A FORMULA RELATING ENTROPY MONOTONICITY TO HARNACK INEQUALITIES

Transcription

1 A FORMULA RELATING ENTROPY MONOTONICITY TO HARNACK INEQUALITIES KLAUS ECKER 1. Introuction In [P], Perelman consiere the functional Wg, f, τ = τ f + R + f n + 1 u V X for τ > 0 an smooth functions f on a close n imensional Riemannian manifol X, g where e f u = 4πτ n+1 an efine an associate entropy by { µg, τ = inf Wg, f, τ, X } u V = 1. His ingenious realization was that when τt > 0 satisfies τ t = 1, X, gt evolves by the Ricci flow t g ij = R ij an f satisfies the equation f t + f + R = f + n + 1 which preserves the conition u V = 1 X then Wgt, ft, τt = R ij + i j f g ij u V. t X This implies in particular that µgt, τt 0 t with equality exactly for homothetically shrinking solutions of Ricci flow. An important consequence of this entropy formula is a lower volume ratio boun for solutions of Ricci flow on a close manifol for a finite time interval [0, T asserting the existence of a constant κ > 0, only epening on n, T an g0, such that the inequality V t Brx t 0 r n+1 κ Date: revise 6 June,

2 hols for all t [0, T an r [0, T for balls B t rx 0 with respect to gt in which the inequality r Rm 1 for the Riemann tensor of gt hols. This lower volume ratio boun rules out certain collapse metrics as rescaling limits near singularities of Ricci flow such as proucts of Eucliean spaces with the so-calle cigar soliton solution of Ricci flow given by X = R with the metric s = x + y 1 + x + y. In this paper, we aim at aapting Perelman s entropy formula to the situation where a family of boune open regions t t [0,T in R n+1 with smooth bounary hypersurfaces M t = t is evolving with smooth normal spee β Mt = x t ν. Here x enotes the embeing map of M t an ν is the normal pointing out of t. For open subsets R n+1, smooth functions f : R an β : R an τ > 0 we consier the quantity W β, f, τ = τ f + f n + 1 u x + βu S with an the associate entropy µ β, τ = inf u = e f 4πτ n+1 { } W β, f, τ, u x = 1. We then erive a formula which states that if t evolves as above, τt > 0 satisfies τ t = 1, f satisfies the evolution equation in t with Neumann bounary conition f t + f = f + n + 1 f ν = β on M t = t an we introuce a family of iffeomorphisms ϕ t : t with x = ϕ t q, q obeying x = fx, t t then t W β t, ft, τt = t i j f δ ij u x W ν S M t where W = τ f f + f n + 1. For evolving boune regions t insie a fixe Riemannian manifol X, g or insie a Ricci flow solutions one can erive analoguous versions of this formula.

3 The main observation in this paper is that this can be converte to t W β t, ft, τt = i j f δ ij t u x β + t M β M f + A M f, M f β u S M t where A enotes the secon funamental form of M t. For functions β for which the hypersurface integral is nonnegative the inequality t W β t, ft, τt i j f δ ij u x results. When β = 0, that is for a fixe boune region with smooth convex bounary insie a fixe manifol of non-negative Ricci curvature, Lei Ni [N] has previously obtaine this inequality. It implies, as in Perelman s situation, t µ β t, τt 0 an also the following localise lower volume ratio boun: There is a constant κ > 0 epening only on n, 0, T, sup M0 β an c 1 such that t V t B r x 0 r n+1 hols for all t [0, T an r 0, T ] in balls B r x 0 R n+1 satisfying the conitions V t B r/ x 0 > 0 an κ V t B r x 0 + r M t B rx 0 β S V t B r/ x 0 c 1. Since this statement is scaling invariant for suitably homogeneous β it is also vali on any smooth limit of suitably rescale solutions of the flow consisting of smooth, compact embee hypersurfaces, but now for all raii r > 0 as long as the other conitions still hol for the balls B r x 0 we consier. In the important case of mean curvature flow, that is where β Mt is the mean curvature H Mt of the hypersurfaces M t, the right han sie of the formula vanishes on homothetically shrinking solutions an for f = x /4τ. This leas us to the following conjecture: Conjecture. In the case of mean curvature flow in R n+1 for compact embee hypersurfaces M t satisfying H > 0 uring the evolution the inequality H t M H M f + A M f, M f H u S 0 M t hols an therefore t W H t, ft, τt t i j f δ ij u x for τ = a t where a T an t < T. In particular, this leas to the above lower volume ratio boun in this case. 3

4 Note that the expression Z M f H t M H M f + A M f, M f is the central quantity in Hamilton s Harnack inequality for convex solutions of the mean curvature flow see [Ha]. Even though Z M f vanishes on translating solutions for u = e xn+1 τ our calculations will, ue to the non-compactness of t an the non-integrability of all integrans in this case, not lea to Z M f H/ on the right han sie. A irect calculation shows that regions boune by certain eternal solutions of mean curvature flow, such as the prouct of R n 1 with the grim reaper curve given by y = log cos x + t, o not satisfy the lower volume boun statement for large r an hence, shoul the conjecture hol, cannot occur as a rescaling limit in this situation. Similarly, certain stationary zero mean curvature hypersurfaces woul then be rule out as rescaling limits such as for instance the catenoi minimal surface in R 3 an two parallel hyperplanes. In the positive mean curvature case, White [Wh] has previously shown that certain solutions of mean curvature flow, in particular the grim reaper hypersurface, cannot occur as rescaling limits. The embeeness assumption for the hypersurfaces M t is essential. Angenent [A] has shown, that solutions of the curve-shortening flow with self-intersections have the grim reaper curve as rescaling limit near singularities. This paper is organise as follows. In Section, we efine entropies for open subsets of complete possibly non-compact Riemannian manifols with respect to a given smooth function β efine on an establish some of their properties. In Section 3, we erive the entropy formula involving the Harnack expression for evolving omains in R n+1. All of the calculations go through with necessary moifications such as aing Ricci an scalar curvature terms in the appropriate places in the case of a fixe ambient manifol or a backgroun Ricci flow solution. However, at the moment we o not see how they might lea to equally interesting consequences. In Section 4, we state our conjecture an show several consequences it woul lea to, such as a lower local volume ratio boun an non-existence of certain egenerate rescaling limits. In Appenix A, we give some explicit examples of entropy functionals an values in R n+1. In the paper, a version of the logarithmic Sobolev inequality on boune open sets in complete Riemannian manifols is use. In Appenix B, we provie a proof base on the stanar Sobolev inequality, essentially following Gross [G]. In Appenix C, we give a erivation of a Harnack type evolution equation associate with solutions of a backwar heat equation. This equation is one of the central results in [P] an is also one of the main ingreients in the proof of our entropy formula. Details of this calculation first appeare in [KL] an [N]. The work presente in this paper was inspire by a iscussion with Grisha Perelman in January 003 in Berlin. I woul like to thank Richar Hamilton, Gerhar Huisken, Dan Knopf, Oliver Schnürer, Carlo Sinestrari, Peter Topping, Mu-Tao Wang an Brian White for helpful iscussions. I am particularly inebte to Felix Schulze for a number of valuable suggestions. 4

5 . Entropy type functionals for omains in Riemannian manifols For open subsets of an n imensional complete possibly non-compact Riemannian manifol X, g, functions f : R an β : R an τ > 0 we consier the quantity W β, g, f, τ = τ f + R + f n + 1 u V + βu S where u = e f 4πτ n+1 The scalar curvature R, the expression f an the volume an area elements V an S are taken with respect to the metric g. We then efine an associate entropy by { } µ β, g, τ = inf W β, g, f, τ, u V = 1. For β = 0 an = X, W β, g, f, τ an µ β, g, τ reuce to Perelman s functional Wg, f, τ an his entropy quantity µg, τ. We therefore write W for W 0 an µ for µ 0. We use n + 1 instea of n as we will later be intereste mainly in the hypersurface which we prefer to be n-imensional. When we o not inten to vary the metric we consier W β, f, τ = τ f + f n + 1 u V + βu S with infimum µ β, τ. We shall only consier sets with smooth bounaries an smooth functions f an β although the above expressions also make sense for more general sets an functions. In case is unboune we require suitable integrability conitions on f an β. The function β coul be the restriction to of a function on X or be efine only on. An important example of the latter is β = H where H is the mean curvature of with respect to the outer unit normal. In this section, we erive several basic properties for these entropies. Some specific examples incluing calculations of entropy values for some natural choices of sets in R n+1 are iscusse in Appenix A. Proposition.1. Suppose that is boune with smooth bounary an that β is smooth. Then for any τ > 0 we have µ β, g, τ cn,, g 1 + log1 + τ + τ sup R + sup β 1 + sup β. The same lower boun hols for µ β, τ. Remark.. The lower boun for µ β, g, τ an for µ β, τ follows from the logarithmic Sobolev inquality for which in turn can be erive from the stanar Sobolev inequality see Appenix B. The constant cn,, g thus epens on the constant in the Sobolev inequality an the L 1 - trace inequality for C 1 - functions, the latter controlling the bounary integral. The metric enters via bouns for the Riemann curvature tensor on an the explicit boun for the sup R - term arising from the functional. The Proposition hols for more general sets such as boune sets of finite perimeter an for boune β. 5.

6 Proof of Proposition.1. We give the proof only for µ β, g, τ. For µ β, τ simply set the scalar curvature term to zero. We essentially moify the arguments in [KL] an [N]. Setting u = ϕ an using the conition u V = 1 we obtain W β, g, f, τ = τ4 ϕ + Rϕ ϕ log n+1 ϕ V 1 + βϕ S cn with ϕ V = 1. The trace inequality ϕ S c ϕ + ϕ V with c = c, g in combination with Young s inequality yiels βϕ S ϕ V + c 3 τ sup β 1 + sup β where c 3 epens on c. Here we have use again the conition ϕ V = 1. Combining this with 1 yiels W β, g, f, τ ϕ ϕ log n+1 ϕ V c τ sup R + sup β 1 + sup β where c 4 epens on the previous constants. Scaling the metric gives 3 ϕ ϕ log n+1 ϕ V = ϕτ τ ϕ τ log ϕ τ Vτ an ϕ τ V τ = 1 where ϕ τ = n+1 4 ϕ an V τ an ϕ τ τ are taken with respect to g τ = 1 g. By scaling the stanar Sobolev inequality n ψ n+1 n+1 n V cs, g ψ + ψ V we see that the Sobolev constant c S, g τ can be estimate by c S, g1 + τ. Therefore, by the logarithmic Sobolev inequality applie in with respect to the metric g τ see Appenix B ϕτ τ ϕ τ log ϕ τ Vτ cn 1 + log c S, g + log1 + τ. Combining this inequality with an 3, we arrive at W β, g, f, τ c log1 + τ + τ sup R + sup β 1 + sup β with c 5 = c 5 n,, g an for f satisfying u V = 1. This gives the esire lower boun for µ β, g, τ. 6

7 Proposition.3. Let be boune with smooth bounary an assume β to be smooth. Then for every τ > 0 there exist smooth minimizers for µ β, g, τ an for µ β, τ. Proof. We only consier µ β, g, τ again. The argument is analogous as in [FIN]. The necessary semicontinuity an coercivity in W 1, for the transforme functional Eϕ = τ4 ϕ + Rϕ ϕ log ϕ V + βϕ S cn1 + log τ for u = ϕ subject to the conition ϕ V = 1 follow from similar arguments as in the proof of the lower boun for µ β, g, τ given above. The quantity W = W f = τ f f + R + f n + 1 feature in Ch.9 of [P] an in [N]. It arises naturally in the Euler-Lagrange equation for the functional W β, g, f, τ. Proposition.4. A minimizer f min for the functional W β, g, f, τ subject to the constraint u V = 1 satisfies the Euler-Lagrange equation in an the natural bounary conition W f min = µ β, g, τ f min, ν = β on. Here, refers to the metric g. For a minimizer of W β, f, τ we have instea W f min = µ β, τ where W f = τ f f + f n + 1. Proof. Stanar computation using Lagrange multipliers. Remark.5. The Euler-Lagrange equation for the transforme functional Eϕ = τ4 ϕ + Rϕ ϕ log ϕ V + βϕ S cn1 + log τ for ϕ = u subject to the conition ϕ V = 1 is 4τ ϕ ϕ log ϕ + τrϕ = µ β, g, τ + n log 4πτ ϕ in with bounary conition ϕ, ν = βϕ on. Proposition.6. For any function f : R satisfying f, ν = β on with respect to the outer unit normal ν we have W β, g, f, τ = W u V 7

8 with W = W f = τ f f + R + f n + 1 an W β, f, τ = W u V for W = W f = τ f f + f n + 1. Proof. The bounary conition implies u, ν = βu on an hence W β, g, f, τ = τ f + R + f n + 1 u V u V by the ivergence theorem. Since u = u f f. the claim follows. For the next statement we o not require to be boune. Proposition.7. Suppose that µ β, g, r c 0 or µ β, r c 0. Let B r x 0 X, g satisfy V B r/ x 0 > 0, V B r x 0 + r B rx 0 β S V B r/ x 0 c 1 an r Rm c in B r x 0 for the Riemann tensor of g. Then with κ = κn, c 0, c 1, c. V B r x 0 r n+1 κ > 0 Remark.8. We will actually prove that µ β, g, r an µ β, r are boune from above by the expression log V B rx 0 r n+1 + c V B rx 0 + r B rx 0 β S V B r/ x 0 with c = cn, r Rm. From this the claim follows immeiately. Proof of Proposition.7. In the case = X an β = 0 the proof is sketche in Ch.3 of [P] see [KL] an [N] for more etails. We procee along similar lines. If we set e f = aζ the normalisation conition for f becomes a = 4πr n+1 ζ V. The functional W β, f, r can then be expresse as a r ζ + r R ζ ζ logaζ V n r βζ S 4πr n+1 ζ ζ V. By approximation, we may substitute functions ζ C0X into this expression. We choose as ζ a cut-off function for B r/ x 0 that is ζ satisfies χ Br/ x 0 ζ χ Brx 0 as well as r ζ r sup ζ c ζ 8

9 where c is a constant which epens on r sup Brx 0 Rm an is therefore boune by c. Since ζ V V B r/ x 0 > 0 we can thus estimate 1 4πr n+1 r a ζ ζ Jensen s inequality now implies 1 1 aζ logaζ V 4πr n+1 4πr n+1 V spt ζ V c ζ V c V B rx 0 V B r/ x 0. 1 aζ V log V sptζ aζ V. Since spt ζ = B r x 0 an in view of the normalisation conition the right han sie equals V B r x 0 log. 4πr n+1 The scalar curvature integral is estimate using the bouneness assumption on the Riemann tensor in B r x 0. This yiels the upper boun for µ, g, r an µ, r state in Remark.8. Remark.9. In [P], Perelman rule out the occurrence of collapse metrics as rescaling limits of compact, finite time solutions of Ricci flow. A metric g on X is calle collapse if there exists a sequence of balls B rk x k X, g satisfying r k Rm 1 in B r k x k for which V B rk x k r n+1 k 0. An important example of a collapse metric is the so-calle cigar soliton solution of the Ricci flow given by X = R enowe with the metric s = x + y 1 + x + y. On collapse metrics we have inf τ>0 µ β g, τ = by the proposition. The following reformulation of Proposition.7 links a kin of volume collapsing behaviour of subsets of X, g to a property of the entropy µ β, g, τ. Corollary.10. If for some fixe constants c 1 an c we can fin a sequence of balls B rk x k X, g such that V B rk /x k > 0, V B rk x k + rk B rk x k β S c 1, V B rk /x k r k Rm c in B rk x k an V B rk x k r n+1 k 0 then inf τ>0 µ β, g, τ = an inf τ>0 µ β, τ =. 9

10 For compact we can of course always fin such a sequence of balls with raii tening to infinity. In the case of non-compact regions the sitation is more interesting. Examples are the following regions in X = R n+1 : 1 The slab = {x R n+1, < x n+1 < } for some > 0. On the hypersurface M = we have H = 0. The enclose region satisfies V B r/ > 0 an for all balls B r = B r 0. Moreover, V B r V B r/ cn V B r lim r r n+1 = 0. The smaller of the two regions boune by the catenoi minimal surface M = in R 3 given by = {x = ˆx, x 3 R 3, ˆx > 1, x 3 < cosh 1 ˆx }. Note that H = 0 on. One checks that there is a constant c 1 such that for all r > V B r V B r/ c 1 an V B r c 1 r log1 + r so that V B r lim r r 3 = 0. 3 The translating solution of mean curvature flow corresponing to the grim reaper hypersurface M = where = R n 1 G with G = { x n, x n+1 R, π/ < x n < π/, x n+1 > log cos x n }. An explicit calculation shows that the mean curvature satisfies Hx = e xn+1 for any x M =. One therefore checks irectly that there is a sequence of balls B rk x k with r k satisfying V B rk /x k > 0, an V B rk x k V B rk /x k cn, rk B rk x k H S V B rk /x k V B rk x k r n+1 k

11 3. An Entropy type formula for evolving omains in R n+1 In this section we restrict ourselves to omain evolution in R n+1. All the calculations go through for fixe Riemannian manifols or Ricci flow solutions as ambient space if we a Ricci an scalar curvature terms in the appropriate places. Howerer, in this case the formulas o not immeiately seem to lea to any interesting consequences. We evolve boune open subsets t t [0,T with smooth bounary hypersurfaces M t t [0,T in R n+1. More precisely, t = φ t with M t = t = φ t where φ t = φ, t : R n+1, t [0, T is a smooth one-parameter family of iffeomorphisms. We will often abbreviate x = φp, t for p. The normal spee of M t with respect to the inwar pointing normal ν is efine by β = β Mt = x t ν for x M t or expresse in terms of the embeing map φ, t by βp, t = φ p, t νφp, t t for p. We assume the function β to be smooth. If for instance β = H, the mean curvature of M t, this escribes mean curvature flow up to iffeomorphisms tangential to M t. Let us assume more specifically that the family of subsets t t 0,T evolves by the equation 4 x t = fx, t for x t. This flow is compatible with the evolution of the bounaries M t = t with normal spee β if f satisfies the conition f ν = β on M t. Suppose ft satisfies the equation 5 t + f = f + n + 1 in t for t 0, T. The total time erivative of f is given by f 6 t = f x + f t t = f t f. Hence 5 can also be written as 7 t + f = n + 1. If τt > 0 evolves by τ t 8 for = 1 then 5 is equivalent to the equation t + u = 0 u = e f 4πτ n+1 11.

12 The above equations are more precisely expresse in terms of the pull back of the function f via the iffeomorphisms evolving t. In fact, if we set x = φq, t where φ t = φ, t : t, the pulle back function given by satisfies fq, t = fφq, t, t f f x, t = q, t. t t Analogously to Ch.9 in [P] see also [N] the function W = τ f f + f n + 1 satisfies a nice evolution equation: Proposition 3.1. Let t t 0,T be a family of subsets evolving by 4 that is accoring to the negative graient of functions ft satisfying equation 5. Suppose also that τt > 0 evolves by τ t = 1 for t 0, T. Then the function W = τ f f + f n + 1 satisfies the evolution equation t + W = i j f δ ij in t. Proof. We use Perelman s ientity t + W = i j f δ ij + W f + W f from Ch.9 in [P]. A erivation of this can be foun in [KL] an in [N]. In our evolving coorinates x = φq, t we change to total time erivatives for W via W = W W f t t which yiels the result. For the convenience of the reaer, we repeat the etails of the calculation in [N] for the expression t + W on evolving sets t R n+1 in Appenix C. Proposition 3.. Suppose the conitions of the previous proposition hol. Then u x = 0 t t for all t 0, T. If f satisfies aitionally f ν = β on M t = t then 9 t W β t, ft, τt = i j f δ ij u x W ν u S M t where W = τ f f + f n + 1. Proof. In view of the family of iffeomorphisms generate by x t = f = 1 u u the volume element x on the evolving sets t changes by t t x = f x. 1

13 Since also an u t = u t + u u u = f f u we obtain in t 10 u u x = t t + u x = 0 by equation 8. Thus u x = 0. t t Combining the Neumann bounary conition, Proposition.6, ientity 10 an the evolution equation for W in Proposition 3.1 we then calculate t W β t, ft, τt = W u x = t t t t + W u x W u x t = i j f δ ij t u x W u + W u x t where we again use u = u f. The last integral equals iv W u x. t The result then follows by applying the ivergence theorem. Remark 3.3. For a fixe omain that is when β = 0 insie a Riemannian manifol of nonnegative Ricci curvature the inequality W, ft, τt W ν u S t M t for a solution f of the above backwar heat equation appears in [N]. Ni then shows that W, ν = A M f, M f an is therefore non-negative for a convex bounary see below for a generalisation of the corresponing calculation to evolving omains, thus obtaining W, ft, τt 0. t When examining the integran W ν of the above bounary integral more closely, an interesting relation with the expression in Hamilton s Harnack inequality for the mean curvature of a hypersurface evolving by mean curvature flow emerges. To appreciate this one shoul first note that the hypersurfaces M t evolve by the equation x 11 t = βν M f ue to the Neumann bounary conition for f where M enotes the tangential graient on the hypersurfaces M t. 13

14 Proposition 3.4. Uner the above conitions on M t an ft the quantity W satisfies the ientity β 1 W ν = t M β M f + A M f, M f β for all t < T, a T an τ = a t where A enotes the secon funamental form of M t. This implies the inequality β t W β t, ft, τt t M β M f + A M f, M f β u S M t Proof. In view of equation 7 we have W = τ f t + f + f. We now calculate similarly as in Appenix C t f = f f, + f t. A calculation as for instance in [Hu1] using the evolution equation 11 for the hypersurfaces M t yiels ν t = M β A M f, for the outwar unit normal fiel on M t. The secon term arises from the efinition of A in terms of tangential erivatives of ν. Combining these an ifferentiating the ientity β = f ν yiels Since β t = f f, ν + f t ν + M β M f A M f, M f. W ν = τ f t ν + f ν + f ν an f ν = f f, ν we obtain the result by observing β t = β t M β M f in view of 11. The integral inequality then follows from Proposition 3.. Remark 3.5. Let f t0 be a minimizer for µ β t0, τt 0. Since W f t0 constant see Proposition.4 we have M t0 W ν u S = 0 at time t 0. However, even if we assume that ft for t < t 0 satisfies the en conition ft 0 = f t0 we cannot conclue that lim W ν u S = 0 t t 0 M t an that therefore note that W β is ifferentiable at t 0 t t 0 W β t, ft, τt 0. 14

15 The problem occurs since W involves thir erivatives of f which won t behave continuously on the bounary for t t 0 unless we impose some kin of higher orer compatibility conition on the en ata f t0 on M t0 = t0. 4. A conjecture Harnack type inequality for mean curvature flow an its consequences For β = H, the expression Z M f H t M H M f + A M f, M f in Proposition 3.4 is the central quantity in Hamilton s Harnack inequality for convex solutions of the mean curvature flow see [Ha]. Hamilton showe, that ZV vanishes on translating solutions of mean curvature flow for some vector fiel V which is tangential to the hypersurfaces M t. His Harnack inequality states that ZV + H t 0 hols for any tangential vector fiel V on a convex solution of mean curvature flow for t > 0 with equality for a suitable vector fiel on a homothetically expaning solution. We observe that on homothetically shrinking solutions that is where H = x ν the ientity ZV H = 0 hols for V = M f where f = x /4τ. Because of the term H we cannot expect this expression to be nonnegative for a general solution an for a general V. It certainly is negative on translating solutions for a suitable V. However, it seems reasonable to expect that this quantity or its integral over M t has a sign, at least on compact solutions. For non-compact solutions, our calculations o not lea to the integral inequality in Proposition 3. since the integral expressions are usually not well-efine in this case as will see a little later in the case of translating solutions. The above consierations lea us to the following Conjecture. Let M t t<t be a family of compact embee hypersurfaces evolving by their mean curvature. Assume H > 0 uring the flow. Let τ = a t for fixe a T an all t < T. Then the Harnack type inequality H t M H V + AV, V H 0 hols on M t for all t < T an all tangential vector fiels V with equality on homothetically shrinking solutions for V = M f an f = x /4τ. If we rescale mean curvature flow by consiering xs = 1/ t xt for s = log t then the conjecture inequality becomes H s M H V + ÃV, V 0 15

16 for all s > 0. A weaker form of the conjecture which suffices for the applications we have in min is that H C t M H M f + A M f, M f H u S 0 M t where f satisfies t + f = f + n + 1 in t for t < T with the bounary conition f ν = H an the omains evolve by a family of iffeomorphisms generate by f. Note that f blows up like x /4T t an therefore M f like x /T t for t T near a singularity corresponing to a homothetically shrinking solution. Let us give two explicit examples of mean curvature flow solutions which illustrate the situation: First note that the evolution equation for the hypersurfaces M t in the case β = H is x t = Hν M f, which is mean curvature flow up to tangential iffeomorphisms. If t is the interior of a homothetically shrinking solution of mean curvature flow, that is up to translation in time t = 0 for τ = T t, then f = x /4T t is a solution of equation 5. The Neumann bounary conition above becomes simply H = x ν. In this situation, H t M H M f + A M f, M f H = 0 so W ν = 0. For translating solutions of mean curvature flow the quantity W ν is negative for positive H. However, our rate of change formula for W H oes not hol in this case as the entropy calculations are not justifie in this situation: Inee, if t is the interior of a translating solution of mean curvature flow, that is up to rotation in R n+1 t = + te n+1 for some fixe set an for all t R then f = x n+1 + τ log4πτ n+1 solves the bounary value problem. The Neumann bounary conition on M t in this case becomes H = ν n+1. We note that M t an t are necessarily unboune since compact solutions cannot exist for all t R by comparison with spheres shrinking to points in finite time. Moreover, the function u featuring in the integran of the entropy functional as well as in the normalisation conition require for the entropy is given by u = e f 4πτ n+1 16 = e xn+1 τ

17 in our example. In view of the comparison principle for mean curvature flow applie to M t an hyperplanes {x R n+1, x n+1 = a}, which are stationary solutions of mean curvature flow, the sets t have an unboune intersection with the upper half space {x R n+1, x n+1 > 0} for every t R. Therefore, the function u is an illegal choice in the normalisation conition t u x = 1 as it is not integrable on t. There are a number of important consequences of inequality C especially for the open problem of no local volume collapse for mean curvature flow solutions an analogue of Perelman s no local collapsing for Ricci flow solutions an consequently non-existence of certain egenerate rescaling limits. This shoul provie sufficient motivation for settling the conjecture. Proposition 4.1. Suppose that the conjecture inequality C hols. Then t W H t, ft, τt 0 for t < t 0 an therefore the entropy is monotonic that is for 0 t 1 t < T an any a T. µ H t1, a t 1 µ H t, a t Proof. The first inequality follows irectly from Proposition 3.4 applie to β = H an from C. To erive the secon inequality we let f t0 for t 0 < T be a minimizer for µ H t0, τt 0 an let ft in aition to the equation t + f = f + n + 1 in t an the bounary conition f ν = H on t for t < t 0 satisfy the en conition ft 0 = f t0. Since we have t W H t, ft, τt 0 W H t, ft, τt W H t0, ft 0, τt 0 = W H t0, f t0, τt 0 = µ H t0, τt 0. Taking the infimum on the left han sie over all functions satisfying the normalisation conition e f V = 1 t 4πτ n+1 we obtain the esire inequality for the entropies at t an at t 0. Since t an t 0 were arbitrary we are one. Corollary 4.. Let M t t [0,T be a solution of mean curvature flow consisting of smooth, compact, embee hypersurfaces which enclose boune regions t t [0,T in R n+1. Let τ = a t for arbitrary but fixe a T an all t < T. Suppose furthermore that the Harnack type inequality C hols. Then for every r > 0 an every t [0, T µ H t, r µ H 0, t + r. Since T < we have for every t [0, T an r 0, T ] µ H t, r c 0 17

18 where c 0 epens only on n, 0, T an sup M0 H. In particular, there is a constant κ > 0 epening only on n, 0, T, sup M0 H an c 1 such that the inequality V t B r x 0 r n+1 κ hols for all t [0, T an r 0, T ] in balls B r x 0 satisfying the conitions V t B r/ x 0 > 0 an V t B r x 0 + r M t B rx 0 H S V t B r/ x 0 c 1. Proof. By the monotonicity of the entropy an Proposition.1 applie with a = r + t, t 1 = 0 an t = t we have for all r T an t < T. Proposition.7 applie to t. µ H t, r µ H 0, t + r cn, T, sup M 0 H, 0 The lower volume ratio bouns then follow from For λ j 0, t j T an x j R n+1 we efine a sequence j s of rescale flows j s = 1 λ j λ j s+t j x j where s λ j t j, λ j T t j a j, b j. Definition 4.3. Let M t t [0,T be a compact, smooth, embee solution of mean curvature flow enclosing boune regions t t [0,T in R n+1. We call a smooth, embee solution M s s,b of mean curvature flow enclosing not necessarily boune s s,b a rescaling limit of M t t 0,T if there are sequences λ j 0, t j T an x j in R n+1 such that j s s aj,b j s s,b smoothly in compact subsets in space-time that is in particular, the hypersurfaces M j s = j s converge smoothly. Remark 4.4. For a solution M t t [0,T which becomes singular for t T, that is sup t<t sup Mt A = for the secon funamental form A on M t one can always fin a rescaling limit for a suitable choice of sequences x j in R n+1 an λ j 0 for example the reciprocal of the maximum of A at an appropriately chosen sequence of times t j T. The smooth convergence follows from stanar a priori estimates for mean curvature flow see for instance [Hu]. Rescaling limits are so-calle ancient solutions which means that they have existe forever. Examples of ancient solutions are all homothetically shrinking solutions of mean curvature flow such as the shrinking spheres given by M s = B ns for s, 0. If the solution M t t 0,T has a so-calle type II - singularity, that is sup t<t T t sup M t A =, then by a rescaling process escribe in [HS] one can even fin a limit flow which is an eternal solution, that is b =. Examples of eternal solutions are all stationary 18

19 solutions, that is solutions with s = for all s R. In this case, the hypersurface M = is minimal that is satisfies H = 0. Other eternal solutions are translating solutions of mean curvature flow for which s = + sω for s R where R n+1 an ω is a fixe unit vector in R n+1. The corresponing hypersurfaces M = satisfy the equation H + ν ω = 0. The statement of Corollary 4. is scaling invariant. Hence the rescale solution M j s s aj,b j satisfies V j s B r x 0 r n+1 κ > 0 for all s a j, b j an r 0, T /λ j in balls with V j s B r/ x 0 > 0 an V j s B r x 0 + r Ms j B rx 0 H S V j s B r/ x 0 c 1. The constant κ = κn, 0, T, sup M0 H, c 1 is the same as for the unscale solution. As a consequence we obtain a lower volume ratio boun for rescaling limits, but without the raius restriction: Corollary 4.5. Let M t t [0,T be a solution of mean curvature flow consisting of compact smooth, embee hypersurfaces which enclose boune regions t t [0,T in R n+1. Suppose furthermore that inequality C hols. Then there is a constant κ > 0 epening only on n, 0, T, sup M0 H an c 1 such that any rescaling limit M s s,b of M t t [0,T with limiting enclose regions s s,b satisfies V s B r x 0 κ r n+1 for every s, b an r > 0 in balls B r x 0 with V s B r/ x 0 > 0 an V s B r x 0 + r H S M s Brx0 V s B r/ x 0 c 1. This Corollary rules out certain solutions of mean curvature flow as rescaling limits uner the assumption that our conjecture is vali: Corollary 4.6. If the conitions of the above corollary are satisfie then the following eternal solutions of mean curvature flow cannot occur as rescaling limits of a compact, smooth embee mean-convex solution M t t [0,T of mean curvature flow which encloses boune regions t t [0,T in R n+1 : 1 The stationary solution corresponing to a pair of parallel hyperplanes that is given by s = for all s R where = {x R n+1, < x n+1 < } for some > 0. The stationary solution of mean curvature flow corresponing to the catenoi minimal surface M = in R 3 given by = {x = ˆx, x 3 R 3, ˆx 1, x 3 cosh 1 ˆx }. 3 The translating solution corresponing to the grim reaper hypersurface M = where = R n 1 G with G = { x n, x n+1 R, π/ < x n < π/, x n+1 > log cos x n }. 19

20 Proof. All three examples amit sequences of balls for raii increasing to infinity for which the volume ratio tens to zero while the other quantities are controlle. This was iscusse in Corollary.10. Remark In the special situation where the original solution M t is mean convex, that is H > 0 for M 0 an subsequently for all M t by the maximum principle, White [Wh] rule out the grim reaper hypersurface as a rescaling limit using techniques from minimal surface theory an geometric measure theory. His methos exten also to non-smooth limit flows of generalize mean curvature flow solutions in the mean-convex case. In view of Corollary.10, the first two examples satisfy inf τ>0 µ, τ = an the thir one inf τ>0 µ H, τ =. 3 The embeeness assumption on the hypersurfaces M t is essential. In [A], it is prove that rescaling limits of non-embee planar curves near singularities are given by the grim reaper curve Γ = G efine above. 4 Some other translating solutions can occur as rescaling limits such as for instance a rotationally symmetric translating bowl see for instance [Wa]. The region boune by this translating bowl opens up quaratically so one can show that it satisfies the conclusions of the above corollary. 5 For the shrinking solution s = B ns there is no lower boun of the form V s B r r n+1 κ > 0 with a fixe κ for all s < 0 an all r > 0 since the balls s shrink to the origin for s 0. This oes not contraict the corollary though as κ epens on c 1 an in this case c 1 behaves like cnr s 1 since for s [ 1/n, 0 an r 1 r H S M s Br V s B r/ = M s H S V s = cn r s. Appenix A. Some basic properties of entropies in R n+1 In this appenix, we iscuss some explicit examples of entropies in R n+1. 1 When β = 0 an = R n+1 we have see [P] WR n+1, f, τ = τ f + f n + 1 u x 0 R n+1 for all f satisfying u x = 1 R n+1 with equality when fx = x 4τ. In particular therefore µr n+1, τ = 0 for all τ > 0. This is the Gaussian logarithmic Sobolev inequality ue to L.Gross [G]. Scaling by x = y, setting f = y log ϕ as in [P] an using the ientity y n + 1 γ n+1 y = iv yγ n+1 y = 0 R n+1 R n+1 0

21 for the Gaussian we obtain its stanar form y e γ n+1 y = π n+1 ϕ log ϕ γ n+1 y 1 R n+1 for all ϕ satisfying R n+1 ϕ γ n+1 y R n+1 ϕ γ n+1 x = 1. For x R n+1 we set x = λy + x 0 where λ > 0 an x 0 R n+1. We then obtain an W β, f, τ = Wλ 1 x 0, fλ + x 0, λ τ + λ τ λβλy + x 0 e fλy+x0 Sy λ 1 x 0 4πλ τ n+1 Therefore 1 = ux x = λ 1 x 0 uλy + x 0 y. µ β, τ = µ λβλ +x0λ 1 x 0, λ τ. Suppose that β : R n+1 R satisfies x x0 β = λβx λ for x, x 0 R n+1 an λ > 0 or that β = β is a geometric quantity which behaves like βy = λβx where x = λy +x 0 for y 1 λ x 0 such as for example the mean curvature of. Then µ β, τ = µ β λ 1 x 0, λ τ. For x 0 = 0, λ =, replace by an such functions β this yiels 3 If x 0 then µ β, τ = µ β, 1/. λ 1 x 0 R n+1. Using this, the scaling ientity for µ β with λ = as well as the ientity µr n+1, 1/ = 0 we expect that µ β, τ 0 for τ 0. This shoul follow along the same lines as in [N]. 4 A natural example is β = x ν where ν is the unit outwar pointing normal to. By the above scaling property we have µ x ν, τ = µ y ν, 1/ where x = y an y. 1

22 An example of a function f on R n+1 satisfying the normalisation conition e f x = 1 4πτ n+1 is f = x 4τ log c where 1 c = e x 4τ x. 4πτ n+1 For this f an β = x ν one calculates W β, f, τ = c x n + 1 e x 4τ 4πτ n+1 x + x ν x e 4τ 4πτ n+1 S +log c. Since iv xe x x 4τ = n + 1 e x 4τ this implies W β, f, τ = log c by the ivergence theorem. Note that for = R n+1 we have c = 1 an hence W β, f, τ = W, f, τ = 0 for f = x 4τ. For the half-space H a = {x R n+1, x n+1 < a}, a R an β = x ν 1 c = a e z z. we calculate This implies that µ x ν H a, τ for a as well as lim τ 0 W x ν H a, f, τ = 0 an lim τ 0 W x ν H a, f, τ = log < 0 for fixe a R. By the scaling an translation property above we have W x ν, x 1 log c, τ = W y ν, y 4τ log c, 1 = log c for x = y with the conition 1 γ n+1 y = 1 c. If the n + 1 -imensional volume of a set insie large balls grows like V B R cr p for R R 0 an p < n + 1 one checks that γ n+1 y 0 1 for τ. Therefore c an hence µ x ν, τ. Such sets inclue for instance all boune sets but also unboune sets which lie in a slab in R n+1. In the latter case the volume in balls grows like R n.

23 Appenix B. Sobolev an logarithmic Sobolev inequalities For the convenience of the reaer who is unfamiliar with logarithmic Sobolev inequalities we show how these can be erive from the stanar Sobolev inequality. We essentially follow a proof given in [G]. Theorem Logarithmic Sobolev inequality. For any open subset of a Riemannian manifol X, g which satisfies the Sobolev inequality n ψ n+1 n+1 n V cs, g ψ + ψ V for all ψ C 1 there also hols a logarithmic Sobolev inequality of the form ɛ ϕ ϕ log ϕ V cn1 + log c S, g 1 ɛ for functions ϕ satisfying ϕ V = 1 an every ɛ > 0. Proof. By a stanar approximation argument it will be sufficient to prove the theorem for non-negative functions. We abbreviate 1 ψ p ψ p p V for p > 0. The interpolation inequality for L p -norms says for functions ψ satisfying ψ 1 = 1 that ψ q ψ n n q n n 1 for 1 q n n 1. Since for q = 1 we have equality, ifferentiation with respect to q at q = 1 preserves the inequality an leas to ψ log ψ V n log ψ n. n 1 In view of the Sobolev inequality ψ n c n 1 S ψ for such functions this yiels ψ log ψ V n log c S ψ = n log n ψ n lognc S. The inequality log x x 1 implies ψ log ψ V ψ 1 + cn1 + log c S. Setting ψ = ϕ with ϕ V = 1 gives ϕ log ϕ V ϕ V + cn1 + log c S. Using Young s inequality, we finally arrive ϕ log ϕ V ɛ ϕ V + 1 ɛ + cn1 + log c S where we again use ϕ V = 1. 3

24 Appenix C. Proof of the Evolution equation for W For the convenience of the reaer we give a etaile proof of the evolution equation of Proposition 3.1 in Section 3. In Section 3, we merely moifie the appropriate formulas in [P] an [N] by transforming to total time erivatives. Let us briefly recall the set-up given in Section 3 in the case of evolving omains in R n+1. We consier a family of subsets t t 0,T in R n+1 which evolve by the equation 13 x t = fx, t for x t where ft satisfies the equation 14 t + f = f + n + 1 in t for t 0, T. The total time erivative of f is given by f 15 t = f t + f, x = f t t f an so 14 can also be written as 16 t + f = n + 1. We also assume that τt > 0 evolves by τ t = 1. Proposition. In the above setting, the function W = τ f f + f n + 1 satisfies the evolution equation t + W = i j f 1 δ ij + W f. Proof. We aapt the computation in [N] to the case of omains evolving by 13 the ifferent sign in Ni s Lemma. stems from the fact that he consiers the forwar heat equation by interchanging the roles of τ an t. In a general Riemannian manifol X, g an aitional Ricci term arises when we interchange thir erivatives of f. In the Ricci flow case this expression is balance by terms coming from the time erivative of the metric. Details of the latter can be foun in [KL]. If we write above x = φq, t where φ t = φ, t : t are the iffeomorphisms evolving t, the pulle back function f given by satisfies fq, t = fφq, t, t f f x, t = q, t. t t The evolution equation 13 written in terms of fq, t = fφq, t, t looks like φ t q, t = fq, t 4

25 where is the graient with respect to the pull-back of the Eucliean metric uner φ t on R n+1 given by In these coorinates we have One now calculates an the inverse metric satisfies g ij q, t = φ q i q, t φ q j q, t. f ij f = g f. q i q j t g ij = i j f, t gij = i j f. Furthermore one computes for the Christoffel symbols of the g ij 17 g ij t Γk ij = k f. One then checks from this an fx, t = fq, t with f = g ij f Γ k f ij q i q j q k that the ientities 18 an 19 t f = i j f i f j f + f f t f = f t t + f + f f hol. We now follow [N] exactly, except for working with f f t instea of t f. The latter of the above ientities in combination with 16 an the relation τ t = 1 implies 0 f t + t = n + 1 f f f. Combining 18 an 16 with the Bochner ientity we fin 1 f = f + f f t + f = f + i j f i f j f. To break up the calculation for W, we rewrite W = τ f f +f n+1 using 16 as W = τw + f where w = f t f = f f n + 1. τ 5

26 From 0 an 1 we calculate t + w = f n + 1 τ Since we thus arrive at i j f i f j f + f f. i j f i f j f + f f = f w t + w = f n + 1 τ + f w. Using again τ t = 1 we now compute t + W = t + τw + f = w + f n + 1 τ + f τw + n + 1 = f W w f + f n + 1. Substituting the ientities f = i j f δ ij + f n + 1 an w = f f n + 1 τ yiels the esire evolution equation for W. References [A] S.B. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom. 33, [FIN] M. Felman, T. Ilmanen an L. Ni, Entropy an reuce istance for Ricci expaners, arxiv: math.dg/ [G] L. Gross, Logarithmic Sobolev inequalities, Amer. J. of Math. 97, [Ha] R. Hamilton, Harnack estimate for the mean curvature flow, J. Diff. Geom. 41, [Hu1] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom. 0, [Hu] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom. 31, J.Differ.Geom. 0, [HS] G. Huisken an C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE 8, [KL] B. Kleiner an J. Lott, Notes on Perelman s paper on the entropy formula for the Ricci flow..., [N] L. Ni, The entropy formula for linear heat equation, Journal Geom. Analysis 14, [P] G. Perelman, The entropy formula for the Ricci flow an its geometric applications, arxiv:math.dg/011159v1 11Nov00 [Wa] X.J. Wang, Translating solutions to the mean curvature flow, preprint, Australian National University [Wh] B. White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math.Soc 16, No 1,

27 Fachbereich Mathematik un Informatik Freie Universität Berlin Arnimallee Berlin Germany 7