SOME REMARKS ON HUISKEN S MONOTONICITY FORMULA FOR MEAN CURVATURE FLOW

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1 SOE REARKS ON HUISKEN S ONOTONICITY FORULA FOR EAN CURVATURE FLOW ANNIBALE AGNI AND CARLO ANTEGAZZA ABSTRACT. We iscuss a monotone quantity relate to Huisken s monotonicity formula an some technical consequences for mean curvature flow. CONTENTS. aximizing Huisken s onotonicity Formula. Applications 6.. A No Breathers Result 6.. Singularities 7 References 9. AXIIZING HUISKEN S ONOTONICITY FORULA For an immerse hypersurface R n+, we call A an H respectively its secon funamental form an its mean curvature. Let t = ϕ(, t) be the mean curvature flow (CF) of an n imensional compact hypersurface in R n+, efine by the smooth family of immersions ϕ : [0, T ) R n+ which satisfies t ϕ = Hν where ν is the inner unit normal vector fiel to the hypersurface. Huisken in [7] foun his funamental monotonicity formula (.) t e x p 4(C t) [4π(C t)] µ t(x) = n/ e x p 4(C t) [4π(C t)]) n/ x p ν H + (C t) µ t (x) 0, for every p R n+, in the time interval [0, min{c, T }). Here µ t is the canonical measure on associate to the metric inuce by the immersion at time t. We call the quantity e x p 4(C t) µ [4π(C t)] n/ t (x), the Huisken s functional. Such formula was generalize by Hamilton in [5, 6] as follows, suppose that we have a positive smooth solution of u t = u in R n+ [0, C) then, in the time interval [0, min{c, T }), there hols (.) [ (C t) t ] u µ t = (C t) (C t) u H log u ν µ t ( u u u ) + µ t u (C t) where enotes the covariant erivative along the normal irection. Date: June 0, 00. Key wors an phrases. ean Curvature Flow.

2 ANNIBALE AGNI AND CARLO ANTEGAZZA Definition.. Let ϕ : R n+ be a smooth, compact, immerse hypersurface. Given τ > 0, we consier the family F τ of smooth positive functions u : R n+ R such that R n+ u x = an there exists a smooth positive solution of the problem { v t = v in R n+ [0, τ), v(x, 0) = u(x) for every p R n+. Then, we efine the following quantity σ(ϕ, τ) = sup u F τ 4πτ u µ. x p Remark.. The heat kernel K R n+(x, p, τ) = e of R n+ at time τ > 0 an (n+)/ point p R n+ clearly belongs to the family F τ. It is immeiate to see by this remark that the quantity σ(ϕ, τ) is positive an precisely, for every p R n+ an τ > 0, σ(ϕ, τ) 4πτ e x p µ(x) = (n+)/ e x p µ(x) > 0, n/ which is the quantity of the classical Huisken s monotonicity formula. Hence, (.3) σ(ϕ, τ) sup p R n+ e x p µ(x) > 0. n/ We want to see that actually this inequality is an equality, that is, we can take the sup only on heat kernels. oreover, the sup is a maximum. We work out some properties of the functions u F τ. We recall the integrate version of Li Yau Harnack inequality (see []). Proposition.3 (Li Yau integral Harnack inequality). Let u : R n+ (0, T ) R be a smooth positive solution of heat equation, then for every 0 < t s < T we have ( s ) (n+)/ x y u(x, t) u(y, s) e 4(s t). t Since the functions v : R n+ [0, τ) R associate to any u F τ are positive solutions of the backwar heat equation, such inequality reas, for 0 s t < τ, ( ) (n+)/ τ s v(x, t) v(y, s) e x y 4(t s). τ t This estimate, together with the uniqueness theorem for positive solution of the heat equation (see again []), implies that the function u = v(, 0) is obtaine by convolution of the function v(, t) with the forwar heat kernel at time t > 0. This fact implies that the conition R n+ v(x, t) x = hols for every t [0, τ), an that every erivative of every function v is boune in the strip [0, τ ε], for every ε > 0. The functions v(, t) weakly converge as probability measures, as t τ, to some positive unit measure λ on R n+ such that (.4) v(x, t) = R n+ e x y 4(τ t) λ(y). [4π(τ t)] (n+)/ Conversely, every probability measure λ, by convolution with the heat kernel, gives rise to a function v such that v(, τ) F τ, the most interesting case being

3 REARKS ON ONOTONICITY FORULA FOR CF 3 λ = δ p for p R n+. Inee, we know that for every t [0, τ) an s (t, τ), v(x, t) = v(y, s) R [4π(s t)] n+ e x y 4(t s) (n+)/ x hence, choosing a sequence of times s i τ such that the measures v(, s i )L n+ weakly converge to some measure λ, we get equality (.4), since x y e 4(t s) [4π(s t)] (n+)/ e converges uniformly to x y 4(τ t) on R n+, as s τ. [4π(τ t)] (n+)/ This representation formula also implies that the limit measure λ is unique an that actually lim s τ v(, s)l n+ = λ in the weak convergence of measures on R n+. Finally, we show that λ =. This follows by Fubini Tonelli s theorem for positive prouct measures, as u(x) x =, R n+ Rn+ e x y = u(x) x = λ(y) x R n+ R [4πτ] n+ (n+)/ Rn+ e x y = x λ(y) R [4πτ] n+ (n+)/ = λ(y) = λ. R n+ By this iscussion it follows that the family F τ consists of u(x, t) = Rn+ e x y λ(y) (n+)/ [4πτ] where λ varies among the convex set of Borel probability measures on R n+ (which is weak compact). A consequence of this fact is that since the integral 4πτ u µ is a linear functional in the function u, the sup in efining σ(ϕ, τ) can be taken consiering only the extremal points of the above convex, which are the elta measures in R n+. Consequently, the functions u to be consiere can be restricte to be heat kernels at time τ > 0. It is then easy to conclue that as the hypersurface is compact in R n+, the sup is actually a maximum. Proposition.4. The quantity σ(ϕ, τ) is given by We have also easily that σ(ϕ, τ) = sup p R n+ σ(ϕ, τ) = max p R n+ e x p µ(x) n/ e x p µ(x). n/ Area() µ(x) n/. n/ Proposition.5 (Rescaling Invariance). For every λ > 0 we have σ(λϕ, λ τ) = σ(ϕ, τ).

4 4 ANNIBALE AGNI AND CARLO ANTEGAZZA Proof. Let u F τ with associate solution of backwar heat equation v : R n+ [0, τ) R an consier the rescale function ũ(y) = u(y/λ)λ (n+). It is easy to see that ũ(y) y = λ (n+) R n+ u(y/λ) y = R n+ u(x) x = R n+ with the change of variable x = λ (n+) y, moreover the function ṽ(y, s) = v(y/λ, s/λ )λ (n+) is a positive solution of the backwar heat equation on the time interval λ τ, hence ũ F λ τ. It is now a straightforwar computation to see that 4πλ τ ũ µ λϕ = 4πτ u µ ϕ, for every smooth immersion of a compact hypersurface ϕ : R n+. The statement clearly follows. By formula (.), as the secon term vanishes when v is a backwar heat kernel, it follows that if ϕ : [0, T ) R n+ is the CF of a compact hypersurface, we have t [ (C t) = (C t) ] K R n+(x, p, C t) µ t (x) K R n+(x, p, C t) H + x p ν /(C t) µ t (x) which is clearly negative in the time interval [0, min{c, T }). Proposition.6 (onotonicity an Differentiability). Along a CF, ϕ : [0, T ) R n+, if τ(t) = C t for some constant C > 0, the quantity σ(ϕ t, τ) is monotone nonincreasing in the time interval [0, min{c, T }), hence it is ifferentiable almost everywhere. oreover, letting p τ a point in R n+ such that K R n+(x, p τ, τ) is one of maximizer for σ(ϕ t, τ(t)) of Proposition.4, we have for almost every t [0, min{c, T }), (.5) t σ(ϕ t, τ) x pτ e n/ H + x p τ ν τ or, since this inequality has to be intene in istributional sense, for every 0 r < t min{c, T }, (.6) σ(ϕ r, τ(r)) σ(ϕ t, τ(t)) t r e x p τ(s) (s) (4πτ(s)) n/ µ t H + x p τ(s) ν τ(s) µ s s Proof. As the function σ(ϕ t, τ) is the maximum of monotone nonincreasing smooth functions, it also must be monotone nonincreasing, hence, ifferentiable at almost every time t [0, min{c, T }). The last assertion is stanar, using Hamilton s trick (see [4]) to exchange the sup an erivative operations. Remark.7. Notice that the quantity σ can be efine also for any n imensional countably rectifiable subset S of R n+, by substituting in the efinition the term u µ with S u Hn, where H n is the n imensional Hausorff measure (possibly counting multiplicities). If then S is the support of a compact rectifiable varifol, with finite Area, moving by mean curvature accoring to Brakke s efinition (see []), Huisken s monotonicity formula (.) hols, hence, also this proposition.

5 REARKS ON ONOTONICITY FORULA FOR CF 5 Definition.8. We efine, in the same hypothesis, for τ = C t with C T, an Σ = Σ(T ). Σ(C) = lim t C σ(ϕ t, τ), By the previous iscussion, Σ sup p R n+ Θ(p), where this latter quantity is efine as e x p 4(T t) (.7) Θ(p) = lim t T [4π(T t)] µ t(x), n/ the existence of this limit for every p R n+ is an obvious consequence of Huisken s monotonicity formula. oreover, it is easy to see also the existence of max p R n+ Θ(p). Definition.9. Let ϕ : R n+ be a smooth, compact, immerse hypersurface. Then we efine ν(ϕ) = sup σ(ϕ, τ). τ>0 Proposition.0. The quantity ν(ϕ) is finite an actually reache by some τ ϕ. Proof. Inee, we have lim σ(ϕ, τ) = Θ(ϕ) > 0, τ 0 + where Θ(ϕ) is the maximum (which clearly exists as is compact) of the n imensional ensity of ϕ() in R n+. Then, if ϕ is an embeing, Θ(ϕ) =, otherwise it will be the highest multiplicity of the points of ϕ(). We show then that lim σ(ϕ, τ) = 0. By the rescaling property of σ, we have σ(ϕ, τ) = σ(ϕ/ 4πτ, /4π), hence we nee to show that lim sup sup u µ = 0. 4πτ u F But we alreay saw that any function u F satisfies 0 u(x) (4π) (n+)/ lim sup sup u F 4πτ u µ lim sup Vol(/ 4πτ) (4π) (n+)/ = lim sup hence, Vol() (4π) (n+)/ τ n/ = 0. The following statement can be prove by the same argument of the proof of Proposition.6. Proposition. (onotonicity an Differentiability II). Along a CF, ϕ : [0, T ) R n+, the quantity above ν(ϕ t ) is monotone non increasing in the time interval [0, T ), hence it is ifferentiable almost everywhere. oreover, letting p ϕ R n+ an τ ϕ to be some of the maximizers whose existence is grante by Propositions.4 an.0, we have for almost every t [0, T ), (.8) t ν(ϕ t) e x pϕ t ϕ t (4πτ ϕt ) n/ H + x p ϕ t ν τ ϕt µ t (x) or, since this inequality has to be intene in istributional sense, for every 0 r < t < T, (.9) ν(ϕ r ) ν(ϕ t ) t r e x pϕs ϕs (4πτ ϕs ) n/ H + x p ϕ s ν µ s (x) s. τ ϕs

6 6 ANNIBALE AGNI AND CARLO ANTEGAZZA Remark.. One can repeat all this analysis for a compact, immerse hypersurface of a flat Riemannian manifol T. oreover, if the original hypersurface ϕ : R n+ is immerse in R n+, we can choose a Riemannian covering map I : R n+ T an consier the immersion ϕ = I ϕ : T. Then, we efine as above, for every τ > 0, the family F T,τ of smooth positive functions u : T R such that u x = an there exists a smooth positive solution of the problem T { v t = v in T [0, τ) v(p, 0) = u(x) for every p T. Then, we efine the following quantity σ T (ϕ, τ) = sup 4πτ u F T,τ f u µ where refers to the fact that we are consiering the immersion ϕ : T. Notice that another possibility is simply to embe isometrically a convex set Ω R n+ containing ϕ() in a flat Riemannian manifol T (uring the mean curvature flow a hypersurface ϕ initially containe in Ω stays insie for all the evolution). As before, these quantities are well efine, finite, positive an monotonically ecreasing if ϕ t moves by mean curvature... A No Breathers Result.. APPLICATIONS Definition.. A breather (following Perelman []) for mean curvature flow in R n+ is a smooth n imensional hypersurface evolving by mean curvature ϕ : [0, T ) R n+, such that there exists a time t > 0, an isometry L of R n+ an a positive constant λ R with ϕ(, t) = λl(ϕ(, 0)). Remark.. It is useless to consier nonshrinking (steay or expaning) compact breather of CF, by the comparison with evolving spheres, they simply o not exist. To authors knowlege, it is unknown if there exist nonhomothetic, noncompact steay or expaning breathers. Theorem.3. Every compact breather is a homothetic solution to CF. Proof. By the rescaling property of σ in Proposition.5, fixing C > 0 we have σ(ϕ 0, C) σ(ϕ t, C t) = σ(λϕ 0, C t) = σ(ϕ 0, (C t)/λ ) hence, if we choose C = follows that t λ we have C > t, as λ < an (C t)/λ = C. It σ(ϕ 0, C) = σ(ϕ t, C t) an (for such special C), by Proposition.6, if τ(t) = C t t 0 e x p τ(t) (t) (4πτ(t)) n/ H + x p τ(t) ν τ(t) µ t t = 0. This implies that for every t (0, t) we have H(x, t) = x p τ(t) ν (C t) for some p τ(t) R n+, which is the well known equation characterizing a homothetically shrinking solution of CF.

7 REARKS ON ONOTONICITY FORULA FOR CF 7 Remark.4. This is the same argument to show that compact shrinking breathers of Ricci flow are actually Ricci graient solitons. Recalling the monotone nonecreasing quantity µ of Perelman in [], along a Ricci flow g(t) of a compact, n imensional Riemannian manifol, µ(g, τ) = inf R u=, u>0 ( τ [R + u u ] u log u un log [4πτ] un ) V. By the rescaling property µ(λg, λτ) = µ(g, τ), if we have that g(t) = λl g(0) for some iffeomorphism L : an 0 < λ <, fixing C > 0 we have µ(g(0), C) µ(g(t), C t) = µ(λl g(0), C t) = µ(λg(0), C t) = µ(g(0), (C t)/λ) hence, if we choose C = t λ we have C > t, as λ < an (C t)/λ = C. It follows that µ(g(0), C) = µ(g(t), C t) an by the results of Perelman, g(t) is a shrinking soliton... Singularities. If ϕ : [0, T ) R n+ is a CF of a smooth, compact, embee hypersurface, it is well known that uring the flow it remains embee an there exists a finite maximal time T > 0 of smooth existence when the curvature A is unboune as t T. oreover for every t [0, T ) sup A(p, t). p (T t) If there exists a constant C > 0 such that also C sup A(p, t). p (T t) we say that at T we have a type I singularity, otherwise we say the singularity is of type II. We want to show that if at time T we have a singularity, the associate quantity Σ = lim t T σ(ϕ t, τ) is larger than one. Inee, for every p R n+ such that there exists a sequence of points q i an times t i T with p = lim i ϕ(q i, t i ), we consier the function Θ(p) efine in equation (.7). By a simple semicontinuity argument, we can see that Θ(p) for every p R n+ like above, see [, Corollary 4.0], hence, as Σ sup p R n+ Θ(p) we get Σ. If then Σ =, it forces Θ(p) = for all such points p which implies, by the local regularity result of White [5], that the flow cannot evelop a singularity at time T (see also Ecker []). Suppose now to have a type I singularity at time T. By Proposition.6 we know that along this flow, for C = T, hence, τ = T t, σ(ϕ r, T r) σ(ϕ t, T t) t for every 0 r t T, hence, (.) T C(ϕ 0 ) σ(ϕ 0, T ) Σ 0 r e x p T s 4(T s) [4π(T s)] n/ e x p T s 4(T s) [4π(T s)] n/ H + x p T s ν (T s) H + x p T s ν (T s) µ s (x) s µ s (x) s Rescaling every hypersurface ϕ t as in [7], aroun the point p T t as follows, ϕ s (q) = ϕ(q, t(s)) p T t(s) s = s(t) = log(t t) (T t(s))

8 8 ANNIBALE AGNI AND CARLO ANTEGAZZA an changing variables in formula (.), we get + (.) C e y µ log T log T e y H + y ν µ s (y) s. It follows that reasoning like in [7] an [3] (or [4]), if the singularity is of type I, the curvature of the rescale hypersurfaces ϕ s : R n+ is uniformly boune an any sequence converges (up to a subsequence) to a limit embee hypersurface satisfying H = x ν which is the efining equation for a homothetic solution of CF. oreover, By the estimates of Stone [3, Lemma.9], this limit hypersurface satisfies e y (π) n/ H n (y) = lim σ(ϕ t, T t) = Σ >. f t T Clearly, by this equation, this embee limit hypersurface cannot be empty. oreover, it cannot be flat also, as it woul be an hyperplane for the origin of R n+ (the only hyperplanes satisfying H = x ν must pass through the origin) as the above integral woul be one. Proposition.5. At a singular time T of the CF of an embee compact hypersurface the quantity Σ is larger than one. If the singularity of the flow is of type I, any sequence of rescale hypersurfaces (with the maximal curvature) aroun the maximizer points for the Huisken s functional at times t i T converges, up to a subsequence, to a nonempty an nonflat, smooth embee limit hypersurface, satisfying H = x ν. Suppose now that we are ealing with the special case of an embee close curve γ t evolving in the plane R. Rescaling as above, without assuming anything about the type of a singularity at some time T, we can extract a subsequence γ si of rescale curves such that: y H + y ν H (y) 0 γ si e with locally equiboune lengths. This implies that the curves γ si have also locally equiboune L norms of the curvature. Possibly passing to another subsequence (not relabele) we can assume that the curves γ si converges in Cloc to a limit curve γ with equiboune curvatures k in L loc ; the curve γ satisfies k = x ν istributionally; (π) n/ eγ e y there hols H (y) = Σ > ; finally, the curve γ is embee, that is, without self intersections, by the geometric argument of Huisken in [8]. By a bootstrap argument, using the conition k = x ν, it follows that γ is a smooth curve an since the only embee curves in the plane satisfying such conition are the lines through the origin an the unit circle, γ has to be among them. Then, the curve γ cannot be a line through the origin, because for all of them the (π) n/ eγ e y value of the integral H (y) is one. Hence, γ must be the unit circle. This implies that at some time the curve γ t has become C close, hence a graph, over a roun circle (in particular, it is starshape). It is then straightforwar to see by means of maximum principle that this last property is preserve uring the evolution.

9 REARKS ON ONOTONICITY FORULA FOR CF 9 Then, by means of the interior estimates of Ecker an Huisken [3], one can fin a close (in time) sequence of rescale curves converging in Cloc to the unit circle. Then, as this implies that at some time the curve has become convex, the singularity can only be a type I vanishing singularity. As a consequence, type II singularities for embee close curves are not possible. Remark.6. It woul be very interesting if this argument can be extene in higher imensions, that is, if rescaling the moving hypersurface aroun the points maximizing the Huisken s functional coul help to prouce homothetic blowups also in the case of a Type II singularity. Some results in this irection has been obtaine by Ilmanen in [9, 0]. Acknowlegement. Annibale agni is partially supporte by the ESF Programme ethos of Integrable Systems, Geometry, Applie athematics (ISGA) an arie Curie RTN European Network in Geometry, athematical Physics an Applications (ENIGA). REFERENCES. K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, NJ, K. Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations an their Applications, 57, Birkhäuser Boston Inc., Boston, A, K. Ecker an G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. ath. 05 (99), no. 3, R. S. Hamilton, Four manifols with positive curvature operator, J. Diff. Geom. 4 (986), no., , A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. (993), no., , onotonicity formulas for parabolic flows on manifols, Comm. Anal. Geom. (993), no., G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom. 3 (990), , A istance comparison principle for evolving curves, Asian J. ath. (998), T. Ilmanen, Singularities of mean curvature flow of surfaces, ilmanen/papers/sing.ps, , Lectures on mean curvature flow an relate equations, ilmanen/papers/notes.ps, P. Li an S.-T. Yau, On the parabolic kernel of the Schröinger operator, Acta ath. 56 (986), no. 3 4, G. Perelman, The entropy formula for the Ricci flow an its geometric applications, ArXiv Preprint Server A. Stone, A ensity function an the structure of singularities of the mean curvature flow, Calc. Var. Partial Differential Equations (994), , Singular an Bounary Behaviour in the ean Curvature Flow of Hypersurfaces, Ph.D. thesis, Stanfor University, B. White, A local regularity theorem for mean curvature flow, Ann. of ath. () 6 (005), no. 3, (Annibale agni) SISSA INTERNATIONAL SCHOOL FOR ADVANCED STUDIES, VIA BEIRUT 4, TRIESTE, ITALY, aress, A. agni: magni@sissa.it (Carlo antegazza) SCUOLA NORALE SUPERIORE DI PISA, P.ZA CAVALIERI 7, PISA, ITALY, 566 aress, C. antegazza: c.mantegazza@sns.it