Mean Curvature Flow in Higher

Codimension

Summer School on Differential Geometry

Konkuk University, Seoul

August 2013

Knut SmoczykLeibniz Universität Hannover

Introduction

Mean curvature flow is perhaps the most important geometric evolution equation ofsubmanifolds in Riemannianmanifolds. Intuitively, a family of smooth submanifoldsevolves under mean curvature flow, if the velocity at each point of the submanifoldis given by the mean curvature vector at that point. For example, round spheres ineuclidean space evolve under mean curvature flow while concentrically shrinkinginward until they collapse in finite time to a single point, the common center of thespheres.

Mullins [Mul56] proposed mean curvature flow to model the formation of grainboundaries in annealing metals. Later the evolution of submanifolds by their meancurvature has been studied by Brakke [Bra78] from the viewpoint of geometric mea-sure theory. Among the first authors who studied the corresponding nonparametricproblem were Temam [Tem76] in the late 1970’s and Gerhardt [Ger80] and Ecker[Eck82] in the early 1980’s. Pioneering work was done by Gage [Gag84], Gage &Hamilton [GH86] and Grayson [Gra87] who proved that the curve shortening flow(more precisely, the “mean" curvature flow of curves in R2) shrinks embedded closedcurves to “round" points. In his seminal paper Huisken [Hui84] proved that closedconvex hypersurfaces in euclidean space Rm+1,m > 1 contract to single round pointsin finite time (later he extended his result to hypersurfaces in Riemannian mani-folds that satisfy a suitable stronger convexity, see [Hui86]). Then, until the mid1990’s, most authors who studied mean curvature flow mainly considered hypersur-faces, both in euclidean and Riemannian manifolds, whereas mean curvature flow inhigher codimension did not play a great role. There are various reasons for this, oneof them is certainly the much different geometric situation of submanifolds in highercodimension since the normal bundle and the second fundamental tensor are morecomplicated. But also the analysis becomes more involved and the algebra of thesecond fundamental tensor is much more subtle since for hypersurfaces there usu-ally exist more scalar quantities related to the second fundamental form than in caseof submanifolds in higher codimension. Some of the results previously obtained formean curvature flow of hypersurfaces carry over without change to submanifolds ofhigher codimension but many do not and in addition even new phenomena occur.

Among the first results in this direction are the results on mean curvature flow ofspace curves by Altschuler and Grayson [Alt91,AG92], measure-theoretic approachesto higher codimension mean curvature flows by Ambrosio & Soner [AS97], existenceand convergence results for the Lagrangian mean curvature flow [Smo96, Smo00,Smo02,TY02], mean curvature flow of symplectic surfaces in codimension two [CL04,Wan02] and long-time existence and convergence results of graphic mean curvatureflows in higher codimension [CLT02, SW02, Smo04,Wan02, Xin08]. Recently there

i

has been done quite some work on the formation and classification of singularitiesin mean curvature flow [Anc06,CL10,CCH09a,CSS07,CM09,GSSZ07,HL09,HS09,JLT10, LS10a, LS10b, LXYZ11, SW03], partially motivated by Hamilton’s and Perel-man’s [Ham95a,Per02,Per03a,Per03b] work on the Ricci flow that in many ways be-haves akin to the mean curvature flow and vice versa.

The results in mean curvature flow can be roughly grouped into two categories: Thefirst category contains results that hold (more or less) in general, i.e. that are inde-pendent of dimension, codimension or the ambient space. In the second class wefind results that are adapted to more specific geometric situations, like results forhypersurfaces, Lagrangian or symplectic submanifolds, graphs, etc..

These notes are based on my lectures on mean curvature flow in higher codimensionthat I held during the Summer School on Differential Geometry at Konkuk Universityin Seoul, August 2013. My aim in these lectures was to give an introduction to meancurvature flow for the beginner. For those interested in a more detailed description,let me refer to several nice monographs on mean curvature flow that can be foundin the literature, e.g. a well written introduction to the regularity of mean curva-ture flow of hypersurfaces is [Eck04]. For the curve shortening flow see [CZ01]. Formean curvature flow in higher codimension there exist some lecture notes by Wang[Wan08b] and a detailed introduction to higher codimensional mean curvature flowby myself [Smo12].

Konkuk University in Seoul, August 2013

Knut Smoczyk

ii

Contents

Lecture 1 1

§1 Immersions and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 1§2 Tangent and normal bundles . . . . . . . . . . . . . . . . . . . . . . . . . 1§3 1st fundamental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2§4 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3§5 2nd fundamental form and mean curvature vector . . . . . . . . . . . . 4

Lecture 2 7

§6 Structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7§7 Local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8§8 First variation of volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 10§9 Mean curvature flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12§10 Invariance under ambient isometries . . . . . . . . . . . . . . . . . . . . 13

Lecture 3 15

§11 Invariance under the diffeomorphism group . . . . . . . . . . . . . . . . 15§12 Analytic nature of the mean curvature flow . . . . . . . . . . . . . . . . 15§13 Short-time existence and uniqueness . . . . . . . . . . . . . . . . . . . . 16§14 Long-time existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16§15 Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Lecture 4 19

§16 Maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19§17 Comparison principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20§18 Lagrangian submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 20§19 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23§20 Self-similar solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Bibliography 27

Index 33

iii

Lecture 1

Throughout these lectures, M will be an oriented smooth manifold of dimensionm and (N,g) will be a complete (either compact or non-compact) smooth Rieman-nian manifold of dimension n > m.

§1 Immersions and embeddings

For 1 ≤ k ≤ ∞ we will consider Ck-Immersions of M into N , i.e. maps F :M → N ofclass Ck for which the Differential

DF|p : TpM→ TF(p)N

is injective for any p ∈M. An immersion F :M → N will be called an embedding, ifF(M) ⊂N is an embedded submanifold.

a) b)

Figure 2.1: a) An immersion ofM = S1 into N =R2. b) Embedding of S1 into R2.

§2 Tangent and normal bundles

If F :M→N is an immersion and p ∈M we define

T ⊤p M :=w ∈ TF(p)N : ∃v ∈ TpM with DF|p(v) = w

1

Lecture 1

andT ⊥p M :=

w ∈ TF(p)N : g|F(p)(w,w) = 0, ∀w ∈ T ⊤p M

,

i.e. T ⊥p M is the orthogonal complement w.r.t. g|F(p) of T ⊤p M within TF(p)N . The normalbundle ofM is the bundle T ⊥M overM whose fibers are given by T ⊥p M. Analogouslywe define the bundle T ⊤M as the bundle over M with fibers T ⊤p M. Both bundlesare sub-bundles of the tangent bundle of N along M, i.e. of the bundle F∗TN overM whose fiber at p ∈ M is given by TF(p)N . Sometimes F∗TN is called the pull-backbundle of TN via F. Since

a) b)

Figure 2.2: a) T ⊤p M can be identified with TpM but is not the same vector space. b)The two projections of w ∈ TqN onto w⊥ ∈ T ⊥p M and w⊤ ∈ T ⊤p M withq = F(p).

DF|p : TpM→ T ⊤p M

is an isomorphism for each p ∈M, one usually identifies T ⊤M with TM. We obtaintwo natural projections

⊥: (F∗TN )p→ T ⊥p M w 7→ w⊥

⊤ : (F∗TN )p→ T ⊤p M w 7→ w⊤

and for any p ∈ F−1(q) we obtain the orthogonal decomposition

TqN = T ⊤p M ⊕ T ⊥p M =DF|p(TpM

)⊕ T ⊥p M.

§3 1st fundamental form

3.1 Definition (Induced Riemannian metric (1st fundamental form))If F : M → N is an immersion into a Riemannian manifold (N,g), then F induces aRiemannian metric F∗g onM, via

(F∗g)|p(v1,v2) := g|F(p)(DF|p(v1),DF|p(v2)), ∀p ∈M and ∀v1,v2 ∈ TpM.

2

§4 Connections

It is important that F is an immersion since otherwise F∗g might be degenerated.

§4 Connections

The connections we use are all induced by the Riemannian metric g and its Levi-Civita connection ∇TN on TN . We recall that the Levi-Civita connection on TN isdefined by the formula

g(∇TNV X,Y ) =12

(V(g(X,Y )

)+X

(g(V ,Y )

)−Y

(g(V ,X)

)−g(V , [X,Y ])− g(X, [V ,Y ]) + g(Y , [V ,X])

),

where V ,X,Y ∈ X(N ) are arbitrary smooth vector fields. Let F :M → N be a smoothimmersion. On F∗TN we use the pull-back connection ∇F∗TN . This can be describedas follows.

Let W ∈ Γ (F∗TN ) be a smooth section in the pull-back bundle, i.e. let W : M →F∗TN be a smooth map with W (p) ∈ TF(p)N for all p ∈M. Let p ∈M be arbitraryand set q := F(p). In an open neighborhood V ⊂ N of q we choose a local trivial-ization of TN , e.g. (eα)α=1,...,n, eα ∈ X(T V ). If U ⊂M is a sufficiently small openset containing p, thenW|U can be written in the form

W (p′) =W α(p′)eα(F(p′)), ∀p′ ∈U

with smooth functionsW α ∈ C∞(U ). For v ∈ TpM we then set(∇F∗TNv W

)(p) :=DW α

|p (v)eα(F(p)) +Wα(p)

(∇TNv eα

)(F(p)).

This is well-defined, i.e. independent of the choice of the frame (eα)α=1,...,n.

Since T ⊤M and T ⊥M are both sub-bundles of F∗TN we obtain connections on themby

∇⊤vW :=(∇F∗TNv W

)⊤, ∇⊥v ν :=

(∇F∗TNv ν

)⊥,

where v ∈ TpM and W ∈ Γ (T ⊤M) and ν ∈ Γ (T ⊥M) are smooth sections. Note that bydefinition of the Levi-Civita connection ∇ w.r.t. the induced metric F∗g on TM andby definition of the pull-back connection and bundle we have for any X ∈ X(M) thatDF(X) ∈ Γ (T ⊤M) and

(4.1) DF(∇vX

)= ∇⊤v

(DF(X)

).

3

Lecture 1

Since in general it is clear how the connections are induced on all bundles overMthat naturally appear in our context, we will sometimes omit the superscript, i.e.we will then simply use ∇ for all these connections. We make an exception for theconnection ∇⊥ on the normal bundle and on product bundles E ⊗ T ⊥M containigT ⊥M as a factor.

§5 2nd fundamental form and mean curvature vector

If F :M → N is an immersion, then the differential DF maps TpM to T ⊤M ⊂ TF(p)Nand hence DF ∈ Γ (T ⊤M ⊗ T ∗M) can also be considered as a section DF ∈ Γ (F∗TN ⊗T ∗M). We extend the projections ⊥,⊤ to product bundles containing either T ⊥M orT ⊤M as a factor. In this way we have for any vector field X ∈ X(M)(

∇T⊤M⊗T ∗M

v DF)(X) = ∇⊤v (DF(X))−DF(∇vX)

(4.1)= 0

and (∇F∗TN⊗T ∗Mv DF

)(X) = ∇F

∗TNv (DF(X))−DF(∇vX)

=(∇F∗TNv (DF(X))

)⊥+(∇F∗TNv (DF(X))

)⊤−DF(∇vX)

=(∇F∗TNv (DF(X))

)⊥+∇⊤v (DF(X))−DF(∇vX)

=(∇F∗TNv (DF(X))

)⊥.(5.1)

2nd fundamental tensorThe second fundamental tensor A of an immersion F :M→N is given by

A := ∇F∗TN⊗T ∗MDF ∈ Γ (F∗TN ⊗ T ∗M ⊗ T ∗M), A|p(v1,v2) =

(∇F∗TN⊗T ∗Mv1 DF

)|p(v2)

for all v1,v2 ∈ TpM.

5.1 LemmaLet A be the second fundamental tensor of an immersion F :M→N .

a) A is normal, i.e.

g|F(p)(A|p(v1,v2),DF|p(v3)

)= 0, ∀v1,v2,v3 ∈ TpM.

Consequently, A can also be considered as a section in T ⊥M ⊗ T ∗M ⊗ T ∗M.

b) A is symmetric, i.e.

A|p(v1,v2) = A|p(v2,v1), ∀v1,v2 ∈ TpM and ∀p ∈M.

4

§5 2nd fundamental form and mean curvature vector

Proof:

a) Directly from equation (5.1).

b) Let V1,V2 ∈ X(M) be smooth vector fields. Since ∇TN and ∇ are torsion free we get

∇F∗TNV1

(DF(V2))−∇F∗TNV2

(DF(V1)) =DF([V1,V2]).

Hence

A(V1,V2)−A(V2,V1) =(∇F∗TNV1

(DF(V2)))⊥−(∇F∗TNV2

(DF(V1)))⊥

=(DF([V1,V2])

)⊥= 0

2nd fundamental formIf ν ∈ T ⊥p M is a normal vector, then the second fundamental form Aν w.r.t. ν is thesymmetric bilinear form on TpM defined by

Aν(v1,v2) := g|F(p)(A|p(v1,v2),ν), for all v1,v2 ∈ TpM.

Mean curvature vectorThe mean curvature vector field H of an immersion F : M → N is the trace of thesecond fundamental tensor A. At p ∈M the vecor H|p is therefore given by

H|p =m∑k=1

A|p(ek , ek),

where (ek)k=1,...,m is an arbitrary orthonormal basis of TpM. Since A is normal thisholds for H as well, i.e. H ∈ Γ (T ⊥M) is a smooth normal vector field.

5

Lecture 2

§6 Structure equations

A connection ∇ on a vector bundle E over a manifold induces a curvature tensor

R(V ,W )Φ = ∇V∇WΦ −∇W∇VΦ −∇[V ,W ]Φ ,

where Φ ∈ Γ (E) and V ,W are smooth vector fields on that manifold. Since R(V ,W )Φis C∞-linear in each argument, i.e.

f R(V ,W )Φ = R(f V ,W )Φ = R(V ,f W )Φ = R(V ,W )(f Φ), for all smooth f ,

one gets that(R(V ,W )Φ

)|pdepends only on v = V (p),w = W (p) and ϕ = Φ(p). This

means that at p the quantity R(v,w)ϕ is well-defined (just extend v,w,ϕ to smoothsections V ,W ,Φ and compute

(R(V ,W )Φ

)|p).

The structure equations of an immersion F : M → N of a smooth manifold M into aRiemannian manifold (N,g) give relations between the curvature tensors RM , RN , R⊥

and the second fundament form A of F. Here, RM is the Riemannian curvature tensorof TM w.r.t. the Levi-Civita connection ∇ and RN is the curvature tensor of TN w.r.t.∇TN . Moreover, R⊥ denotes the curvature tensor of the normal bundle T ⊥M w.r.t.∇⊥.

Gauß equations

RM(v1,w1,v2,w2)−RN(DF|p(v1),DF|p(w1),DF|p(v2),DF|p(w2)

)=n−m∑k=1

(g|F(p)

(Aνk (v1,v2),A

νk (w1,w2))− g|F(p)

(Aνk (v1,w2),A

νk (w1,v2))),

where (νk)k=1,...,n−m is an orthonormal basis of T ⊥p M and v1,v2,w1,w2 ∈ TpM.

7

Lecture 2

Ricci equations

RM(v,w)ν −(RN

(DF|p(v),DF|p(w)

)ν)⊥

=m∑k=1

(Aν(w,ek)A(v,ek)−Aν(v,ek)A(w,ek)

),

where (ek)k=1,...,m is an orthonormal basis of TpM and ν ∈ T ⊥p M, v,w ∈ TpM.

Codazzi equations

(∇⊥uA)(v,w)− (∇⊥v A)(u,w) =(RN

(DF|p(u),DF|p(v)

)(DF|p(w)

))⊥,

where ν ∈ T ⊥p M, u,v,w ∈ TpM.

§7 Local coordinates

For computations one often needs local expressions of tensors. Whenever we uselocal expressions and F : M → N is an immersion we make the following generalassumptions and notations.

i) (U,x,Ω) and (V ,y,Λ) are local coordinate charts around p ∈ U ⊂ M and F(p) ∈V ⊂N such that F|U :U → F(U ) is an embedding and such that F(U ) ⊂ V . Fromthe coordinate functions

(xi)i=1,...,m :U →Ω ⊂Rm , (yα)α=1,...,n : V →Λ ⊂Rn

we obtain a local expression for F,

y F x−1 :Ω→Λ , Fα := yα F x−1, α = 1, . . . ,n.

ii) The Christoffel symbols of the Levi-Civita connections on M resp. N will bedenoted by

Γ ijk , i, j,k = 1, . . . ,m, resp. Γ αβγ , α,β,γ = 1, . . . ,n .

iii) All indices referring toM will be denoted by Latin minuscules and those relatedtoN by Greek minuscules. Moreover, we will always use the Einstein conventionto sum over repeated indices from 1 to the corresponding dimension.

8

§7 Local coordinates

Figure 3.1: Local description of a smooth map F :M→N .

Example 1The local expressions of g,DF and F∗g are

g = gαβdyα ⊗ dyβ ,

DF = Fαi∂∂yα⊗ dxi , Fαi :=

∂Fα

∂xi,

F∗g = gijdxi ⊗ dxj , gij := gαβF

αiFβj .

9

Lecture 2

Example 2The local expression for the 2nd fundamental tensor A is

A = Aijdxi ⊗ dxj = Aαij

∂∂yα⊗ dxi ⊗ dxj ,

where the coefficients Aαij are given by the Gauß formula

(7.1) Aαij =∂2Fα

∂xi∂xj− Γ kij

∂Fα

∂xk+ Γ αβγ

∂Fβ

∂xi∂Fγ

∂xj.

Let (g ij) denote the inverse matrix of (gij) so that g ikgkj = δij gives the Kronecker

symbol. (g ij) defines the metric on T ∗M dual to F∗g. For the mean curvaturevector we get

(7.2) H =Hα ∂∂yα

, Hα := g ijAαij .

§8 First variation of volume

Let us assume that F0 : M → N is a smooth immersion of an oriented manifold Minto a Riemannian manifold (N,g). Then F0 induces a volume form µ0 onM. In localpositively oriented coordinates (xi)i=1,...,m the volume form takes the form

µ0 =√det(F∗0g) dx

1 ∧ · · · ∧ dxm.

If K ⊂M is compact, the volume of K w.r.t. µ0 is

vol0(K) :=∫K

µ0.

10

§8 First variation of volume

A compactly supported variation of F0 is a smooth map F :M × (−ϵ,ϵ)→ N , ϵ > 0,such that

(i) F(p,0) = F0(p) for all p ∈M.

(ii) For each t ∈ (−ϵ,ϵ) the map Ft : M → N , Ft(p) := F(p, t) defines a smoothimmersion into N .

(iii) There exists a compact subset K ⊂M such that for each t0 ∈ (−ϵ,ϵ) we have

supp(ddt |t0Ft

)⊂ K.

For a compactly supported variation let us set

ϕt :=ddtFt.

Then ϕt ∈ Γ (F∗tTN ) has compact support for any t ∈ (−ϵ,ϵ).

We want to compute ddt (F

∗tg). Let V1,V2 ∈ X(M) be time independent smooth vector

field onM. Then(ddt

(F∗tg))(V1,V2) =

ddt

((F∗tg)(V1,V2)

)(8.1)

=ddt

(g(DFt(V1),DFt(V2))

)= g

(∇F∗tTNV1

ϕt,DFt(V2))+ g

(DFt(V1),∇

F∗tTNV2

ϕt)

= g(∇F∗tTNV1

ϕ⊥t ,DFt(V2))+ g

(DFt(V1),∇

F∗tTNV2

ϕ⊥t)

+g(∇F∗tTNV1

ϕ⊤t ,DFt(V2))+ g

(DFt(V1),∇

F∗tTNV2

ϕ⊤t)

= −g(ϕ⊥t ,∇

F∗tTNV1

(DFt(V2)

))− g

(∇F∗tTNV2

(DFt(V1)

),ϕ⊥t

)+g

(∇F∗tTNV1

ϕ⊤t ,DFt(V2))+ g

(DFt(V1),∇

F∗tTNV2

ϕ⊤t)

= −2g(ϕ⊥t ,At(V1,V2)

)+g

(∇F∗tTNV1

ϕ⊤t ,DFt(V2))+ g

(DFt(V1),∇

F∗tTNV2

ϕ⊤t).

Consequently we derive for the variation of the volume form µt w.r.t. F∗tg

ddtµt =

12trace

(ddt

(F∗tg))µt

= −g(ϕ⊥t ,Ht)µt + d(τtyµt),

11

Lecture 2

where τt ∈ X(M) is the tangent vector field with DFt(τt) = ϕ⊤t and Ht denotes themean curvature vector field at time t. Therefore, by Stokes theorem and since ϕt iscompactly supported we obtain

ddt

(volt(K)

)= −

∫K

g(ϕ⊥t ,Ht)µt.

From this it follows

The L2-gradient of the volume functional is given by −H . F :M → N is called aminimal immersion, if H = 0.

§9 Mean curvature flow

DefinitionA smooth family of immersions Ft : M → N , t ∈ [0,T ), 0 < T ≤ ∞, is called asolution of the mean curvature flow, if Ft satisfies the evolution equation

ddtFt =Ht,

where Ht is the mean curvature vector field w.r.t. Ft.

Figure 3.2: Spheres shrink homothetically to points in finite time

ExampleRound spheres Sm(R) ⊂ Rn shrink by a family of round spheres Sm(r(t)) centeredat the same point with r(t) =

√R2 − 2mt. In particular, in this case the flow exists

only on a finite time interval [0,T ) with T = R2/2m.

12

§10 Invariance under ambient isometries

§10 Invariance under ambient isometries

The mean curvature flow is isotropic, i.e. invariant under isometries of the ambientspace. This property follows from the invariance of the first and second fundamentalforms under isometries.

Invariance under isometriesSuppose F :M × [0,T )→ N is a smooth solution of the mean curvature flow andassume thatϕ is an isometry of the ambient space (N,g). Then the family F := ϕFis another smooth solution of the mean curvature flow. In particular, if the initialimmersion is invariant under ϕ, then it will stay invariant for all t ∈ [0,T ).

13

Lecture 3

§11 Invariance under the diffeomorphism group

Writing a solution F :M → N of H = 0 locally as the graph over its tangent plane atF(p), we see that we need as many height functions as there are codimensions, i.e. weneed k = n−m functions. On the other hand the system H = 0 consists of n coupledequations and is therefore overdetermined with a redundancy of m equations. Thesem redundant equations correspond to the diffeomorphism group of the underlyingm-dimensional manifold M. This fact also applies to the mean curvature flow andimplies the following:

Invariance under the diffeomorphism groupIf F :M × [0,T )→ N is a solution of the mean curvature flow, and ψ ∈ Diff(M) afixed diffeomorphism ofM, then F :M × [0,T )→ N , F(p, t) := F(ψ(p), t) is anothersolution. In particular, for each t ∈ [0,T ) the immersed submanifolds Mt := F(M,t)andMt := F(M,t) coincide.

§12 Analytic nature of the mean curvature flow

Using local coordinates, wemay easily get insight into the analytic nature of the meancurvature flow. From Gauß’ equation we see that locally

ddtFα(x, t) = g ij(x, t)

( ∂2Fα∂xi∂xj

(x, t)− Γ kij(x, t)∂Fα

∂xk(x, t) + Γ αβγ

(F(x, t)

)∂Fβ∂xi

(x, t)∂Fγ

∂xj(x, t)

).

Thus the mean curvature flow is a degenerate quasilinear parabolic system of secondorder, where the k = n −m degenerecies stem from the invariance under the diffeo-morphism group. Since H = traceA = trace(∇DF), we may also consider the meancurvature flow as the heat equation

15

Lecture 3

Mean curvature flow = heat equation on the space of immersions

ddtFt =Ht = ∆tFt

on the space of smooth immersion of a givenmanifoldM into a Riemannianmanifold(N,g), where ∆t denotes the Laplace-Beltrami operator w.r.t. F∗tg.

§13 Short-time existence and uniqueness

The following theorem is well-known and in particular forms a special case of a the-orem by Richard Hamilton [Ham82b], based on the Nash-Moser implicit functiontheorem treated in another paper by Hamilton [Ham82a].

Short-time existence and uniquenessLet M be a smooth closed manifold and F0 : M → N a smooth immersion intoa smooth Riemannian manifold (N,g). Then the mean curvature flow admits aunique smooth solution on a some short time interval [0,ϵ), ϵ > 0.

Figure 4.1: Embedded closed curves in R2 shrink to "round" points.

§14 Long-time existence

In general one does not have long-time existence of a solution.

16

§15 Evolution equations

ExampleSuppose F0 :M→ Rn is a smooth immersion of a closed m-dimensional manifoldM. Then the maximal time T of existence of a smooth solution F :M × [0,T )→Rn

of the mean curvature flow with initial immersion F0 is finite.

The next well known theorem holds in any case.

Long-time existence criterionLet M be a closed manifold and F : M × [0,T )→ (N,g) a smooth solution of themean curvature flow in a complete (compact or non-compact) Riemannian mani-fold (N,g). Suppose the maximal time of existence T is finite. Then

limsupt→T

(maxMt

|A|2)=∞ .

§15 Evolution equations

From the main evolution equation

ddtFt =Ht

we obtain the evolution equations of all relevant geometric quantities, e.g. the evolu-tion equation of the induced metric F∗tg is

ddtF∗tg = −2A

Htt = −2g(Ht,At).

This immediately follows from (8.1). Moreover the volume form evolves by

ddtµt = −|Ht |2µt,

so that the volume is always decreasing, if Ft is not a minimal immersion.

17

Lecture 4

§16 Maximum principle

One key technique in mean curvature flow are maximum principles. To demonstratethis we will give an example.

ExampleLet Ft :M → Rn, t ∈ [0,T ), be a mean curvature flow and suppose M is compact.Then the maximal time of existence T is finite.

Proof: We compute the evolution equation of f := ||Ft ||2.

ddtf = 2

⟨Ft,

ddtFt

⟩= 2⟨Ft,Ht⟩ = 2⟨Ft,∆tFt⟩

= ∆t ||Ft ||2 − 2||∇tFt ||2

= ∆tf − 2m,

because

||∇tFt ||2 = gijt

⟨∂Ft∂xi

,∂Ft∂xj

⟩︸ ︷︷ ︸

=(gt)ij

= dimM =m.

Therefore the function p := f +2mt satisfies

ddtp = ∆tp.

The weak parabolic maximum principle now states that for any t0 ∈ (0,T ) we have

max(x,t)∈M×[0,t0]

p(x, t) ≤maxx∈M

p(x,0).

Therefore in particular for all x ∈M

p(x, t0) = ||Ft0(x)||2 +2mt0 ≤max

x∈M||F0(x)||2.

Since this holds for any t0 < T we obtain T <∞.

19

Lecture 4

RemarkIn general one can say the following: If a function f satisfies an evolution equationof the form

ddtf = ∆tf + ⟨∇tf ,Vt⟩+ϕ(f )

for a smooth function ϕ and a smooth vector field Vt, then f behaves in the worstcase as the solution of the ODE

ddtf = ϕ(f ).

§17 Comparison principles

From the maximum principle one can deduce the following comparison principle.

Comparison principleLet M1,M2 be m-dimensional and let N have dimension n = m + 1. If Fi : Mi ×[0,Ti) → N , i = 1,2, are two (immersed) mean curvature flows and F1(M1,0) ∩F2(M2,0) = ∅, then this holds for all t ∈ [0,minT1,T2). i.e.

F1(M1, t)∩F2(M2, t) = ∅,

provided at least one of the manifoldsM1,M2 is compact.

In the same way one can prove that embeddedness is preserved, if the codimensionis again one.

EmbeddednessSuppose F : M × [0,T )→ N is a mean curvature flow of a compact hypersurfaceand suppose F(M,0) is embedded. Then F(M,t) is embedded for all t ∈ [0,T ).

§18 Lagrangian submanifolds

Let (N,g = ⟨·, ·⟩, J) be a Kähler manifold, i.e. J ∈ End(TN ) is a parallel complex struc-ture compatible with g. Then N becomes a symplectic manifold with the symplecticform ω given by the Kähler form ω(V ,W ) = ⟨JV ,W ⟩. An immersion F : M → Nis called Lagrangian, if F∗ω = 0 and n = dimN = 2m = 2dimM. For a Lagrangianimmersion we define a section

ν ∈ Γ (T ⊥M ⊗ T ∗M) , ν := JDF ,

20

§18 Lagrangian submanifolds

where J is applied to the F∗TN -part of DF. ν is a 1-form with values in T ⊥M sinceby the Lagrangian condition J induces a bundle isomorphism (actually even a bundleisometry) between DF(TM) and T ⊥M. In local coordinates ν can be written as

ν = νidxi = ναi

∂∂yα⊗ dxi

with

νi = JFi = JαβF

βi∂∂yα

, ναi = JαβF

βi .

Since J is parallel, we have∇ν = J∇DF = JA.

Second fundamental formWe may define a second fundamental form as a tri-linear form

h(X,Y ,Z) := ⟨ν(X),A(Y ,Z)⟩ .

It turns out that h is fully symmetric.

Mean curvature formTaking a trace, we obtain a 1-form H ∈Ω1(M), called the mean curvature form,

H(X) := traceh(X, ·, ·) .

In local coordinates

h = hijkdxi ⊗ dxj ⊗ dxk , H =Hidx

i , Hi = gklhikl .

The second fundamental tensor A and the mean curvature vector−→H can be written

in the form

Aαij = hk

ij ναk ,

−→H =Hkνk .

Since J gives an isometry between the normal and tangent bundle ofM, the equationsof Gauß and Ricci coincide, so that we get the single equation

Rijkl = RN (Fi ,Fj ,Fk ,Fl) + hikmh

mjl − hilmh

mjk .

21

Lecture 4

Since ∇J = 0 and J2 = − Id we also get

∇iναj = ∇i(JαβF

βj) = J

αβ∇iF

βj = J

αβA

βij = J

αβν

βkh

kij = −h k

ij Fαk .

Similarly as above we conclude

∇ihjkl −∇jhikl = ∇i⟨Ajk ,νl⟩ −∇j⟨Aik ,νl⟩(∇νl∈DF(TM))

= ⟨∇iAjk −∇jAik ,νl⟩= RN (νl ,Fk ,Fi ,Fj) .

Taking a trace over k and l, we deduce

∇iHj −∇jHi = RN (νk ,Fk ,Fi ,Fj)

and if we take into account that N is Kähler and M Lagrangian, then the RHS is aRicci curvature, so that the exterior derivative dH of the mean curvature form H isgiven by

(dH)ij = ∇iHj −∇jHi = −RicN (νi ,Fj).

Lagrangian angleIf (N,g, J) is Kähler-Einstein, then H is closed (since RicN (νi ,Fj) = c ·ω(Fi ,Fj) = 0)and defines a cohomology class on M. In this case any (in general only locallydefined) function α with dα =H is called a Lagrangian angle.

In some sense the Lagrangian condition is an integrability condition. If we repre-sent a Lagrangian submanifold locally as the graph over its tangent space, then them “height" functions are not completely independent but are related to a commonpotential. An easy way to see this, is to consider a locally defined 1-form λ onM (in aneighborhood of some point of F(M)) with dλ = ω. Then by the Lagrangian condition

0 = F∗ω = F∗dλ = dF∗λ.

So F∗λ is closed and by Poincaré’s Lemma locally integrable. By the implicit functiontheorem this potential for λ is related to the height functions ofM (cf. [Smo00]). Notealso that by a result of Weinstein for any Lagrangian embedding M ⊂ N there existsa tubular neighborhood of M which is symplectomorphic to T ∗M with its canonicalsymplectic structure ω = dλ induced by the Liouville form λ.

22

§19 Graphs

§19 Graphs

GraphsLet (M,gM), (K,gK ) be two Riemannian manifolds and f :M → K a smooth map.f induces a graph

Γf := F(M) ⊂M ×K,

whereF :M→N :=M ×K , F(p) := (p,f (p)) .

Since N is also a Riemannian manifold equipped with the product metric g = gM ×gKone may consider the geometry of such graphs. It is clear that the geometry of F mustbe completely determined by f , gM and gK . Local coordinates (xi)i=1,...,m, (zA)A=1,...,kforM resp. K induce local coordinates (yα)α=1,...,n=m+k on N by y = (x,z). Then locally

Fi(x) =∂

∂xi+ f Ai(x)

∂

∂zA,

where similarly as before f A = zA f x−1 and f Ai =∂f A

∂xi.

First fundamental form of graphsFor the induced metric F∗g = gijdxi ⊗ dxj we get

F∗g = gM + f ∗gK .

Since this is obviously positive definite and F is injective, graphs F :M →M ×K ofsmooth mappings f : M → K are always embeddings. From the formula for DF =Fidx

i and the Gauß formula one may then compute the second fundamental tensorA = ∇DF.

Pseudo-Riemannian metricThe tensor s = gM × (−gK ) defines a pseudo-Riemannian metric on the productmanifold. The tensor

F∗s = gM − f ∗gK

turns out to be very important in the analysis of the flow. The eigenvalues µk ofF∗s w.r.t. F∗g are given by

µk =1−λ2k1+λ2k

where λk are the singular values of f .

23

Lecture 4

Length decreasing maps (contractions)A map f :M→ K is called length decreasing, if

f ∗gK ≤ gM .

Equivalently F∗s ≥ 0.

Area decreasingA map f :M→ K is called area decreasing, if

|df (v)∧ df (w)|gK ≤ |v ∧w|gM , ∀v,w ∈ TM.

Equivalently, F∗s is two-positive, i.e. µk +µl ≥ 0 for all k , l.

IsotopiesUnder certain conditions on the curvatures of M, K one can prove that area de-creasing (or length decreasing) maps can be deformed into constant maps throughmean curvature flow isotopies.

§20 Self-similar solutions

Let F :M→Rn be an immersion.

Self-shrinkerF is called a self-shrinker, if

H = −F⊥.

Self-expanderF is called a self-expander, if

H = F⊥.

TranslatorF is called a translator, if

H = V ⊥,

for some fixed non-zero vector V ∈Rn.

24

§20 Self-similar solutions

These submanifolds appear as special solutions of the mean curvature flow and theyserve as models for certain singularities.

ExampleAny minimal immersion F :M→ Sn is a self-shrinker in Rn+1.

25

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31

Index

Area decreasing, 24

Codazzi equations, 8Compactly supported variation, 10

Gauß equations, 7Gauß formula, 9Graph, 23

Heat equation, 15

Invariance under ambient isometries, 13Invariance under diffeomorphisms, 15Isotropic, 13

Lagrangian angle, 22Length decreasing, 24

Mean curvature flow, 12Mean curvature form, 21Mean curvature vector, 5Minimal immersion, 12

Normal bundle of M, 2

Pull-back bundle, 2

Ricci equations, 7

Second fundamental form, 5Second fundamental tensor, 4Self-expander, 24Self-shrinker, 24Structure equations, 7

Tangent bundle of N along M, 2Translator, 24Two-positive, 24

33