Mean Curvature Flow in Higher Codimension

Mean Curvature Flow in Higher Codimension Summer School on Differential Geometry Konkuk University, Seoul August 2013 Knut Smoczyk Leibniz Universität Hannover

Introduction Mean curvature flow is perhaps the most important geometric evolution equation of submanifolds in Riemannian manifolds. Intuitively, a family of smooth submanifolds evolves under mean curvature flow, if the velocity at each point of the submanifold is given by the mean curvature vector at that point. For example, round spheres in euclidean space evolve under mean curvature flow while concentrically shrinking inward until they collapse in finite time to a single point, the common center of the spheres. Mullins [Mul56] proposed mean curvature flow to model the formation of grain boundaries in annealing metals. Later the evolution of submanifolds by their mean curvature has been studied by Brakke [Bra78] from the viewpoint of geometric measure theory. Among the first authors who studied the corresponding nonparametric problem 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 R 2 ) shrinks embedded closed curves to round" points. In his seminal paper Huisken [Hui84] proved that closed convex hypersurfaces in euclidean space R m+1,m > 1 contract to single round points in finite time (later he extended his result to hypersurfaces in Riemannian manifolds that satisfy a suitable stronger convexity, see [Hui86]). Then, until the mid 1990 s, most authors who studied mean curvature flow mainly considered hypersurfaces, both in euclidean and Riemannian manifolds, whereas mean curvature flow in higher codimension did not play a great role. There are various reasons for this, one of them is certainly the much different geometric situation of submanifolds in higher codimension since the normal bundle and the second fundamental tensor are more complicated. But also the analysis becomes more involved and the algebra of the second fundamental tensor is much more subtle since for hypersurfaces there usually exist more scalar quantities related to the second fundamental form than in case of submanifolds in higher codimension. Some of the results previously obtained for mean curvature flow of hypersurfaces carry over without change to submanifolds of higher 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 of space curves by Altschuler and Grayson [Alt91,AG92], measure-theoretic approaches to higher codimension mean curvature flows by Ambrosio & Soner [AS97], existence and 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 curvature flows in higher codimension [CLT02, SW02, Smo04, Wan02, Xin08]. Recently there i

has been done quite some work on the formation and classification of singularities in mean curvature flow [Anc06, CL10, CCH09a, CSS07, CM09, GSSZ07, HL09, HS09, JLT10, LS10a, LS10b, LXYZ11, SW03], partially motivated by Hamilton s and Perelman s [Ham95a, Per02, Per03a, Per03b] work on the Ricci flow that in many ways behaves akin to the mean curvature flow and vice versa. The results in mean curvature flow can be roughly grouped into two categories: The first category contains results that hold (more or less) in general, i.e. that are independent of dimension, codimension or the ambient space. In the second class we find results that are adapted to more specific geometric situations, like results for hypersurfaces, Lagrangian or symplectic submanifolds, graphs, etc.. These notes are based on my lectures on mean curvature flow in higher codimension that I held during the Summer School on Differential Geometry at Konkuk University in Seoul, August 2013. My aim in these lectures was to give an introduction to mean curvature 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 found in the literature, e.g. a well written introduction to the regularity of mean curvature flow of hypersurfaces is [Eck04]. For the curve shortening flow see [CZ01]. For mean curvature flow in higher codimension there exist some lecture notes by Wang [Wan08b] and a detailed introduction to higher codimensional mean curvature flow by 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 dimension m and (N,g) will be a complete (either compact or non-compact) smooth Riemannian manifold of dimension n > m. 1 Immersions and embeddings For 1 k we will consider C k -Immersions of M into N, i.e. maps F : M N of class C k for which the Differential DF p : T p M T F(p) N is injective for any p M. An immersion F : M N will be called an embedding, if F(M) N is an embedded submanifold. a) b) Figure 2.1: a) An immersion of M = S 1 into N = R 2. b) Embedding of S 1 into R 2. 2 Tangent and normal bundles If F : M N is an immersion and p M we define T p M := { w T F(p) N : v T p M with DF p (v) = w } 1

Lecture 1 and T p M := { w T F(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 T F(p) N. The normal bundle of M is the bundle T M over M whose fibers are given by T p M. Analogously we define the bundle T M as the bundle over M with fibers T p M. Both bundles are sub-bundles of the tangent bundle of N along M, i.e. of the bundle F T N over M whose fiber at p M is given by T F(p) N. Sometimes F T N is called the pull-back bundle of T N via F. Since a) b) Figure 2.2: a) T p M can be identified with T p M but is not the same vector space. b) The two projections of w T q N onto w T p M and w T p M with q = F(p). DF p : T p M T p M is an isomorphism for each p M, one usually identifies T M with T M. We obtain two natural projections : (F T N) p T p M w w : (F T N) p T p M w w and for any p F 1 (q) we obtain the orthogonal decomposition T q N = T p M T p M = DF p ( Tp M ) 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 a Riemannian metric F g on M, via (F g) p (v 1,v 2 ) := g F(p) (DF p (v 1 ),DF p (v 2 )), p M and v 1,v 2 T p M. 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 T N on T N. We recall that the Levi-Civita connection on T N is defined by the formula g( T V N X,Y ) = 1 (V ( g(x,y ) ) + X ( g(v,y ) ) Y ( g(v,x) ) 2 ) 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 smooth immersion. On F T N we use the pull-back connection F T N. This can be described as follows. Let W Γ (F T N) be a smooth section in the pull-back bundle, i.e. let W : M F T N be a smooth map with W (p) T F(p) N for all p M. Let p M be arbitrary and set q := F(p). In an open neighborhood V N of q we choose a local trivialization of T N, e.g. (e α ) α=1,...,n, e α X(T V ). If U M is a sufficiently small open set containing p, then W U can be written in the form W (p ) = W α (p )e α (F(p )), p U with smooth functions W α C (U). For v T p M we then set ( F T N v W ) (p) := DW α p (v)e α(f(p)) + W α (p) ( ) T v N e α (F(p)). This is well-defined, i.e. independent of the choice of the frame (e α ) α=1,...,n. Since T M and T M are both sub-bundles of F T N we obtain connections on them by v W := ( F T N v W ), v ν := ( F T N v ν ), where v T p M and W Γ (T M) and ν Γ (T M) are smooth sections. Note that by definition of the Levi-Civita connection w.r.t. the induced metric F g on T M and by definition of the pull-back connection and bundle we have for any X X(M) that DF(X) Γ (T M) and (4.1) DF ( v X ) ( ) = v DF(X). 3

Lecture 1 Since in general it is clear how the connections are induced on all bundles over M that 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 the connection on the normal bundle and on product bundles E T M containig T M as a factor. 5 2nd fundamental form and mean curvature vector If F : M N is an immersion, then the differential DF maps T p M to T M T F(p) N and hence DF Γ (T M T M) can also be considered as a section DF Γ (F T N T M). We extend the projections, to product bundles containing either T M or T M as a factor. In this way we have for any vector field X X(M) and ( T M T M v DF ) (X) = v (DF(X)) DF( v X) (4.1) = 0 (5.1) ( F T N T M v DF ) (X) = F T N v (DF(X)) DF( v X) = ( F T N v (DF(X)) ) ( + F T N v (DF(X)) ) DF( v X) = ( F T N v (DF(X)) ) + v (DF(X)) DF( v X) = ( F T N v (DF(X)) ). 2nd fundamental tensor The second fundamental tensor A of an immersion F : M N is given by A := F T N T M DF Γ (F T N T M T M), A p (v 1,v 2 ) = ( F T N T M v 1 DF ) p (v 2) for all v 1,v 2 T p M. 5.1 Lemma Let A be the second fundamental tensor of an immersion F : M N. a) A is normal, i.e. g F(p) ( A p (v 1,v 2 ),DF p (v 3 ) ) = 0, v 1,v 2,v 3 T p M. Consequently, A can also be considered as a section in T M T M T M. b) A is symmetric, i.e. A p (v 1,v 2 ) = A p (v 2,v 1 ), v 1,v 2 T p M and p M. 4

= ( DF([V 1,V 2 ]) ) = 0 5 2nd fundamental form and mean curvature vector Proof: a) Directly from equation (5.1). b) Let V 1,V 2 X(M) be smooth vector fields. Since T N and are torsion free we get Hence F T N V 1 (DF(V 2 )) F T N V 2 (DF(V 1 )) = DF([V 1,V 2 ]). A(V 1,V 2 ) A(V 2,V 1 ) = ( F T N V 1 (DF(V 2 )) ) ( F T N V 2 (DF(V 1 )) ) 2nd fundamental form If ν T p M is a normal vector, then the second fundamental form A ν w.r.t. ν is the symmetric bilinear form on T p M defined by A ν (v 1,v 2 ) := g F(p) (A p (v 1,v 2 ),ν), for all v 1,v 2 T p M. Mean curvature vector The mean curvature vector field H of an immersion F : M N is the trace of the second fundamental tensor A. At p M the vecor H p is therefore given by H p = m A p (e k,e k ), k=1 where (e k ) k=1,...,m is an arbitrary orthonormal basis of T p M. Since A is normal this holds 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 W Φ W V Φ [V,W ] Φ, where Φ Γ (E) and V,W are smooth vector fields on that manifold. Since R(V,W )Φ is 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 ) depends only on v = V (p),w = W (p) and ϕ = Φ(p). This p means that at p the quantity R(v,w)ϕ is well-defined (just extend v,w,ϕ to smooth sections 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 a Riemannian manifold (N,g) give relations between the curvature tensors R M, R N, R and the second fundament form A of F. Here, R M is the Riemannian curvature tensor of T M w.r.t. the Levi-Civita connection and R N is the curvature tensor of T N w.r.t. T N. Moreover, R denotes the curvature tensor of the normal bundle T M w.r.t.. Gauß equations R M (v 1,w 1,v 2,w 2 ) R ( N DF p (v 1 ),DF p (w 1 ),DF p (v 2 ),DF p (w 2 ) ) n m ( ( = g F(p) A ν k (v 1,v 2 ),A ν k (w 1,w 2 ) ) ( g F(p) A ν k (v 1,w 2 ),A ν k (w 1,v 2 ) )), k=1 where (ν k ) k=1,...,n m is an orthonormal basis of T p M and v 1,v 2,w 1,w 2 T p M. 7

Lecture 2 Ricci equations ( R M (v,w)ν R ( N DF p (v),df p (w) ) ) m ( ) ν = A ν (w,e k )A(v,e k ) A ν (v,e k )A(w,e k ), where (e k ) k=1,...,m is an orthonormal basis of T p M and ν T p M, v,w T p M. k=1 Codazzi equations ( u A)(v,w) ( v A)(u,w) = where ν T p M, u,v,w T p M. ( R N ( DF p (u),df p (v) )( DF p (w) )), 7 Local coordinates For computations one often needs local expressions of tensors. Whenever we use local expressions and F : M N is an immersion we make the following general assumptions 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. From the coordinate functions (x i ) i=1,...,m : U Ω R m, (y α ) α=1,...,n : V Λ R n we obtain a local expression for F, y F x 1 : Ω Λ, F α := y α F x 1, α = 1,...,n. ii) The Christoffel symbols of the Levi-Civita connections on M resp. N will be denoted by Γjk i, i,j,k = 1,...,m, resp. Γ βγ α, α,β,γ = 1,...,n. iii) All indices referring to M will be denoted by Latin minuscules and those related to N by Greek minuscules. Moreover, we will always use the Einstein convention to 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 1 The local expressions of g,df and F g are g = g αβ dy α dy β, DF = F α i y α dxi, F α i := Fα x i, F g = g ij dx i dx j, g ij := g αβ F α i Fβ j. 9

Lecture 2 Example 2 The local expression for the 2nd fundamental tensor A is A = A ij dx i dx j = A α ij y α dxi dx j, where the coefficients A α ij are given by the Gauß formula (7.1) A α ij = 2 F α x i x j Γ ij k F α x k + Γ βγ α F β F γ x i x j. Let (g ij ) denote the inverse matrix of (g ij ) so that g ik g kj = δ i j gives the Kronecker symbol. (g ij ) defines the metric on T M dual to F g. For the mean curvature vector we get (7.2) H = H α y α, Hα := g ij A α ij. 8 First variation of volume Let us assume that F 0 : M N is a smooth immersion of an oriented manifold M into a Riemannian manifold (N,g). Then F 0 induces a volume form µ 0 on M. In local positively oriented coordinates (x i ) i=1,...,m the volume form takes the form µ 0 = det(f 0 g) dx1 dx m. If K M is compact, the volume of K w.r.t. µ 0 is vol 0 (K) := µ 0. K 10

8 First variation of volume A compactly supported variation of F 0 is a smooth map F : M ( ϵ,ϵ) N, ϵ > 0, such that (i) F(p,0) = F 0 (p) for all p M. (ii) For each t ( ϵ,ϵ) the map F t : M N, F t (p) := F(p,t) defines a smooth immersion into N. (iii) There exists a compact subset K M such that for each t 0 ( ϵ,ϵ) we have ( ) d supp F dt t K. t 0 For a compactly supported variation let us set ϕ t := d dt F t. Then ϕ t Γ (Ft T N) has compact support for any t ( ϵ,ϵ). We want to compute d dt (F t g). Let V 1,V 2 X(M) be time independent smooth vector field on M. Then ( ) d dt (F t g) (V 1,V 2 ) = d ( ) (8.1) (Ft g)(v dt 1,V 2 ) = d ( ) g(df dt t (V 1 ),DF t (V 2 )) = g ( F t T N V 1 ϕ t,df t (V 2 ) ) + g ( DF t (V 1 ), F t T N ) V 2 ϕ t = g ( F t T N V 1 +g ( F t T N V 1 = g ( ϕt, F t T N V 1 +g ( F t T N V 1 = 2g ( ϕt,a t(v 1,V 2 ) ) +g ( F t T N V 1 ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ϕt ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ) ϕ t ) ( DFt (V 1 ) ),ϕt ) ( DFt (V 2 ) )) g ( F t T N V 2 ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N V 2 ϕt ϕt,df t(v 2 ) ) + g ( DF t (V 1 ), F t T N ) V 2 ϕt. Consequently we derive for the variation of the volume form µ t w.r.t. Ft g d dt µ t = 1 ( ) d 2 trace dt (F t g) µ t = g(ϕ t,h t)µ t + d(τ t µ t ), ) 11

Lecture 2 where τ t X(M) is the tangent vector field with DF t (τ t ) = ϕt and H t denotes the mean curvature vector field at time t. Therefore, by Stokes theorem and since ϕ t is compactly supported we obtain d ( volt (K) ) = g(ϕt dt,h t)µ t. From this it follows The L 2 -gradient of the volume functional is given by H. F : M N is called a minimal immersion, if H = 0. 9 Mean curvature flow Definition A smooth family of immersions F t : M N, t [0,T ), 0 < T, is called a solution of the mean curvature flow, if F t satisfies the evolution equation K d dt F t = H t, where H t is the mean curvature vector field w.r.t. F t. Figure 3.2: Spheres shrink homothetically to points in finite time Example Round spheres S m (R) R n shrink by a family of round spheres S m (r(t)) centered at the same point with r(t) = R 2 2mt. In particular, in this case the flow exists only on a finite time interval [0,T ) with T = R 2 /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 ambient space. This property follows from the invariance of the first and second fundamental forms under isometries. Invariance under isometries Suppose F : M [0,T ) N is a smooth solution of the mean curvature flow and assume that ϕ is an isometry of the ambient space (N,g). Then the family F := ϕ F is another smooth solution of the mean curvature flow. In particular, if the initial immersion 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 at F(p), we see that we need as many height functions as there are codimensions, i.e. we need k = n m functions. On the other hand the system H = 0 consists of n coupled equations and is therefore overdetermined with a redundancy of m equations. These m redundant equations correspond to the diffeomorphism group of the underlying m-dimensional manifold M. This fact also applies to the mean curvature flow and implies the following: Invariance under the diffeomorphism group If F : M [0,T ) N is a solution of the mean curvature flow, and ψ Diff(M) a fixed diffeomorphism of M, then F : M [0,T ) N, F(p,t) := F(ψ(p),t) is another solution. In particular, for each t [0,T ) the immersed submanifolds M t := F(M,t) and M t := F(M,t) coincide. 12 Analytic nature of the mean curvature flow Using local coordinates, we may easily get insight into the analytic nature of the mean curvature flow. From Gauß equation we see that locally ( d dt Fα (x,t) = g ij 2 F α (x,t) x i x j (x,t) Γ ij k (x,t) Fα x k (x,t) + Γ ( ) F βγ α β ) F(x,t) x i (x,t) Fγ x j (x,t). Thus the mean curvature flow is a degenerate quasilinear parabolic system of second order, where the k = n m degenerecies stem from the invariance under the diffeomorphism group. Since H = tracea = trace( DF), we may also consider the mean curvature flow as the heat equation 15

Lecture 3 Mean curvature flow = heat equation on the space of immersions d dt F t = H t = t F t on the space of smooth immersion of a given manifold M into a Riemannian manifold (N,g), where t denotes the Laplace-Beltrami operator w.r.t. F t g. 13 Short-time existence and uniqueness The following theorem is well-known and in particular forms a special case of a theorem by Richard Hamilton [Ham82b], based on the Nash-Moser implicit function theorem treated in another paper by Hamilton [Ham82a]. Short-time existence and uniqueness Let M be a smooth closed manifold and F 0 : M N a smooth immersion into a smooth Riemannian manifold (N, g). Then the mean curvature flow admits a unique smooth solution on a some short time interval [0,ϵ), ϵ > 0. Figure 4.1: Embedded closed curves in R 2 shrink to "round" points. 14 Long-time existence In general one does not have long-time existence of a solution. 16

15 Evolution equations Example Suppose F 0 : M R n is a smooth immersion of a closed m-dimensional manifold M. Then the maximal time T of existence of a smooth solution F : M [0,T ) R n of the mean curvature flow with initial immersion F 0 is finite. The next well known theorem holds in any case. Long-time existence criterion Let M be a closed manifold and F : M [0,T ) (N,g) a smooth solution of the mean curvature flow in a complete (compact or non-compact) Riemannian manifold (N,g). Suppose the maximal time of existence T is finite. Then lim sup t T ( max A 2) =. M t 15 Evolution equations From the main evolution equation d dt F t = H t we obtain the evolution equations of all relevant geometric quantities, e.g. the evolution equation of the induced metric F t g is d dt F t g = 2A H t t = 2g(H t,a t ). This immediately follows from (8.1). Moreover the volume form evolves by d dt µ t = H t 2 µ t, so that the volume is always decreasing, if F t is not a minimal immersion. 17

Lecture 4 16 Maximum principle One key technique in mean curvature flow are maximum principles. To demonstrate this we will give an example. Example Let F t : M R n, 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 := F t 2. d dt f = 2 F t, d dt F t = 2 F t,h t = 2 F t, t F t = t F t 2 2 t F t 2 = t f 2m, because t F t 2 = g ij Ft t x i, F t x j } {{ } = dimm = m. =(g t ) ij Therefore the function p := f + 2mt satisfies d dt p = tp. The weak parabolic maximum principle now states that for any t 0 (0,T ) we have Therefore in particular for all x M max p(x,t) max p(x,0). (x,t) M [0,t 0 ] x M p(x,t 0 ) = F t0 (x) 2 + 2mt 0 max x M F 0(x) 2. Since this holds for any t 0 < T we obtain T <. 19

Lecture 4 Remark In general one can say the following: If a function f satisfies an evolution equation of the form d dt f = tf + t f,v t + ϕ(f ) for a smooth function ϕ and a smooth vector field V t, then f behaves in the worst case as the solution of the ODE d f = ϕ(f ). dt 17 Comparison principles From the maximum principle one can deduce the following comparison principle. Comparison principle Let M 1,M 2 be m-dimensional and let N have dimension n = m + 1. If F i : M i [0,T i ) N, i = 1,2, are two (immersed) mean curvature flows and F 1 (M 1,0) F 2 (M 2,0) =, then this holds for all t [0,min{T 1,T 2 }). i.e. F 1 (M 1,t) F 2 (M 2,t) =, provided at least one of the manifolds M 1,M 2 is compact. In the same way one can prove that embeddedness is preserved, if the codimension is again one. Embeddedness Suppose F : M [0,T ) N is a mean curvature flow of a compact hypersurface and 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(T N) is a parallel complex structure compatible with g. Then N becomes a symplectic manifold with the symplectic form ω given by the Kähler form ω(v,w ) = JV,W. An immersion F : M N is called Lagrangian, if F ω = 0 and n = dimn = 2m = 2dimM. For a Lagrangian immersion we define a section ν Γ (T M T M), ν := JDF, 20

18 Lagrangian submanifolds where J is applied to the F T N-part of DF. ν is a 1-form with values in T M since by the Lagrangian condition J induces a bundle isomorphism (actually even a bundle isometry) between DF(T M) and T M. In local coordinates ν can be written as with Since J is parallel, we have ν = ν i dx i = ν α i y α dxi ν i = JF i = J α β Fβ i y α, να i = Jα β Fβ i. ν = J DF = JA. Second fundamental form We may define a second fundamental form as a tri-linear form It turns out that h is fully symmetric. h(x,y,z) := ν(x),a(y,z). Mean curvature form Taking a trace, we obtain a 1-form H Ω 1 (M), called the mean curvature form, In local coordinates H(X) := traceh(x,, ). h = h ijk dx i dx j dx k, H = H i dx i, H i = g kl h ikl. The second fundamental tensor A and the mean curvature vector H can be written in the form A α ij = h k ij να k, H = H k ν k. Since J gives an isometry between the normal and tangent bundle of M, the equations of Gauß and Ricci coincide, so that we get the single equation R ijkl = R N (F i,f j,f k,f l ) + h ikm h m jl h ilm h m jk. 21

Lecture 4 Since J = 0 and J 2 = Id we also get i ν α j = i(j α β Fβ j ) = Jα β if β j = Jα β Aβ ij = Jα β νβ k h k ij Similarly as above we conclude i h jkl j h ikl = i A jk,ν l j A ik,ν l Taking a trace over k and l, we deduce ( ν l DF(T M)) = i A jk j A ik,ν l = R N (ν l,f k,f i,f j ). i H j j H i = R N (ν k,f k,f i,f j ) = h k ij Fα k. and if we take into account that N is Kähler and M Lagrangian, then the RHS is a Ricci curvature, so that the exterior derivative dh of the mean curvature form H is given by (dh) ij = i H j j H i = Ric N (ν i,f j ). Lagrangian angle If (N,g,J) is Kähler-Einstein, then H is closed (since Ric N (ν i,f j ) = c ω(f i,f j ) = 0) and defines a cohomology class on M. In this case any (in general only locally defined) function α with dα = H is called a Lagrangian angle. In some sense the Lagrangian condition is an integrability condition. If we represent a Lagrangian submanifold locally as the graph over its tangent space, then the m height" functions are not completely independent but are related to a common potential. An easy way to see this, is to consider a locally defined 1-form λ on M (in a neighborhood 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 function theorem this potential for λ is related to the height functions of M (cf. [Smo00]). Note also that by a result of Weinstein for any Lagrangian embedding M N there exists a tubular neighborhood of M which is symplectomorphic to T M with its canonical symplectic structure ω = dλ induced by the Liouville form λ. 22

19 Graphs 19 Graphs Graphs Let (M,g M ), (K,g K ) be two Riemannian manifolds and f : M K a smooth map. f induces a graph Γ f := F(M) M K, where F : M N := M K, F(p) := (p,f (p)). Since N is also a Riemannian manifold equipped with the product metric g = g M g K one may consider the geometry of such graphs. It is clear that the geometry of F must be completely determined by f, g M and g K. Local coordinates (x i ) i=1,...,m, (z A ) A=1,...,k for M resp. K induce local coordinates (y α ) α=1,...,n=m+k on N by y = (x,z). Then locally F i (x) = x i + f A i (x) z A, where similarly as before f A = z A f x 1 and f A i = f A x i. First fundamental form of graphs For the induced metric F g = g ij dx i dx j we get F g = g M + f g K. Since this is obviously positive definite and F is injective, graphs F : M M K of smooth mappings f : M K are always embeddings. From the formula for DF = F i dx i and the Gauß formula one may then compute the second fundamental tensor A = DF. Pseudo-Riemannian metric The tensor s = g M ( g K ) defines a pseudo-riemannian metric on the product manifold. The tensor F s = g M f g K turns out to be very important in the analysis of the flow. The eigenvalues µ k of F s w.r.t. F g are given by where λ k are the singular values of f. µ k = 1 λ2 k 1 + λ 2 k 23

Lecture 4 Length decreasing maps (contractions) A map f : M K is called length decreasing, if Equivalently F s 0. f g K g M. Area decreasing A map f : M K is called area decreasing, if df (v) df (w) g K v w g M, v,w T M. Equivalently, F s is two-positive, i.e. µ k + µ l 0 for all k l. Isotopies Under certain conditions on the curvatures of M, K one can prove that area decreasing (or length decreasing) maps can be deformed into constant maps through mean curvature flow isotopies. 20 Self-similar solutions Let F : M R n be an immersion. Self-shrinker F is called a self-shrinker, if H = F. Self-expander F is called a self-expander, if H = F. Translator F is called a translator, if for some fixed non-zero vector V R n. H = V, 24

20 Self-similar solutions These submanifolds appear as special solutions of the mean curvature flow and they serve as models for certain singularities. Example Any minimal immersion F : M S n is a self-shrinker in R n+1. 25

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