arXiv:1402.5087v3 [math.DG] 25 Feb 2015 · 2015. 2. 26. · Cjyj jr xfjand jf(x)j C 1 jyj 3 (0.6) 2 for some positive constant Cand, hence, jf(x)j23 Cjr (0.7) xfj: Therefore, we only

LOJASIEWICZ INEQUALITIES AND APPLICATIONS

TOBIAS HOLCK COLDING AND WILLIAM P MINICOZZI II

Abstract In real algebraic geometry Lojasiewiczrsquos theorem asserts that any integral curveof the gradient flow of an analytic function that has an accumulation point has a uniquelimit Lojasiewicz proved this result in the early 1960s as a consequence of his gradientinequality

Many problems in calculus of variations are questions about critical points or gradi-ent flow lines of an infinite dimensional functional Perhaps surprisingly even blowups atsingularities of many nonlinear PDErsquos can in a certain sense be thought of as limits ofinfinite dimensional gradient flows of analytic functionals Thus the question of uniquenessof blowups can be approached as an infinite dimensional version of Lojasiewiczrsquos theoremThe question of uniqueness of blowups is perhaps the most fundamental question aboutsingularities

This approach to uniqueness was pioneered by Leon Simon thirty years ago for the areafunctional and many related functionals using an elaborate reduction to a finite dimensionalsetting where Lojasiewiczrsquos arguments applied

Recently the authors proved two new infinite dimensional Lojasiewicz inequalities atnoncompact singularities where it was well-known that a reduction to Lojasiewiczrsquos argu-ments is not possible but instead entirely new techniques are required As a consequencethe authors settled a major long-standing open question about uniqueness of blowups formean curvature flow (MCF) at all generic singularities and for mean convex MCF at allsingularities Using this the authors have obtained a rather complete description of thespace-time singular set for MCF with generic singularities In particular the singular setof a MCF in Rn+1 with only generic singularities is contained in finitely many compactLipschitz submanifolds of dimension at most nminus 1 together with a set of dimension at mostnminus 2

0 Finite and infinite dimensional inequalities

01 Lojasiewicz inequalities In real algebraic geometry the Lojasiewicz inequality [L1][L2] [L4] from the late 1950s named after Stanislaw Lojasiewicz gives an upper bound forthe distance from a point to the nearest zero of a given real analytic function Specificallylet f U rarr R be a real-analytic function on an open set U in Rn and let Z be the zerolocus of f Assume that Z is not empty Then for any compact set K in U there exist α ge 2and a positive constant C such that for all x isin K

infzisinZ|xminus z|α le C |f(x)| (01)

Here α can be large

The authors were partially supported by NSF Grants DMS 11040934 DMS 1206827 and NSF FRG grantsDMS 0854774 and DMS 0853501

2 TOBIAS HOLCK COLDING AND WILLIAM P MINICOZZI II

Equation (01) was the main inequality in Lojasiewiczrsquos proof of Laurent Schwarzrsquos divisionconjecture1 in analysis Around the same time Hormander [Ho] independently provedSchwarzrsquos division conjecture in the special case of polynomials and a key step in his proofwas also (01) when f is a polynomial

A few years later Lojasiewicz solved a conjecture of Whitney2 in [L3] using the followinginequality3 With the same assumptions on f for every p isin U there are a possibly smallerneighborhood W of p and constants β isin (0 1) and C gt 0 such that for all x isin W

|f(x)minus f(p)|β le C |nablaxf | (02)

Note that this inequality is trivial unless p is a critical point for f One immediate consequence of (02) is that every critical point of f has a neighborhood

where every other critical point has the same value It is easy to construct smooth functionswhere this is not the case

02 First Lojasiewicz implies the second In this subsection we will explain how thesecond Lojasiewicz inequality for a function f in a neighborhood of an isolated critical pointfollows from the first To make things concrete we will show that the second holds withβ = 2

3when the first holds with α = 2

Suppose that f Rn rarr R is smooth function with f(0) = 0 and nablaf(0) = 0 without lossof generality we may assume that the Hessian is in diagonal form at 0 and we will write thecoordinates as x = (y z) where y are the coordinates where the Hessian is nondegenerateBy Taylorrsquos formula in a small neighborhood of 0 we have that

f(x) =ai2y2i +O(|x|3) (03)

fyi(x) = ai yi +O(|x|2) (04)

fzi(x) = O(|x|2) (05)

It follows from this that the second of the two Lojasiewicz inequalities holds for f and β = 23

provided that |z|2 le ε |y| for some sufficiently small ε gt 0 Namely if |z|2 le ε |y| then

C |y| le |nablaxf | and |f(x)| le Cminus1 |y|32(06)

for some positive constant C and hence

|f(x)|23 le C |nablaxf | (07)

Therefore we only need to prove the second Lojasiewicz inequality for f in the region|z|2 ge ε |y| We will do this using the first Lojasiewicz inequality for nablaf Since 0 is anisolated critical point for f the first Lojasiewicz inequality for nablaf gives that

|nablaxf | ge C |x|2 (08)

By assumption on the region and the Taylor expansion for f we get that in this region

|f(x)| le C |y|2 + C |z|3 le C |z|3 le C |x|3 (09)

1L Schwartz conjectured that if f is a non-trivial real analytic function and T is a distribution then thereexists a distribution S satisfying f S = T

2Whitney conjectured that if f is analytic in an open set U of Rn then the zero set Z is a deformationretract of an open neighborhood of Z in U

3Lojasiewicz called this inequality the gradient inequality

LOJASIEWICZ INEQUALITIES AND APPLICATIONS 3

Combining these two inequalities gives

|f(x)|23 le C |x|2 le |nablaxf | (010)

This proves the second Lojasiewicz inequality for f with β = 23

Lojasiewicz used his second inequality to show the ldquoLojasiewicz theoremrdquo If f Rn rarr Ris an analytic function x = x(t) [0infin)rarr Rn is a curve with xprime(t) = minusnablaf and x(t) has alimit point xinfin then the length of the curve is finite and

limtrarrinfin

x(t) = xinfin (011)

Moreover xinfin is a critical point for f In contrast it is easy to construct smooth functions even on R2 where the Lojasiewicz

theorem fails ie where there are negative gradient flow lines that have more than one limitpoint (and thus also have infinite length) see Figure 1

Figure 1 There are smooth functions vanishing on an open (compact) setfor which the gradient flow lines spiral around the zero locus The flow lineshave infinite length and the Lojasiewicz theorem fails

03 The Lojasiewicz Theorem Next we will explain how the second Lojasiewicz in-equality is typically used to show uniqueness Before we do that observe first that inthe second inequality we always work in a small neighborhood of p so that in particular|f(x)minus f(p)| le 1 and hence smaller powers on the left hand side of the inequality imply theinequality for higher powers As it turns out we will see that any positive power strictly lessthan one would do for uniqueness

Suppose that f Rn rarr R is a differentiable function Let x = x(t) be a curve in Rn

parametrized on [0infin) whose velocity xprime = minusnablaf We would like to show that if the second

inequality of Lojasiewicz holds for f with a power 1 gt β gt 12 then the Lojasiewicz theoremmentioned above holds That is if x(t) has a limit point xinfin then the length of the curve isfinite and limtrarrinfin x(t) = xinfin Since xinfin is a limit point of x(t) and f is non-increasing alongthe curve xinfin must be a critical point for f

The length of the curve x(t) isint|nablaf | so we must show that

intinfin1|nablaf | ds is finite Assume

that f(xinfin) = 0 and note that if we set f(t) = f(x(t)) then f prime = minus|nablaf |2 Moreover bythe second Lojasiewicz inequality we get that f prime le minusf 2β if x(t) is sufficiently close to xinfin(Assume for simplicity below that x(t) stays in a small neighborhood xinfin for t sufficientlylarge so that this inequality holds the general case follows with trivial changes) Then thisinequality can be rewritten as (f 1minus2β)prime ge (2β minus 1) which integrates to

f(t) le C tminus1

2βminus1 (012)

We need to show that (012) implies thatintinfin

1|nablaf | ds is finite This shows that x(t)

converges to xinfin as trarrinfin To see thatintinfin

1|nablaf | ds is finite observe by the Cauchy-Schwarz

inequality thatint infin1

|nablaf | ds =

int infin1

radicminusf prime ds le

(minusint infin

f prime s1+ε ds

) 12(int infin

sminus1minusε ds

It suffices therefore to show that

minusint T

f prime s1+ε ds(014)

is uniformly bounded Integrating by parts givesint T

f prime s1+ε ds = |f s1+ε|T1 minus (1 + ε)

f sε ds (015)

If we choose ε gt 0 sufficiently small depending on β then we see that this is boundedindependent of T and hence

intinfin1|nablaf | ds is finite

04 Infinite dimensional Lojasiewicz inequalities and applications Many problemsin geometry and the calculus of variations are essentially questions about functionals oninfinite dimensional spaces such as the energy functional on the space of mappings or thearea functional on the space of graphs over a hypersurface Infinite dimensional versionsof Lojasiewicz inequalities proven in a celebrated work of Leon Simon [Si1] have playedan important role in these areas over the last 30 years Clearly the infinite dimensionalinequalities have immediate applications to uniqueness of limits for gradient flows butperhaps surprisingly they also have implications for singularities of nonlinear PDErsquos

Once singularities occur one naturally wonders what the singularities are like A standardtechnique for analysing singularities is to magnify around them Unfortunately singularitiesin many of the interesting problems in Geometric-PDE looked at under a microscope willresemble one blowup but under higher magnification it might (as far as anyone knows)resemble a completely different blowup Whether this ever happens is perhaps the mostfundamental question about singularities see eg [Si2] and [Hr] By general principles theset of blowups is connected and thus the difficulty for uniqueness is when the blowups arenot isolated in the space of blowups

One of the first major results on uniqueness was by Allard-Almgren in 1981 [AA] whereuniqueness of tangent cones with smooth cross-section for minimal varieties is proven underan additional integrability assumption on the cross-section The integrability condition ap-plies in a number of important cases but it is difficult to check and is not satisfied in manyother important cases

Perhaps surprisingly blowups for a number of important Geometric PDErsquos can essentiallybe reformulated as infinite dimensional gradient flows of analytic functionals Thus theuniqueness question would follow from an infinite dimensional version of Lojasiewiczrsquos the-orem for gradient flows of analytic functionals Infinite dimensional versions of Lojasiewiczinequalities were proven in a celebrated work of Leon Simon [Si1] for the area energy andrelated functionals and used in particular to prove a fundamental result about uniquenessof tangent cones with smooth cross-section of minimal surfaces This holds for instance atall singular points of an area-minimizing hypersurface in R8

Lojasiewicz inequalities follow easily near a critical point where the Hessian is uniformlynon-degenerate (this is the infinite dimensional analog of a non-degenerate critical pointwhere the Hessian is full rank) The difficulty is dealing with the directions in the kernel ofthe Hessian In the cases that Simon considers the Hessian has finite dimensional kernel byelliptic theory The rough idea of his approach is to use the easy argument in the (infinitelymany) directions where the Hessian is invertible and use the classical Lojasiewicz inequalitieson the finite dimensional kernel He makes this rigorous by reducing the infinite dimensionalversion to the classical Lojasiewicz inequality using Lyapunov-Schmidt reduction Infinitedimensional Lojasiewicz inequalities proven using Lyapunov-Schmidt reduction as in thework of Simon have had a profound impact on various areas of analysis and geometry andare usually referred to as Lojasiewicz-Simon inequalities

The cross-sections of the tangent cones at the singularities in these cases are assumed tobe smooth and compact and this is crucial This means that nearby cross-sections can bewritten as graphs over the cross-section and thus can be identified with functions on thecross-section of the cone The problem is then to prove a Lojasiewicz-Simon inequality for ananalytic functional on a Banach space of functions where 0 is a critical point correspondingto the cross-section

Uniqueness of tangents has important applications to regularity of the singular set seeSection 5 and cf eg [Si3] [Si4] [Si5] [BrCoL] and [HrLi] and cf Figure 2

1 Uniqueness of blowups for mean curvature flow

In the next few sections we will explain why at each generic singularity of a mean curvatureflow the blowup is unique that is independent of the sequence of rescalings see Figure 3This very recent result settled a major open problem that was open even in the case of meanconvex hypersurfaces where it was known that all singularities are generic Moreover itis the first general uniqueness theorem for blowups to a Geometric-PDE at a non-compactsingularity

As already mentioned uniqueness of blowups is perhaps the most fundamental questionthat one can ask about singularities and is known to imply regularity of the singular set

The proof of this uniqueness result relies on two completely new infinite dimensional Lo-jasiewicz type inequalities that unlike all other infinite dimensional Lojasiewicz inequalities

k=infin

k=0 lsquoinitiatorrsquo

lsquogeneratorrsquo

Figure 2 The Koch curve is close to a line on all scales yet the line that itis close to changes from scale to scale It is not rectifiable but admits a Holderparametrization It also illustrates that uniqueness of blowups is closely relatedto rectifiability

Figure 3 The essence of uniqueness of tangent flows Can the flow be closeto a cylinder at all times right before the singular time yet the axis of thecylinder changes as the time gets closer to the singular time

we know of do not follow from reduction to the classical finite-dimensional Lojasiewicz in-equalities but rather are proven directly and do not rely on Lojasiewiczrsquos arguments orresults

It is well-known that to deal with non-compact singularities requires entirely new ideasand techniques as one cannot argue as in Simonrsquos work and all the later work that uses hisideas Partly because of this it is expected that the techniques and ideas described herehave applications to other flows

The rest of this paper focuses on mean curvature flow (or MCF) of hypersurfaces This isa non-linear parabolic evolution equation where a hypersurface evolves over time by locallymoving in the direction of steepest descent for the volume element It has been used andstudied in material science for almost a century Unlike some of the other earlier papersin material science both von Neumannrsquos 1952 paper and Mullins 1956 paper had explicitequations In his paper von Neumann discussed soap foams whose interface tend to haveconstant mean curvature whereas Mullins is describing coarsening in metals in which inter-faces are not generally of constant mean curvature Partly as a consequence Mullins mayhave been the first to write down the MCF equation in general Mullins also found some ofthe basic self-similar solutions like the translating solution now known as the Grim ReaperTo be precise suppose that Mt sub Rn+1 is a one-parameter family of smooth hypersurfacesthen we say that Mt flows by the MCF if

(11) xt = minusH n

where H and n are the mean curvature and unit normal respectively of Mt at the point x

11 Tangent flows By definition a tangent flow is the limit of a sequence of rescalings ata singularity where the convergence is uniform on compact subsets4 For instance a tangentflow to Mt at the origin in space-time is the limit of a sequence of rescaled flows 1

δiMδ2i t

where δi rarr 0 A priori different sequences δi could give different tangent flows and thequestion of the uniqueness of the blowup - independent of the sequence - is a major questionin many geometric problems By a monotonicity formula of Huisken [H1] and an argumentof Ilmanen and White [I] [W3] tangent flows are shrinkers ie self-similar solutions ofMCF that evolve by rescaling The only generic shrinkers are round cylinders by [CM1]

We will say that a singular point is cylindrical if at least one tangent flow is a multiplicityone cylinder Sk times Rnminusk The main application of the new Lojasiewicz type inequality of[CM2] is the following theorem that shows that tangent flows at generic singularities areunique

Theorem 12 [CM2] Let Mt be a MCF in Rn+1 At each cylindrical singular point thetangent flow is unique That is any other tangent flow is also a cylinder with the same Rk

factor that points in the same direction

4This is analogous to a tangent cone at a singularity of a minimal variety cf [FFl]

This theorem solves a major open problem see eg page 534 of [W2] Even in the case ofthe evolution of mean convex hypersurfaces where all singularities are cylindrical uniquenessof the axis was unknown see [HS1] [HS2] [W1] [W4] [SS] [An] and [HaK]5

In recent joint work with Tom Ilmanen [CIM] we showed that if one tangent flow at asingular point of a MCF is a multiplicity one cylinder then all are However [CIM] left openthe possibility that the direction of the axis (the Rk factor) depended on the sequence ofrescalings The proof of Theorem 12 and in particular the first Lojasiewicz type inequalityof [CM2] has its roots in some ideas and inequalities from [CIM] and in fact implicitly usethat cylinders are isolated among shrinkers by [CIM]

The results of [CM2] are the first general uniqueness theorems for tangent flows to ageometric flow at a non-compact singularity (In fact not only are the singularities that[CM2] deal with non-compact but they are also non-integrable) Some special cases ofuniqueness of tangent flows for MCF were previously analyzed assuming either some sort ofconvexity or that the hypersurface is a surface of rotation see [H1] [H2] [HS1] [HS2] [W1][SS] [AAG] section 32 in the book [GGS] and [GK] [GKS] [GS] In contrast uniquenessfor blowups at compact singularities is better understood cf [AA] [Si1] [H3] [Sc] [KSy]and [Se]

One of the significant difficulties that [CM2] overcomes and sets it apart from all otherwork we know of is that the singularities are noncompact This causes major analyticaldifficulties and to address them requires entirely new techniques and ideas This is not somuch because of the subtleties of analysis on noncompact domains though this is an issuebut crucially because the evolving hypersurface cannot be written as an entire graph overthe singularity no matter how close we get to the singularity Rather the geometry of thesituation dictates that only part of the evolving hypersurface can be written as a graph overa compact piece of the singularity6

2 Lojasiewicz inequalities for non-compact hypersurfaces and MCF

The infinite dimensional Lojasiewicz-type inequalities that [CM2] showed are for the F -functional on the space of hypersurfaces

The F -functional is given by integrating the Gaussian over a hypersurface Σ sub Rn+1This is also often referred to as the Gaussian surface area and is defined by

F (Σ) = (4π)minusn2int

eminus|x|24 dmicro (21)

The entropy λ(Σ) is the supremum of the Gaussian surface areas over all centers and scales

λ(Σ) = supt0gt0 x0isinRn

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

The entropy is a Lyapunov functional for both MCF and rescaled MCF (it is monotonenon-increasing under the flows)

5The results of [CM2] not only give uniqueness of tangent flows but also a definite rate where the rescaledMCF converges to the relevant cylinder The distance to the cylinder is decaying to zero at a definite rateover balls whose radii are increasing at a definite rate to infinity

6In the end what comes out of the analysis in [CM2] is that the domain the evolving hypersurface is agraph over is expanding in time and at a definite rate but this is not all all clear from the outset see alsofootnote 3

It follows from the first variation formula that the gradient of F is

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

Thus the critical points of F are shrinkers ie hypersurfaces with H = 〈xn〉2

The mostimportant shrinkers are the generalized cylinders C these are the generic ones by [CM1]The space C is the union of Ck for k ge 1 where Ck is the space of cylinders SktimesRnminusk wherethe Sk is centered at 0 and has radius

radic2k and we allow all possible rotations by SO(n+ 1)

A family of hypersurfaces Σs evolves by the negative gradient flow for the F -functional ifit satisfies the equation

(partsx)perp = minusH n + xperp2 (24)

This flow is called the rescaled MCF since Σs is obtained from a MCF Mt by setting Σs =1radicminustMt s = minus log(minust) t lt 0 By (23) critical points for the F -functional or equivalently

stationary points for the rescaled MCF are the shrinkers for the MCF that become extinctat the origin in space-time A rescaled MCF has a unique asymptotic limit if and only if thecorresponding MCF has a unique tangent flow at that singularity

The paper [CM2] proved versions of the two Lojasiewicz inequalities for the F -functionalon a general hypersurface Σ Roughly speaking [CM2] showed that

dist(Σ C)2 le C |nablaΣF | (25)

(F (Σ)minus F (C))23 le C |nablaΣF | (26)

Equation (25) corresponds to Lojasiewiczrsquos first inequality fornablaF whereas (26) correspondsto his second inequality for F The precise statements of these inequalities are much morecomplicated than this but they are of the same flavor

As noted earlier a consequence of the classical Lojasiewicz gradient inequality for ananalytical function on Euclidean space is that near a critical point there is no other criticalvalues This consequence of a Lojasiewicz gradient inequality for the F -functional near around cylinder (and in fact the corresponding consequence of (25)) was established in earlierjoint work with Tom Ilmanen (see [CIM] for the precise statement)

Theorem 27 [CIM] Any shrinker that is sufficiently close to a round cylinder on a largebut compact set must itself be a round cylinder

In [CM2] an infinite dimensional analog of the first Lojasiewicz inequality is proven directlyand used together with an infinite dimensional analog of the argument in Subsection 02 toshow an analog of the second Lojasiewicz inequality As mentioned the reason why onecannot argue as in Simonrsquos work and all the later work that makes use his ideas comes fromthat the singularities are noncompact

21 The two Lojasiewicz inequalities We will now state the two Lojasiewicz-type in-equalities for the F -functional on the space of hypersurfaces

Suppose that Σ sub Rn+1 is a hypersurface and fix some sufficiently small ε0 gt 0 Given alarge integer ` and a large constant C` we let r`(Σ) be the maximal radius so that

bull Br`(Σ) cap Σ is the graph over a cylinder in Ck of a function u with uC2α le ε0 and|nabla`A| le C`

In the next theorem we will use a Gaussian L2 distance dC(R) to the space Ck in the ballof radius R To define this given Σk isin Ck let wΣk Rn+1 rarr R denote the distance to theaxis of Σk (ie to the space of translations that leave Σk invariant) Then we define

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

L2(BR) equiv infΣkisinCk

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

The Gaussian Lp norm on the ball BR is upLp(BR) =intBR|u|p eminus

Given a general hypersurface Σ it is also convenient to define the function φ by

φ =〈xn〉

2minusH (29)

so that φ is minus the gradient of the functional F

The main tools that [CM2] developed are the following two analogs for non-compacthypersurfaces of Lojasiewiczrsquos inequalities The first of these inequalities is really for thegradient whereas the second is for the function

Theorem 210 (A Lojasiewicz inequality for non-compact hypersurfaces [CM2]) If Σ subRn+1 is a hypersurface with λ(Σ) le λ0 and R isin [1 r`(Σ)minus 1] then

d2C(R) le C Rρ

φb`nL1(BR) + eminus

b`n R2

where C = C(n ` C` λ0) ρ = ρ(n) and b`n isin (0 1) satisfies lim`rarrinfin b`n = 1

The theorem bounds the L2 distance to Ck by a power of φL1 with an error term thatcomes from a cutoff argument since Σ is non-compact and is not globally a graph of thecylinder7 This theorem is essentially sharp Namely the estimate (211) does not hold forany exponent b`n larger than one but Theorem 210 lets us take b`n arbitrarily close to one

In [CM2] it is shown that the above inequality implies the following gradient type Lo-jasiewicz inequality This inequality bounds the difference of the F functional near a criticalpoint by two terms The first is essentially a power of nablaF while the second (exponentiallydecaying) term comes from that Σ is not a graph over the entire cylinder

Theorem 212 (A gradient Lojasiewicz inequality for non-compact hypersurfaces [CM2])If Σ sub Rn+1 is a hypersurface with λ(Σ) le λ0 β isin [0 1) and R isin [1 r`(Σ)minus 1] then

|F (Σ)minus F (Ck)| le C Rρ

c`n3+β2+2β

L2(BR) + eminusc`n (3+β)R2

8(1+β) + eminus(3+β)(Rminus1)2

where C = C(n ` C` λ0) ρ = ρ(n) and c`n isin (0 1) satisfies lim`rarrinfin c`n = 1

7This is a Lojasiewicz inequality for the gradient of the F functional (φ is the gradient of F ) This followssince by [CIM] cylinders are isolated critical points for F and thus dC locally measures the distance to thenearest critical point

When the theorem is applied the parameters β and ` is chosen to make the exponentgreater than one on the nablaF term essentially giving that |F (Σ)minus F (Ck)| is bounded by apower greater than one of |nablaF | A separate argument is needed to handle the exponentiallydecaying error terms

The paper [CM2] showed that when Σt are flowing by the rescaled MCF then both termson the right-hand side of (213) are bounded by a power greater than one of φL2 (thecorresponding statement holds for Theorem 210) Thus one essentially get the inequalities

d2C le C |nablaΣtF | (214)

(F (Σt)minus F (C))23 le C |nablaΣtF | (215)

These two inequalities can be thought of as analogs for the rescaled MCF of Lojasiewiczinequalities cf (25) and (26)

3 Cylindrical estimates for a general hypersurface

The proof of the two Lojasiewicz inequalities relies on some equations and estimates ongeneral hypersurfaces Σ sub Rn+1 Particularly important are bounds for nablaA

Hwhen the mean

curvature H is positive on a large set This will be discussed in this section

31 A general Simons equation An important point for the proof of the Lojasiewicztype inequalities is that the second fundamental form A of Σ satisfies an elliptic differentialequation similar to Simonsrsquo equation for minimal surfaces The elliptic operator will be theL operator from [CM1] given by

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

where we have the following

Proposition 32 [CM2] If φ = 12〈xn〉 minusH then

LA = A+ Hessφ + φA2 (33)

where the tensor A2 is given in orthonormal frame by (A2)ij = Aik Akj

Note that φ vanishes precisely when Σ is a shrinker and in this case we recover theSimonsrsquo equation for A for shrinkers from [CM1]

32 An integral bound when the mean curvature is positive One of the keys in theproof of the first Lojasiewicz type inequality is that the tensor τ = AH is almost parallelwhen H is positive and φ is small This generalizes an estimate from [CIM] in the case whereΣ is a shrinker (ie φ equiv 0) with H gt 0

Given f gt 0 define a weighted divergence operator divf and drift Laplacian Lf by

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

Lf u equiv divf (nablau) = Lu+ 〈nabla log fnablau〉 (35)

Here u may also be a tensor in this case the divergence traces only with nabla Note thatL = L1 We recall the quotient rule (see lemma 43 in [CIM])

Lemma 36 Given a tensor τ and a function g with g 6= 0 then

g=gL τ minus τ L g

g2=g L τ minus τ L g

g2(37)

Proposition 38 [CM2] On the set where H gt 0 we have

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

〈Hessφ + φA2 A〉H2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

The proposition follows easily from Proposition 32 and the Leibniz rule of Lemma 36see [CM2] for details

The next proposition gives exponentially decaying integral bounds for nabla(AH) when His positive on a large ball It will be important that these bounds decay rapidly

Proposition 311 [CM2] If BR cap Σ is smooth with H gt 0 then for s isin (0 R) we haveintBRminusscapΣ

∣∣∣∣nablaAH∣∣∣∣2 H2 eminus

|x|24 le 4

s2supBRcapΣ

|A|2 Vol(BR cap Σ) eminus(Rminuss)2

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

Proof Set τ = AH and u = |τ |2 = |A|2H2 It will be convenient within this proof to use

square brackets [middot] to denote Gaussian integrals over BR cap Σ ie [f ] =intBRcapΣ

f eminus|x|24

Let ψ be a function with support in BR Using the divergence theorem the formula fromProposition 38 for LH2u and the absorbing inequality 4ab le a2 + 4b2 we get

0 =[divH2

(ψ2nablau

)H2 ]=[

(ψ2 LH2u+ 2ψ〈nablaψnablau〉

[2ψ2 |nablaτ |2 + 2ψ2

(〈Hessφ + φA2 A〉

H2+|A|2 (∆φ+ φ |A|2)

)+ 4ψ〈nablaψ τ middot nablaτ〉

]ge[(ψ2 |nablaτ |2 minus 4 |τ |2 |nablaψ|2

+ 2[ψ2〈Hessφ + φA2 A〉

[ψ2 |A|2 (∆φ+ φ |A|2)

from which we obtain[ψ2 |nablaτ |2 H2

[|nablaψ|2 |A|2

]minus 2

[ψ2 〈Hessφ + φA2 A〉

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

The proposition follows by choosing ψ equiv 1 on BRminuss and going to zero linearly on partBR

This proposition has the the following corollary

Corollary 313 [CM2] If BRcapΣ is smooth with H gt δ gt 0 and |A| le C1 then there existsC2 = C2(n δ C1) so that for s isin (0 R) we haveint

BRminusscapΣ

∣∣∣∣nablaAH∣∣∣∣2 eminus

|x|24 le C2

s2Vol(BR cap Σ) eminus

(Rminuss)24 + C2

intBRcapΣ

|Hessφ|+ |φ| eminus|x|24 (314)

Remark 315 Corollary 313 essentially bounds the distance squared to the space of cylin-ders by φL1 This is sharp it is not possible to get the sharper bound where the powersare the same This is a general fact when there is a non-integrable kernel Namely if weperturb in the direction of the kernel then φ vanishes quadratically in the distance

The next corollary combines the Gaussian L2 bound on nablaτ from Corollary 313 withstandard interpolation inequalities to get pointwise bounds on nablaτ and nabla2τ

Corollary 316 [CM2] If BR cap Σ is smooth with H gt δ gt 0 |A| +∣∣nabla`+1A

∣∣ le C1 and

λ(Σ) le λ0 then there exists C3 = C3(n λ0 δ ` C1) so that for |y|+ 11+|y| lt Rminus 1 we have∣∣∣∣nabla A

∣∣∣∣ (y) +

∣∣∣∣nabla2 A

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

where the exponent d`n isin (0 1) has lim`rarrinfin d`n = 1

See [CM2] for the proof of Corollary 316

4 Distance to cylinders and the first Lojasiewicz inequality

Finally we will briefly outline how one get from Corollary 316 to the proof of the firstLojasiewicz type inequality for the F -functional This inequality will follow from the boundson the tensor τ = A

Hin the previous section together with the following proposition

Proposition 41 [CM2] Given n δ gt 0 and C1 there exist ε0 gt 0 ε1 gt 0 and C so that ifΣ sub Rn+1 is a hypersurface (possibly with boundary) that satisfies

(1) H ge δ gt 0 and |A|+ |nablaA| le C1 on BR cap Σ(2) B5

radic2n cap Σ is ε0 C

2-close to a cylinder in Ck for some k ge 1

then for any r isin (5radic

2nR) with

r2 supB5radic2n

(|φ|+ |nablaφ|) + r5 supBr

(|nablaτ |+ |nabla2τ |

)le ε1 (42)

we have that Bradicr2minus3k cap Σ is the graph over (a subset of) a cylinder in Ck of u with

(43) |u|+ |nablau| le C

r2 sup

B5radic2n

This proposition shows that Σ must be close to a cylinder as long as H is positive φ issmall τ is almost parallel and Σ is close to a cylinder on a fixed small ball Together withTom Ilmanen we proved a similar result in proposition 22 in [CIM] in the special case whereΣ is a shrinker (ie when φ equiv 0) and this proposition is inspired by that one

To prove Proposition 41 we make use of the following result from [CIM] (see corollary422 in [CIM])

Lemma 44 [CIM] If Σ sub Rn+1 is a hypersurface (possibly with boundary) with

bull 0 lt δ le H on Σbull the tensor τ equiv AH satisfies |nablaτ |+ |nabla2τ | le ε le 1bull At the point p isin Σ τp has at least two distinct eigenvalues κ1 6= κ2

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

From this lemma we see that if the assumption of the lemma holds for a hypersurface thenthe principal curvatures divide into two groups One group consists of principal curvaturesthat are close to zero and the other group consists of principal curvatures that cluster arounda non-zero real number Thus we get flatness for any two-plane containing a principaldirection in the first group while any two-plane spanned by principal directions in thesecond group is umbilic This is the starting point for the proof of Proposition 41

5 The singular set of MCF with generic singularities

A major theme in PDErsquos over the last fifty years has been understanding singularities andthe set where singularities occur In the presence of a scale-invariant monotone quantityblowup arguments can often be used to bound the dimension of the singular set see eg[Al] [F] Unfortunately these dimension bounds say little about the structure of the setHowever using the results of the previous sections [CM3] gave a rather complete descriptionof the singular set for MCF with generic singularities

The main result of [CM3] is the following

Theorem 51 ([CM3]) Let Mt sub Rn+1 be a MCF of closed embedded hypersurfaces withonly cylindrical singularities then the space-time singular set satisfies

bull It is contained in finitely many (compact) embedded Lipschitz8 submanifolds each ofdimension at most (nminus 1) together with a set of dimension at most (nminus 2)bull It consists of countably many graphs of 2-Holder functions on spacebull The time image of each subset with finite parabolic 2-dimensional Hausdorff measure

has measure zero each such connected subset is contained in a time-slice

In fact [CM3] proves considerably more than what is stated in Theorem 51 see theorem418 in [CM3] For instance instead of just proving the first claim of the theorem theentire stratification of the space-time singular set is Lipschitz of the appropriate dimensionMoreover this holds without ever discarding any subset of measure zero of any dimensionas is always implicit in any definition of rectifiable To illustrate the much stronger versionconsider the case of evolution of surfaces in R3 In that case this gives that the space-timesingular set is contained in finitely many (compact) embedded Lipschitz curves with cylindersingularities together with a countable set of spherical singularities In higher dimensionsthe direct generalization of this is proven

Theorem 51 has the following corollaries

Corollary 52 ([CM3]) Let Mt sub Rn+1 be a MCF of closed embedded mean convexhypersurfaces or a MCF with only generic singularities then the conclusion of Theorem 12holds

8In fact Lipschitz is with respect to the parabolic distance on space-time which is a much strongerassertion than Lipschitz with respect to the Euclidean distance Note that a function is Lipschitz when thetarget has the parabolic metric on R is equivalent to that it is 2-Holder for the standard metric on R

More can be said in dimensions three and four

Corollary 53 ([CM3]) If Mt is as in Theorem 51 and n = 2 or 3 then the evolvinghypersurface is completely smooth (ie without any singularities) at almost all times Inparticular any connected subset of the space-time singular set is completely contained in atime-slice

Corollary 54 ([CM3]) For a generic MCF in R3 or R4 or a flow starting at a closedembedded mean convex hypersurface in R3 or R4 the conclusion of Corollary 53 holds

The conclusions of Corollary 54 hold in all dimensions if the initial hypersurface is 2- or3-convex A hypersurface is said to be k-convex if the sum of any k principal curvatures isnonnegative

A key technical point in [CM3] is to prove a strong parabolic Reifenberg property for MCFwith generic singularities In fact the space-time singular set is proven to be (parabolically)Reifenberg vanishing In Analysis a subset of Euclidean space is said to be Reifenberg (orReifenberg flat) if on all sufficiently small scales it is after rescaling to unit size close toa k-dimensional plane The dimension of the plane is always the same but the plane itselfmay change from scale to scale Many snowflakes like the Koch snowflake are Reifenbergwith Hausdorff dimension strictly larger than one A set is said to be Reifenberg vanishingif the closeness to a k-plane goes to zero as the scale goes to zero It is said to have thestrong Reifenberg property if the k-dimensional plane depends only on the point but not onthe scale Finally one sometimes distinguishes between half Reifenberg and full Reifenbergwhere half Reifenberg refers to that the set is close to a k-dimensional plane whereas fullReifenberg refers to that in addition one also has the symmetric property The plane on thegiven scale is close to the set

Using the results from [CM2] described earlier in this paper [CM3] shows that the singularset in space-time is strong (half) Reifenberg vanishing with respect to the parabolic Hausdorffdistance This is done in two steps showing first that nearby singularities sit inside aparabolic cone (ie between two oppositely oriented space-time paraboloids that are tangentto the time-slice through the singularity) In fact this parabolic cone property holds withvanishing constant Next in the complementary region of the parabolic cone in space-time(that is essentially space-like) the parabolic Reifenberg essentially follows from the spaceReifenberg that the uniqueness of [CM2] of tangent flows implies

An immediate consequence of independent interest of our parabolic cone property withvanishing constant is that nearby a generic singularity in space-time (nearby is with respectto the parabolic distance) all other singularities happen at almost the same time

These results should be contrasted with a result of Altschuler-Angenent-Giga [AAG] (cf[SS]) that shows that in R3 the evolution of any rotationally symmetric surface obtainedby rotating the graph of a function r = u(x) a lt x lt b around the x-axis is smooth exceptat finitely many singular times where either a cylindrical or spherical singularity formsFor more general rotationally symmetric surfaces (even mean convex) the singularities canconsist of nontrivial curves For instance consider a torus of revolution bounding a regionΩ If the torus is thin enough it will be mean convex Since the symmetry is preserved andbecause the surface always remains in Ω it can only collapse to a circle Thus at the time ofcollapse the singular set is a simple closed curve White showed that a mean convex surface

evolving by MCF in R3 must be smooth at almost all times and at no time can the singularset be more than 1-dimensional (see section 5 in [W2]) In all dimensions White showedthat the space-time singular set of a mean convex MCF has parabolic Hausdorff dimensionat most (nminus 1) see [W1] and cf theorem 115 in [HaK] In fact Whitersquos general dimensionreducing argument gives that the singular set of any MCF with only cylindrical singularitieshas dimension at most (nminus 1)

These results motivate the following conjecture

Conjecture 55 ([CM3]) Let Mt be a MCF of closed embedded hypersurfaces in Rn+1

with only cylindrical singularities Then the space-time singular set has only finitely manycomponents

If this conjecture was true then it would follow from this paper that in R3 and R4 meancurvature flow with only generic singularities is smooth except at finitely many times cfwith the three-dimensional conjecture at the end of section 5 in [W2]

References

[AA] WK Allard and FJ Almgren Jr On the radial behavior of minimal surfaces and the uniqueness oftheir tangent cones Ann of Math (2) 113 (1981) no 2 215ndash265

[AAG] S Altschuler SB Angenent and Y Giga Mean curvature flow through singularities for surfaces ofrotation Journal of geometric analysis Volume 5 Number 3 (1995) 293ndash358

[Al] F J Almgren Jr Q-valued functions minimizing dirichletrsquos integral and the regularity of area mini-mizing rectiable currents up to codimension two preprint

[An] B Andrews Noncollapsing in mean-convex mean curvature flow Geom Topol 16 (2012) no 3 1413ndash1418

[BrCoL] H Brezis J-M Coron and E Lieb Harmonic maps with defects Comm Math Phys 107 (1986)no 4 649ndash705

[CIM] TH Colding T Ilmanen and WP Minicozzi II Rigidity of generic singularities of mean curvatureflow publications IHES to appear arXiv13046356

[CM1] TH Colding and WP Minicozzi II Generic mean curvature flow I generic singularities Annals ofMath Volume 175 (2012) Issue 2 755ndash833

[CM2] Uniqueness of blowups and Lojasiewicz inequalities Annals of Math to appearhttparXiv13124046

[CM3] The singular set of mean curvature flow with generic singularities preprint arXiv14055187[F] H Federer The singular sets of area minimizing rectifiable currents with codimension one and of area

minimizing flat chains modulo two with arbitrary codimension Bull Amer Math Soc 76 (1970) 767ndash771[FFl] H Federer and W Fleming Normal and integral currents Ann of Math 72 (1960) 458ndash520[GK] Z Gang and D Knopf Universality in mean curvature flow neckpinches

httparxivorgabs13085600[GKS] Z Gang D Knopf and IM Sigal Neckpinch dynamics for asymmetric surfaces evolving by mean

curvature flow preprint httparXiv11090939v1[GS] Z Gang and IM Sigal Neck pinching dynamics under mean curvature flow preprint

httparxivorgabs07082938[GGS] M Giga Y Giga and J Saal Nonlinear partial differential equations Asymptotic behavior of solu-

tions and self-similar solutions Progress in Nonlinear Differential Equations and their Applications 79Birkhauser Boston Inc Boston MA 2010

[Hr] RM Hardt Singularities of harmonic maps Bull Amer Math Soc (NS) 34 (1997) no 1 15ndash34[HrLi] RM Hardt and F-H Lin Stability of singularities of minimizing harmonic maps J Differential

Geom 29 (1989) no 1 113ndash123[HaK] R Haslhofer and B Kleiner Mean curvature flow of mean convex hypersurfaces preprint

httparXiv13040926

[Ho] L Hormander On the division of distributions by polynomials Ark Mat 3 (1958) 555ndash568[H1] G Huisken Asymptotic behavior for singularities of the mean curvature flow J Differential Geom 31

(1990) no 1 285ndash299[H2] Local and global behaviour of hypersurfaces moving by mean curvature Differential geometry

partial differential equations on manifolds (Los Angeles CA 1990) 175ndash191 Proc Sympos Pure Math54 Part 1 Amer Math Soc Providence RI 1993

[H3] Flow by mean curvature of convex surfaces into spheres J Differential Geometry 20 (1984)237-266

[HS1] G Huisken and C Sinestrari Convexity estimates for mean curvature flow and singularities of meanconvex surfaces Acta Math 183 (1999) no 1 45ndash70

[HS2] G Huisken and C Sinestrari Mean curvature flow singularities for mean convex surfaces Calc VarPartial Differ Equ 8 (1999) 1ndash14

[I] T Ilmanen Singularities of Mean Curvature Flow of Surfaces preprint 1995httpwwwmathethzch˜paperspubhtml

[KSy] R Kohn and S Serfaty A deterministic-control-based approach to motion by curvature Comm PureAppl Math 59 (2006) no 3 344ndash407

[L1] S Lojasiewicz Ensembles semi-analytiques IHES notes (1965)[L2] Sur le problme de la division Studia Math 18 (1959) 87ndash136

[L3] Une propriete topologique des sous-ensembles analytiques reels Les Equations aux DeriveesPartielles (1963) pp 87ndash89 CNRS Paris

[L4] Division drsquoune distribution par une fonction analytique de variables relles C R Acad SciParis 246 683ndash686 (1958) (Seance du 3 Fevrier 1958)

[Sc] F Schulze Uniqueness of compact tangent flows in mean curvature flow J Reine Angew Math 690(2014) 163172

[Se] N Sesum Rate of convergence of the mean curvature flow Comm Pure Appl Math 61 (2008) no 4464ndash485

[Si1] L Simon Asymptotics for a class of evolution equations with applications to geometric problemsAnnals of Math 118 (1983) 525ndash571

[Si2] A general asymptotic decay lemma for elliptic problems Handbook of geometric analysis No1 381ndash411 Adv Lect Math (ALM) 7 Int Press Somerville MA 2008

[Si3] Rectifiability of the singular set of energy minimizing maps Calc Var Partial DifferentialEquations 3 (1995) no 1 1ndash65

[Si4] Theorems on regularity and singularity of energy minimizing maps Lectures on geometricvariational problems (Sendai 1993) 115ndash150 Springer Tokyo 1996

[Si5] Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizingmaps Surveys in differential geometry Vol II (Cambridge MA 1993) 246ndash305 Int Press CambridgeMA 1995

[SS] H Soner and P Souganidis Singularities and uniqueness of cylindrically symmetric surfaces moving bymean curvature Comm Partial Differential Equations 18 (1993) no 5-6 859ndash894

[T] J Taylor Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in R3Invent Math 22 (1973) 119ndash159

[W1] B White The nature of singularities in mean curvature flow of mean-convex sets J Amer MathSoc 16 (2003) no 1 123ndash138

[W2] Evolution of curves and surfaces by mean curvature Proceedings of the International Congressof Mathematicians Vol I (Beijing 2002) 525ndash538

[W3] Partial regularity of mean-convex hypersurfaces flowing by mean curvature Int Math ResNotices (1994) 185ndash192

[W4] Subsequent singularities in mean-convex mean curvature flow arXiv11031469 2011

MIT Dept of Math 77 Massachusetts Avenue Cambridge MA 02139-4307

E-mail address coldingmathmitedu and minicozzmathmitedu

01 Lojasiewicz inequalities

02 First Lojasiewicz implies the second

03 The Lojasiewicz Theorem

04 Infinite dimensional Lojasiewicz inequalities and applications


11 Tangent flows


21 The two Lojasiewicz inequalities


31 A general Simons equation

32 An integral bound when the mean curvature is positive



References

Equation (01) was the main inequality in Lojasiewiczrsquos proof of Laurent Schwarzrsquos divisionconjecture1 in analysis Around the same time Hormander [Ho] independently provedSchwarzrsquos division conjecture in the special case of polynomials and a key step in his proofwas also (01) when f is a polynomial

A few years later Lojasiewicz solved a conjecture of Whitney2 in [L3] using the followinginequality3 With the same assumptions on f for every p isin U there are a possibly smallerneighborhood W of p and constants β isin (0 1) and C gt 0 such that for all x isin W

|f(x)minus f(p)|β le C |nablaxf | (02)

Note that this inequality is trivial unless p is a critical point for f One immediate consequence of (02) is that every critical point of f has a neighborhood

where every other critical point has the same value It is easy to construct smooth functionswhere this is not the case

02 First Lojasiewicz implies the second In this subsection we will explain how thesecond Lojasiewicz inequality for a function f in a neighborhood of an isolated critical pointfollows from the first To make things concrete we will show that the second holds withβ = 2

3when the first holds with α = 2

Suppose that f Rn rarr R is smooth function with f(0) = 0 and nablaf(0) = 0 without lossof generality we may assume that the Hessian is in diagonal form at 0 and we will write thecoordinates as x = (y z) where y are the coordinates where the Hessian is nondegenerateBy Taylorrsquos formula in a small neighborhood of 0 we have that

f(x) =ai2y2i +O(|x|3) (03)

fyi(x) = ai yi +O(|x|2) (04)

fzi(x) = O(|x|2) (05)

It follows from this that the second of the two Lojasiewicz inequalities holds for f and β = 23

provided that |z|2 le ε |y| for some sufficiently small ε gt 0 Namely if |z|2 le ε |y| then

C |y| le |nablaxf | and |f(x)| le Cminus1 |y|32(06)

for some positive constant C and hence

|f(x)|23 le C |nablaxf | (07)

Therefore we only need to prove the second Lojasiewicz inequality for f in the region|z|2 ge ε |y| We will do this using the first Lojasiewicz inequality for nablaf Since 0 is anisolated critical point for f the first Lojasiewicz inequality for nablaf gives that

|nablaxf | ge C |x|2 (08)

By assumption on the region and the Taylor expansion for f we get that in this region

|f(x)| le C |y|2 + C |z|3 le C |z|3 le C |x|3 (09)

1L Schwartz conjectured that if f is a non-trivial real analytic function and T is a distribution then thereexists a distribution S satisfying f S = T

2Whitney conjectured that if f is analytic in an open set U of Rn then the zero set Z is a deformationretract of an open neighborhood of Z in U

3Lojasiewicz called this inequality the gradient inequality

limtrarrinfin

x(t) = xinfin (011)

f(t) le C tminus1

2βminus1 (012)

|nablaf | ds =

int infin1

(minusint infin

f prime s1+ε ds

) 12(int infin

sminus1minusε ds

minusint T

f sε ds (015)

k=infin

lsquogeneratorrsquo

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

limtrarrinfin

x(t) = xinfin (011)

f(t) le C tminus1

2βminus1 (012)

|nablaf | ds =

int infin1

(minusint infin

f prime s1+ε ds

) 12(int infin

sminus1minusε ds

minusint T

f sε ds (015)

k=infin

lsquogeneratorrsquo

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

f(t) le C tminus1

2βminus1 (012)

|nablaf | ds =

int infin1

(minusint infin

f prime s1+ε ds

) 12(int infin

sminus1minusε ds

minusint T

f sε ds (015)

k=infin

lsquogeneratorrsquo

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

k=infin

lsquogeneratorrsquo

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

k=infin

lsquogeneratorrsquo

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

(11) xt = minusH n

δiMδ2i t

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

(4π t0)minusn2int

eminus |xminusx0|

4t0 dmicro (22)

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

nablaΣF (ψ) =

(H minus 〈xn〉

)ψ eminus

|x|24 (23)

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

d2C(R) = inf

ΣkisinCkwΣk minus

radic2k2

intBRcapΣk

(wΣk minusradic

2k)2 eminus|x|24 (28)

φ =〈xn〉

2minusH (29)

d2C(R) le C Rρ

b`n R2

c`n3+β2+2β

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

Hwhen the mean

L equiv L+ |A|2 +1

2equiv ∆minus 1

2nablaxT + |A|2 +

divf (V ) =1

24 divΣ

(f eminus|x|

24 V)(34)

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

g2(37)

Hessφ + φA2

H+A (∆φ+ φ |A|2)

H2(39)

∣∣∣∣nablaAH∣∣∣∣2 + 2

+ 2|A|2 (∆φ+ φ |A|2)

H3(310)

|x|24 le 4

s2supBRcapΣ

intBRcapΣ

∣∣∣∣〈Hessφ A〉+|A|2

H∆φ

∣∣∣∣+

∣∣∣∣〈A2 A〉+|A|4

∣∣∣∣ |φ| eminus|x|24 (312)

f eminus|x|24

0 =[divH2

(ψ2nablau

)H2 ]=[

[2ψ2 |nablaτ |2 + 2ψ2

H2+|A|2 (∆φ+ φ |A|2)

[ψ2 |A|2 (∆φ+ φ |A|2)

[|nablaψ|2 |A|2

]minus 2

[ψ2 ∆φ

H+ ψ2 φ

BRminusscapΣ

|x|24 le C2

(Rminuss)24 + C2

intBRcapΣ

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

∣∣ le C1 and

∣∣∣∣ (y) +

∣∣∣∣ (y) le C3R2n

eminusd`n

(Rminus1)2

8 + φd`n2

L1(BR)

e|y|28 (317)

2nR) with

r2 supB5radic2n

)le ε1 (42)

r2 sup

B5radic2n

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

|κ1κ2| le2 ε

|κ1 minus κ2|+

|κ1 minus κ2|2

References

httparXiv13040926






11 Tangent flows








References

References

httparXiv13040926






11 Tangent flows








References

References

httparXiv13040926






11 Tangent flows








References

Documents

arXiv:1402.5087v3 [math.DG] 25 Feb 2015 · 2015. 2. 26. · Cjyj jr xfjand jf(x)j C 1 jyj 3 (0.6) 2 for some positive constant Cand, hence, jf(x)j23 Cjr (0.7) xfj: Therefore, we only