )will ∈ × N 0 not intersect the cone . If we then evolve these spheres by mean curvature N C flow at the same time as we evolve , the spheres will exist for a time T := N h h2d2 , (see, e.g. [EC]) and the coCmparison principle (see, e.g. [EC]) tells us 2(n+1)N2 that the evolution of cannot make it past the evolution of the spheres and N C go above the point (0,h) for all t [0,T ]. In particular, we find that at time h ∈ t = 1, the evolution of cannot have gone above the point (0, 2(n+1)N), CN d which is a statement a little stronger than (11). p In conclusion, we have established: Lemma 3.1. For any N 1 there exists a smooth graphical self-expander Σ˜ =graph(v˜ ) with the sa≥me Lipschitz constant as , in particular N N N C sup Dv˜ C(M )N, N 0 Rn+1| |≤ and with controlled infimum 0 minv˜ C(M ,n)N, (12) N 0 ≤Rn+1 ≤ such that √tΣ˜ defines (for t>0) a graphical mean curvature flow starting at N that is smooth throughout the flow, except at the cone point at time 0. In N Cparticular, Σ˜ is asymptotic to . N N C 7 Since Σ˜ = graph(v˜ ) becomes steeper and steeper as N , we squash N N →∞ it down by defining 1 v := v˜ . (13) N N N Now we can take the limit of the functions v as N gets large, and level-set N flow theory gives us a precise notion of what the limit is: Proposition 3.2. There exists a sequence Nn → ∞ such that (vNn)n∈N converges locally in C0 to a continuous limit v∞. The graph of v∞ is the asymptotically conical space-time track defined to be −1 t 2 (x,1): x Mt,t (0,Tmax] . (14) ∈ ∈ n o Proof. Lemma 3.1 provides a suitable derivative bound for v˜ in order to have N a uniform bound on Dv , independently of N. Together with the infimum N | | bound (12) from Lemma 3.1, this implies that there is a sequence N n such that (vNn)n∈N converges in Cl0oc Rn+1 to a continuous limit v∞. → ∞ Lemma 3.1 tells us that √tΣ˜ is a mean curvature flow that is the graph N (cid:0) (cid:1) ofafunction thatwe willcall V˜ :Rn+1 (0, ) [0, ),with V˜ (,1)=v˜ , N N N or more generally V˜ (x,t)= √tv˜ (x/√×t). W∞e m→ake th∞e analogous e·xtensions N N of vN and v∞ to VN and V∞ respectively. The equation of graphical mean curvature flow (see for example [EC]) with respect to a time coordinate s (0, ), is ∈ ∞ ∂ 2 DV˜ ∂sVÑn =r1+(cid:12)(cid:12)(cid:12)DVÑn(cid:12)(cid:12)(cid:12) div r1+ DNVnÑn 2,  (cid:12) (cid:12)  (cid:12) (cid:12) so we see that VNn solves (cid:12) (cid:12) ∂ 1 DV V = + DV 2 div Nn . ∂s Nn sNn2 | Nn|  N1n2 +|DVNn|2 q  We cannowapply anapproximationlemma (see [ES,proofofTheorem4.2])in order to show that the limit V∞ corresponds to a weak solution of the level-set flow equation (see [ES, CGG]) ∂ V∞ = I DV∞⊗DV∞ :D2V∞. ∂s − DV∞ 2 ! | | Thus V∞ is a weak solution of the level-set flow equation with the cone C1 asaninitialcondition. Thereforeforeachheightα>0, the smoothα-levelsets of V∞(,s) agree with the classical smooth evolution of the compact α-level set · ofC with respectto the time parameters (see [ES, Theorem6.1]). Buta level 1 8 set αM of C at height α>0 gives rise to the parabolically rescaled evolution 0 1 αMα−2s, andby setting s=1, we see that the α-levelsetof v∞ willbe αMα−2, whichisalsothe α-levelsetofthe asymptoticallyconicalspace-time track(14). Hence the graph of v∞ must be the asymptotically conical space-time track as desired. 4 Canonical self-expanders and the Harnack quantity For a solution (M ) of (8) there is an associated space-time construction t t∈[0,T] byB.Kotschwar(see[KO]), whichwecallacanonicalself-expanderbyanalogy with the Ricci flow case (see [CT]). The canonical self-expander Γ can be N defined for a parameter N >0 as −1 ΓN = t 2 (x,N): x Mt,t (0,T] . (15) ∈ ∈ n o Suppose nowthat the hypersurfaces(M ) haveuniformly boundedcurva- t t∈[0,T] ture. Then Γ is an approximate self-expander of (8), i.e. N z,νΓN HΓN + 0 (16) 2 ≈ (cid:10) (cid:11) for z ∈ΓN (see [KO, 4.1]). By this we mean that HΓN + hz,ν2ΓNi =EN, where N E is bounded locally uniformly, independently of N (we continue to use N | | this notation). Furthermore Γ is asymptotic to and we have (see Kotschwar [KO]): N N C Theorem4.1. ThesecondfundamentalformofΓ ,whichwecallhΓN,satisfies N ∂ ∂ Z(V,V) Z(V,V) hΓN V + ,V + = (17) ∂t ∂t σ √t ≈ √t (cid:18) (cid:19) N for any tangent vector V TMn, where the constant σ > 0 satisfies σ 1 N N ∈ → as N . →∞ 5 A geometric proof of Hamilton’s Harnack estimate We can now deduce Theorem 2.1 directly from a chain of convexity statements as follows: Proof. The starting assumption is that ⋆ M is convex, 0 which immediately implies that 9 ⋆ The cone over M is convex. N 0 C Sinceconvexityispreservedundermeancurvatureflow(seeinparticular[BBL, Theorem 10.2]) we find that ⋆ Σ˜ =graph(v˜ ) is convex, N N and since convexity is preserved under squashing in one direction, we deduce that ⋆ graph(v )=graph(1v˜ ) is convex. N N N Now, Cl0oc-limits of convex functions are convex,and v∞ is the Cl0oc limit of the convex functions v . Therefore Nn ⋆ v∞ is convex, or equivalently, by Proposition 3.2, ⋆ The asymptotically conical space-time track (14) is convex, and by preservation of convexity under stretching in one direction, this implies that ⋆ The canonical self-expander Γ is convex. N By (17) we can then deduce the Harnack inequality ⋆ Z(V,V) 0, ≥ as desired. Acknowledgements: ThisworkwassupportedbyTheLeverhulmeTrust. The secondauthorwasalsosupportedbyEPSRCgrantEP/K00865X/1. Thanksto Klaus Ecker, Gerhard Huisken and Esther Cabezas-Rivas for conversations on this topic. References [AN] B. Andrews Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994), 179-197 [BBL] G. Barles, S. Biton and O. LeyA geometrical approach to the study ofunboundedsolutionsofquasilinearparabolicequations,Arch.Rat.Mech. An. 162 (2002), 287-325 [CC1] B.Chow, S. Chu A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Let. 2 (1995), 701-718 [CC2] B. Chow, S. Chu Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. An. 11 (2001), 219-231 10