SELF-SHRINKERS OF THE MEAN CURVATURE FLOW IN ARBITRARY CODIMENSION

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1 SELF-SHRINKERS OF THE MEAN CURVATURE FLOW IN ARBITRARY CODIMENSION KNUT SMOCZYK Abstract. In this paper we study self-similar solutions M m R n of the mean curvature flow in arbitrary codimension. Self-similar curves Γ R have been completely classified by Abresch & Langer [AL86] and this result can be applied to curves Γ R n equally well. A submanifold M m R n is called spherical, if it is contained in a sphere. Obviously, spherical self-shrinkers of the mean curvature flow coincide with minimal submanifolds of the sphere. For hypersurfaces M m R m+1, m, Huisken [Hui90] showed that compact self-shrinkers with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution M m R n, m, is spherical, if and only if the mean curvature vector H is non-vanishing and the principal normal ν is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type. 1. Introduction In this article we study an immersion F : M R n of a smooth manifold M of dimension m and codimension p = n m into euclidean space that satisfies the quasilinear elliptic system H = F, (1) where H denotes the mean curvature vector of the immersion and is the projection onto the normal bundle of M. Solutions of (1) give rise to homothetically shrinking solutions M t := F (M, t) of the mean curvature flow F : M [0, T ) R n, d F (x, t) dt = H(x, t). () Therefore they are often called self-shrinkers. While until recently people were mostly interested in mean curvature flow of hypersurfaces, this non-linear evolution equation is getting more and more attention in higher codimension, see for instance [AS96], [Anc00], [BN99], [CLT0], [TW04], [SW0], [TY0]. Date: July 16, Mathematics Subject Classification. Primary 53C44; Partially supported by the DFG priority program SPP 1154, SM/

2 KNUT SMOCZYK Self-similar solutions play an important role in the formation of type-1 singularities of the mean curvature flow. They are characterized by their blow-up behaviour sup M t A c T t, where A is the second fundamental form, c a constant and T denotes the blowup time. It has been shown by Huisken [Hui90] that solutions of () forming a type-1 singularity can be homothetically rescaled so that any resulting limiting submanifold satisfies (1). Consequently, the classification of type-1 blow-ups is equivalent to the classification of self-shrinkers. If M = Γ R is a curve, then all solutions of (1) have been classified by Abresch and Langer [AL86]. Except for the straight lines passing through the origin, the curvature k is positive for all them. In higher codimension the theorem of Abresch and Langer applies as well. To be precise, any self-shrinking curve γ R n lies in a flat linear two-space E R n and coincides with one of the Abresch-Langer curves Γ in E, because then equation (1) becomes an ODE of order two. In [Hui90] it was proved that a compact hypersurface M m R m+1, m satisfying (1) with positive scalar mean curvature H is S m ( m), i.e. a sphere of radius m. Later this was extended in [Hui93] to noncompact hypersurfaces with H > 0 that appear as type-1 singularities of compact hypersurfaces. There it was shown that the blow-up limit belongs to one of the classes i) M = Γ R m 1 ii) M = S m+1 q( m + 1 q ) R q 1, q {1,..., m}, where Γ are the curves found by Abresch and Langer. If one drops the condition H > 0, then there are new examples, e.g. those found by Angenent [Ang9]. In higher codimension the situation becomes more complicated as the codimension increases. In particular, even under the condition > 0 we get new examples. Indeed, if Γ 1,..., Γ m are Abresch-Langer curves, then L := Γ 1 Γ m R m is a self-shrinker with > 0. Moreover, this is a Lagrangian submanifold in C m and since the complex structure J on T C m induces an isometry between the tangent and the normal bundle of L, these examples have a flat normal bundle. Except for the case Γ 1,..., Γ m = S 1 they are not contained in a sphere. Let us also mention that Anciaux [Anc00] classified equivariant Lagrangian self-shrinkers in C m and that all of them satisfy > 0 [GSSZ05]. On the other hand, one easily observes that spherical self-shrinkers coincide with minimal submanifolds of the sphere and that for spherical solutions the principal normal ν := H/ is parallel in the normal bundle. The purpose of this article is to show that the converse holds true as well. We state our main theorem.

3 SELF-SHRINKERS IN ARBITRARY CODIMENSION 3 Theorem 1.1. Let M m R n, m be a compact self-shrinker. Then M is spherical, if and only if H 0 and ν = 0. Remarks. It is trivial that the principal normal of a hypersurface is parallel. So this theorem forms a natural extension of Huisken s result mentioned above. In codimension two we have a torsion 1-form τ defined by τ(x) := Xν, b, where b is the binormal vector b = Jν (J denoting the complex structure on the normal bundle NM). In this case ν = 0 τ = 0. We also point out that in codimension two a parallel principal normal implies the flatness of the normal bundle, since the curvature R of the normal bundle is given by R = dτ. In higher codimensions ν = 0 is weaker than R = 0. We do not believe the condition ν = 0 is preserved under mean curvature flow. The torsion 1-form is a generalization of the usual torsion τ of a biregular space curve γ R 3 and it has been shown by Altschuler [Alt91] that space curves tend to develop singularities with no torsion. Therefore, in higher codimension there is some hope that singularities will have parallel principle normal, at least under additional geometric conditions to be discovered. The idea in Huisken s original paper [Hui90] was to show that the scaling invariant quantity A / is constant on a self-shrinking hypersurface. This will not work in our context and in fact the implications would be too strong. Instead, we look at the quantity P / 4, where P = H, A is the second fundamental form w.r.t. the mean curvature vector itself. In codimension one this coincides with A /. One easily constructs many non-trivial examples of noncompact solutions of (1). In fact, any minimal cone Λ R k is a solution and if M 1 R k, M R l are solutions of (1), then so is M 1 M R k+l. In particular, a product of a minimal submanifold M S n 1 with a minimal cone Λ R k gives a noncompact solution of (1) in R n+k with parallel principal normal ν. Since minimal cones in higher codimension need not be analytic (see e.g. [HL8]), we find non-analytic examples of noncompact self-shrinkers with parallel principal normal. Therefore, the general picture in the noncompact case becomes more subtle. On the other side, minimal cones never form as type-1 singularities of compact submanifolds, since the blow-up must have uniformly bounded curvature. In the complete case we give the following classification. Theorem 1.. Let M m R n be a complete connected self-shrinker with H 0 and parallel principal normal. Suppose further that M has uniformly bounded geometry, i.e. there exist constants c k such that ( ) k A c k holds uniformly on M for any k 0. Then M must belong to one of the following classes. (i) M m = Γ R m 1, (ii) M m = M r R m r.

4 4 KNUT SMOCZYK Here, Γ is one of the curves found by Abresch and Langer and M r is a complete minimal submanifold of the sphere S n m+r 1 ( r) R n m+r, where 0 < r = rank(a ν ) m denotes the rank of the principal second fundamental form A ν = ν, A. Remarks. In codimension one, any self-shrinker is analytic and this was exploited in [Hui93] to classify the noncompact solutions with H 0. In higher codimension, as pointed out above, this might not be the case. It turns out, analyticity is not needed in the proof of the splitting theorem 1.. On the other hand, theorem 1. implies that in codimension two any solution of (1) with H 0 and parallel principle normal is analytic. The uniformly bounded geometry is essential only in the sense that we need to integrate by parts w.r.t. the Gauß kernel and this condition will prevent us from getting boundary terms at infinity. It may actually be replaced by a weaker condition which allows even a certain growth rate at infinity for ( ) k A and is only needed for k. In addition, completeness and bounded geometry (here, bounded second fundamental form in M B(0, R) suffices for some large ball of radius R centered at the origin) will exclude products of spherical self-shrinkers with minimal cones. Any blow-up (possibly noncompact) of a type-1 singularity forming on a compact submanifold will automatically be complete with uniformly bounded geometry, in particular theorem 1. may be applied to those blow-up limits. The organization of the paper is as follows. After recalling the basic geometric structure equations for submanifolds in euclidean space in section two we exploit (1) to derive various elliptic equations for curvature quantities of self-similar solutions, in particular for those with parallel principal normal. This will be done in section three. In section four we give a proof of theorem 1.1, whereas in section five we proceed to classify noncompact solutions as stated in theorem 1.. This work has been initiated while I was supported by DFG as a Heisenberg fellow at the Albert Einstein Institute in Golm and the Max Planck Institute in Leipzig and I am indebted to my hosts, Gerhard Huisken and Jürgen Jost. Special thanks go to Guofang Wang for discussions on this subject.. Geometry in higher codimension Let F : M m R n be a smooth immersion of a submanifold of codimension p = n m. We let (x i ) i=1,...,m denote local coordinates on M and we will always use cartesian coordinates (y α ) α=1,...,n on R n. Doubled greek and latin indices are summed form 1 to m resp. from 1 to n. In local coordinates the differential df of F is given by df = F α i y α dxi,

5 where F α = y α (F ) and Fi α dx j are SELF-SHRINKERS IN ARBITRARY CODIMENSION 5 = F α x i. The coefficients of the induced metric g ij dx i g ij = F i, F j = g αβ F α i F β j, where g αβ = δ αβ is the euclidean metric in cartesian coordinates. As usual, the Christoffel symbols are Γ k ij = 1 ( glj gkl x i + g li x j g ) ij x l. The second fundmental form A Γ(F 1 T R n T M T M) is defined by A := df =: A α ij y α dxi dx j. Here and in the following all canonically induced full connections on bundles over M will be denoted by. We will also use the connection on the normal bundle which will be denoted by. It is easy to check that in cartesian coordinates on R n we have A α ij = F α ij Γ k ijf α k, where F α ij = F α x i x j. By definition, A is a section in the bundle F 1 T R n T M T M and it is well known that A is normal, i.e. A Γ (NM T M T M), where NM denotes the normal bundle of M. This means that The mean curvature vector field H = H α F k, A ij := g αβ Fk α A β ij = 0, i, j, k. (3) y α is defined by H = g ij A ij = g ij A α ij y α. It will be convenient to rise and lower indices using the metric tensors g ij, g ij, for instance A i j = g ki A kj. The Riemannian curvature tensor on the tangent bundle will be denoted by R ijkl, whereas the curvature tensor of the normal bundle, considered as a -form with values in NM NM, has components R ij. We summarize the equations of Gauß, Codazzi and Ricci in the following proposition. Proposition.1. For an immersion F : M m R n holds true R ijkl = A ik, A jl A il, A jk, i A jk = j A ik, R ij = A ik A k j. (Gauß) (Codazzi) (Ricci)

6 6 KNUT SMOCZYK We define P ij := H, A ij, Q ij := A k i, A kj, S ijkl := A ij, A kl. Then by Gauß equation the Ricci curvature is given by R ij = g kl R ikjl = P ij Q ij. Moreover, we have Simons identity Proposition.. k l H = A kl + R kilj A ij RkA i il + Q i la ik S kilj A ij. We use the Simons identity to derive an expression for A, ( ) H. In a first step we compute A, ( ) H = A A + R kilj S ijkl R ij Q ij For the normal curvature tensor R ij = A ik A k j so that by Gauß equation (4) becomes Proposition.3. + Q S ikjl S ijkl. (4) one has R = A αi l A βm i A βi l A αm i = Q S ikjl S ijkl, A, ( ) H = A A + S P, Q + R. If the immersion satisfies > 0, we can define the principal normal ν := H. The principal normal is parallel in the normal bundle, iff This is equivalent to i ν = 0. i H = i ν. 3. Self-similar solutions in arbitrary codimension Suppose F : M m R n is a self-shrinker, i.e. H = F. We define θ := 1 d F and compute i θ j = g ij + F, A ij = g ij P ij. (5)

7 SELF-SHRINKERS IN ARBITRARY CODIMENSION 7 Moreover so that i F = ( i (F θ k F k ) ) = ( F i i θ k F k θ k A ik ), and i F = θ k A ik Taking another covariant derivative we derive i j F = ( i θ k ) A jk i H = θ k A ik. (6) = ( i θ k A jk + θ k i A jk ) = i θ k A jk θ k i A jk = A ij F, A k i A kj θ k k A ij, where we have used (5) and the Codazzi equation in the last step. In particular, since H = F, we conclude and From this we derive i j H = A ij P k i A kj + θ k k A ij (7) H θ k k H + P ij A ij H = 0. (8) H F, + P = 0. (9) For a self-similar solution we may exploit the Simons identity in proposition. and (7) to deduce and A ij θ k k A ij + Q k i A kj + Q k j A ki + (R ikjl S ikjl )A kl A ij = 0 (10) A A F, A + S + R A = 0. (11) We will need the following lemmata. Lemma 3.1. Let M R n be a self-shrinker with > 0 and parallel principal normal, i.e. ν = 0. Then we have and P ij A ij = S ijkl P ij P kl = P ν, P 4 P k i A kj = P k j A ki, S ikjl P ij P kl = P k i P kj Q ij.

8 8 KNUT SMOCZYK Proof. H = ν, k H = kν and (8) imply that P ij A ij is a multiple of ν. But then P ij A ij = P ij P ν, A ij ν = ν and S ijkl P ij P kl = A ij P ij, A kl P kl P 4 =. This proves the first two equations. To prove the third we apply Ricci s equation to H 0 = i j H j i H = H, A k j A ki H, A k i A kj = P k j A ki P k i A kj. It then follows S ikjl P ij P kl = A ik P ij, A jl P kl = A j i P i k, A jl P kl = Q il P i kp kl which is the last equation in the statement of the lemma. Note, that the last two equations in lemma 3.1 hold for any submanifold with parallel principal normal since we did not use that M is a self-shrinker there. Lemma 3.. Let M R n have parallel principal normal, then 4 4 H, A ij P ij = ( ) P 4 P, Proof. From ν = 0 we deduce H, A ij P ij = k ν, k A ij P ij and then the desired equation follows easily. = k ( k ν, Aij ) P ij ( ) = k Pij k P ij = 1, P P A straightforward computation shows 4 i P jk i = ( P and ( Pjk ) ) P 6 ( ) P P P = 4 6 6,, ( ) P 4 ( ) P 4

9 SELF-SHRINKERS IN ARBITRARY CODIMENSION 9 so that 4 i P jk i ( ) Pjk P P = , ( ) P 4 (1) Lemma 3.3. Let M R n be a self-shrinker with > 0 and parallel principal normal ν. Then the following equation holds ( ) P 4 = 4 i P ( ) jk Pjk i ( ) P + F, 4 ( ) P, 4. Proof. First we use (8) and (10) to compute and then also Since P ij = F T, P ij + H, A ij Q k i P kj Q k j P ki + (S ikjl S ijkl )P kl + P ij P = P + F, P + 4 H, A ij P ij ( ) P 4 this and (9) implies ( ) P 4 = = 4P k i P kj Q ij + 4(S ikjl S ijkl )P kl P ij + 4 P. P 4 P 6 8 ( ) P, 4 P 8 6 F, ( P 4 ) P + P 4 6 H 8 P H, A ij P ij ( ) P, 4 (P ki P kj Q ij + (S ijkl S ikjl )P kl P ij ) P 4. We apply lemma 3.1 to the last term and obtain ( ) ( ) P P 4 = F P, H, A ij P ij P ( ) P, 4,

10 10 KNUT SMOCZYK where we have used H = ν + ν = ν to replace H by. In a next step we apply formula (1) and lemma 3. to continue ( ) P 4 = 4 i P ( ) jk Pjk i ( ) P + F, 4 ( ) P, Compact self-shrinkers In this section we shall assume that M is a compact self-shrinker. ready to prove theorem 1.1. We are now Proof of theorem 1.1. As pointed out in the introduction, it is obvious that any spherical self-shrinker is a minimal submanifold of the sphere, that H 0 and that the principle normal is parallel. It remains to prove the converse holds true as well. The strong elliptic maximum principle and lemma 3.3 imply P 4 = c for some constant c > 0 and i P jk i ( ) Pjk = 0. The Codazzi equation and ν = 0 show that ( ) ( ) Pjk i = i ν, A jk = ν, i A jk = ν, Pik j A ik = j. Decomposing i P ( ) jk Pjk i into i P ( ) jk Pjk i = 1 ( i P jk j P ) ik + 1 ( i P jk + j P ) ( ) ik Pjk i we find Consequently i P jk j P ik = 0. P P i kp kj i j = 0. If = 0 at some point p M, then at this point there is only one nonzero eigenvalue of the symmetric tensor P ij and the corresponding eigenvector is. At this point we have P = (trace(p )) = 4 which implies that the constant c from above is 1. Since ν is parallel, equation (9) and P = 4 imply F, + 3 = 0 (13)

11 SELF-SHRINKERS IN ARBITRARY CODIMENSION 11 and as in [Hui90] partial integration gives (m 1) dµ = 0 M which is impossible for m. Hence H = ν = 0 everywhere on M. From the equation F = H one deduces F = m + F, H = (m ), where we used (1) in the last step. Since implies is constant, the maximum principle F = 0 = m. (14) So F is constant, θ = 0 and H = F. This proves M S n 1 ( m). 5. The complete case In the complete case we cannot use the maximum principle equally well as in the last section. Instead we exploit integration by parts w.r.t. the Gauß kernel ρ(y) := e y / on R n. Lemma 5.1. Let M R n be a complete self-shrinker with > 0, parallel principle normal ν and with bounded geometry, i.e. ( ) k A c k for suitable constants c k and all k 0. Then the following integral expression holds ( ) P 4 ρdµ M + M P 6 i P jk i ( ) Pjk ρdµ = 0. (15) Proof. The bounded geometry of M guarantees that partial integration w.r.t. the Gauß kernel does not yield any boundary terms at infinity. Since i ρ = θ i ρ, Lemma 3.3 and partial integration of ( P / ) ( P / 4 )ρdµ gives (15). Corollary 5.. On any self-shrinker as in Lemma 5.1 we must have i P ( ) jk Pjk i = 0, P 4 = const. The next lemma on Abresch-Langer curves will become important in the proof of theorem 1. (see part (ii) below).

12 1 KNUT SMOCZYK Lemma 5.3. Suppose Γ R is one of the self-shrinking curves found by Abresch and Langer. There exists a constant c Γ > 0 such that F k e = c Γ (16) holds true on all of Γ. Moreover, the critical values k 0 of the curvature function k satisfy k 0 e k 0 / = c Γ. (17) If Γ 1, Γ are two such curves with c Γ1 = c Γ, then up to an euclidean motion Γ 1 = Γ. In particular, if Γ 1, Γ are two different self-shrinking curves and k 1, k are critical values of the curvature of Γ 1 resp. of Γ, then k 1 k. Proof. Let ν be the inner unit normal. Then (1) becomes k = F, ν and taking the gradient on both sides gives i k = kθ i. F This implies (k e ) = 0. It is clear that c Γ is positive and hence k > 0. Then at a critical point k 0 we have i k = θ i = 0 and consequently k 0 = F, ν = F. Substituting F in (16) gives (17). The remainder of the lemma follows from the fact that equation (16) is a second order ODE. Proof of theorem 1.. Proceeding as in the proof of theorem 1.1 we first conclude that either H = 0 everywhere, or P admits only one non-zero eigenvalue which is. Let us treat both cases separately. (i) H = 0 : There exist several classification results for submanifolds in euclidean space with parallel mean curvature. In particular, under surprisingly similar assumptions on the relation between the mean curvature vector H and the position vector F, Yano [Yan71] classified compact submanifolds with parallel mean curvature. It seems, an analogous classification to that of Yano for complete submanifolds with parallel mean curvature does not exist. However, in our special situation this can be established as follows. First observe, that in case H = 0, equation (6) implies θ i A ij = 0. (18) Moreover, from H = 0 we deduce P ij = 0 and then with equation (7) P j i = P k i P j k. (19) Hence, if λ is an eigenvalue of P j i, then either λ = 0 or λ = 1. Let us define (P A) ij := P k i A kj. By Lemma 3.1, P A is symmetric. We claim θ k k A ij = 0 and A ij = P l i A lj. (0)

13 SELF-SHRINKERS IN ARBITRARY CODIMENSION 13 To prove (0), we exploit (7), (19) and P = 0 to get in a first step θ k ( ) k P l i A lj = θ k Pi l k A lj (7) = Pi l P m A mj Pi l A lj l (19) = Pi m A mj Pi l A lj = 0. (1) So it suffices to show A P A = 0. From (7), (19) and (1) we conclude θ k k ( A P A ) = θ k A ij, k A ij (7) = A ij, P k i A kj A ij (19) = A ij, P l i P k l A kj A ij = ( A P A ). () If θ vanishes at a point p M, then () implies A ij = P l i A lj and A = P A at p. So suppose θ(p) 0. Let γ be the integral curve of θ with γ(0) = p, i.e. dγ = θ. Then along γ we define the function f(t) := θ (γ(t)) and obtain d dt f = dγ( θ ) = θ k k θ = θ k θ l k θ l. From (5), F = H and (18) we see so that Consequently i θ j = g ij P ij, θ i i θ j = θ j, d f = f. (3) dt θ (γ(t)) = θ (p)e t > 0, t which by the completeness of M implies that the integral curve γ is regular and well-defined for all t R. Along the same curve γ we now define a new function f(t) := ( A P A )(γ(t)) and from () we derive the evolution equation and d dt f = f ( A P A )(γ(t)) = ( A P A )(p)e t. This is the typical behaviour of a cone. If ( A P A )(p) 0, then ( A P A )(γ(t)) becomes unbounded as t. This contradicts the boundedness of the second fundamental form. So A = P A and (0) is valid. The multiplicities of the two eigenvalues of P are constant on M since P ij = 0. Let r denote the multiplicity of λ = 1, i.e. r is

14 14 KNUT SMOCZYK the rank of the second fundamental form A ν = P/. We define the two distributions E p M := {V T p M : P j i V i = V j }, F p M := {V T p M : P j i V i = 0}. so that T p M = E p M F p M. In addition, for the nullspace E p M we have E p M ker(θ) since (18) implies θ(v ) = θ j V j = θ j P j i V i = 0, V E p M. (4) Let e p E p M, f p F p M and let γ(t) be a curve in M with γ(0) = p. If e(t) with e(0) = e p and f(t) with f(0) = f p are varying along γ by parallel transport we obtain from P = 0 d dt P e e = 0 = d dt P f, so that e(t) E γ(t) M and f(t) F γ(t) M. In particular for vector fields f 1, f F M and e 1, e E M we have e1 e, [e 1, e ] E M, (5) f1 f, [f 1, f ] F M, (6) so that E M, F M are invariant under parallel transport and in particular involutive. This means that for each p M there exist two integral leaves E p, F p intersecting orthogonally in p. By Lemma 3.1, we have for any normal vector V perpendicular to ν where P, A V = 0, A V := V, A is the second fundamental form w.r.t. V. At some point p M let us choose an orthonormal basis e 1,..., e r of E p M. Then r tr E (A V ) := V, A(e i, e i ) = P, A V = 0, i=1 V N p M with H, V = 0. (7) From (4) we conclude that each integral leaf of E M is an r - dimensional submanifold of the sphere S n 1 (ρ) R n of radius ρ = F, the radius depending on the leaf (see figure 1). We claim that the leaves F p are (m r) - dimensional affine subspaces of R n and that these affine subspaces are parallel for any p, p in the same connected component of M. To see this, let q F p be arbitrary and note that A ij = P k i A kj implies A(f, X) = 0, f F q M = T q F p, X T q M. (8) The normal space of F p at q decomposes into N q F p = N q M E q M. By (8) the second fundamental form of F p vanishes w.r.t. any normal vector belonging to N q M and by (6) also for all normal vectors belonging to E q M. This shows that the integral leaves of F M are affine subspaces of R n of dimension m r. Now we fix a point p M and suppose that p E p

15 SELF-SHRINKERS IN ARBITRARY CODIMENSION 15 is arbitrary. Let γ E p be any smooth curve with γ(0) = p and γ(1) = p. Suppose f p F p is some vector and f(t) is varying along γ by parallel transport with f(0) = f p. Considering f(t) as a vector field along γ in R n we compute d dt f = ċf + ( d dt f) = ( d dt f) = A(ċ, f). But then by (8) d dt f = 0. Since F M is invariant under parallel transport we conclude that f p F p M as well and since f was arbitrary we must have F p M = F p M. To see that this relation holds for all p, p in the same connected component of M we proceed as follows. The leaf E p is contained in a linear subspace of dimension n m + r which is orthogonal to F p M. Then we define an embedding F : E p R m r R n, F (p, X) = (p, X). Since the integral leaves of F M through p E p coincide with the affine subspace w.r.t. F p M through p, the m - dimensional image of E p R m r under F is contained in the m - dimensional submanifold M and by the completeness of both E p and M we conclude that F (E p R m r ) is a connected component of M. Since the integral leaves of E M are contained in spheres of radii ρ = F and F = F + θ = + θ we observe that θ must be constant along the leaves of E M since H is parallel and hence is constant. Moreover, from (4) we conclude that the projection π : M E p maps the points q in the integral leaves of E M with q = F to points q E p with q Ep = F = and since H = F we see that M := E p must be a minimal submanifold of the sphere S n m 1+r (). On the other hand = g ij P ij = r since all eigenvalues of P are either 0 or 1 and r is the multiplicity of the latter. Thus = r and each connected component of M is isometric to M R m r with a minimal submanifold M S n m 1+r ( r). (ii) The remaining case is H(p 0 ) 0 at a point p 0 M. Then, as in the compact case, we find P = 4 globally. In addition, P admits only one non-zero eigenvalue λ = and spans the eigenspace. Then Lemma 3.1 implies the relations Let us define P k i A kj = P ij ν, P j i i = j. (9) M := {p M : H(p) 0}. Following Huisken s [Hui93] argument closely, we choose a connected component U M and consider the distributions E p U := {X T p U : p U, P X = X}, F p U := {X T p U : p U, P X = 0}. In contrast to part (i), these distributions are defined a-priori only on M. As in the first part, they are invariant under parallel transport and again (5), (6) are valid for them so that we may apply Frobenius theorem.

16 16 KNUT SMOCZYK Figure 1. The sphere intersects the cylinder M in two leaves of the distribution E M which are the circles on the cylinder. The straight lines perpendicular to these circles are the leaves belonging to F M, We claim that the integral leaves of F U are affine subspaces of dimension m 1. This is equivalent to showing that A ij := A ij 1 P ij ν vanishes completely. From (6) we observe i = θ k P ki (30) and then with (9) θ ( ) =. Let θ be the projection of θ onto F U, i.e. θ i = θ i θ ( ) i = θ i 1 i.

17 SELF-SHRINKERS IN ARBITRARY CODIMENSION 17 A straightforward computation shows ( θ k A ki = θ k 1 ) ( k A ki 1 ) P ki ν Then (30) = θ k A ki i ν 1 k A ki + 1 k P ki ν (9) = θ k A ki 1 3 j P k j A ki (9) = θ k A ki 1 j P ij ν (9) = θ k A ki i ν (6) = 0. (31) 0 = l ( θ k ) A ki (5) = A li P k l A ki + 1 l k A ki 1 l k A ki + θ k l A ki (9) = A li 1 l k A ki + θ k l A ki. Since F U is invariant under parallel transport, we have f1 f = f 1 (, f ), f1 f = 0 for all f 1, f F U. Therefore we may write i j = P ij and since (9) implies P k i A kj = 0, we get from the formula above θ k l A ki = A li, (3) in particular both sides in (3) are symmetric in i and j. From (5), (p) E p U, θ (p) Fp U and since F U is invariant under parallel transport we derive ( η θ ) = η, θ, η Fp U. (33) We may now proceed similarly as in part (i). If θ (p) = 0, then (3) implies A = 0. If θ (p) 0, we choose η = θ in (33) and consider the integral curve γ w.r.t. θ starting at p which as before proves that the integral curve is regular and contained in U for all t R. Applying formula (3) to γ we get exponential growth in t for the quantity A if A (p) = 0 which by the boundedness of the second fundamental form would be a contradiction. Therefore A = 0 everywhere and as in the first part we obtain for some fixed point p U that the integral leaf E p U is part of an

18 18 KNUT SMOCZYK Abresch-Langer curve Γ that extends smoothly up to the boundary of U and that all integral leaves F q U for q U are parallel affine subspaces. So as above any connected component U M is a product of a part of an Abresch-Langer curve and some R m 1. To extend this to all of M we have to consider the points, where = 0. We claim that the critical points of are transversally isolated along E U. By this we mean the following. The integral leaves E p U of U are geodesics in M and by the completeness can be extended arbitrarily. So we obtain a complete geodesic γ M that coincides in U with a portion of an Abresch-Langer curve Γ = E p U and which intersects the boundary of U orthogonally. Suppose the fraction of γ U is parametrized by γ : (0, t 0 ) U and that lim t t0 (γ(t)) = 0. Then (8) implies lim = 3 (34) t t 0 On the other hand (γ(t)) coincides in U with the curvature k of the Abresch-Langer curve Γ and lim t t0 k(t) = k(t 0 ) = k 0 is a critical value of the curvature of Γ. Since k 0 = 1 is the critical value of the round circle, lemma 5.3 and (34) would imply that, if (t 0 ) = 0, then Γ is a fraction of S 1 and k = 0 = on all of Γ. Therefore k 0 1 and (t 0 ) 0. This implies that 0 immediately after passing γ(t 0 ) in direction of γ. In particular, the boundary of each connected component U M consists of at most two affine subspaces of dimension m 1 that are parallel to each of the leaves F q U, q U. If U 1, U are two connected components with Ū1 Ū, then the integral leaves F q1 U 1, F q U are parallel. We may thus join two such connected components U 1, U along their common boundary and obtain two portions of Abresch-Langer curves Γ 1, Γ that both intersect the common boundary of U 1, U orthogonally and which meet at a boundary point, for instance at γ(t 0 ). If Γ 1 and Γ would be portions of different Abresch-Langer curves, then again by lemma 5.3 we get a contradiction, since then the critical values of the two curvature functions are different but both satisfy (34). By the completeness of M we may extend this process until we get M = Γ R m 1. References [AL86] [Alt91] [Anc00] [Ang9] [AS96] [BN99] U. Abresch and J. Langer. The normalized curve shortening flow and homothetic solutions. J. Differential Geom., 3(): , Steven J. Altschuler. Singularities of the curve shrinking flow for space curves. J. Differential Geom., 34(): , H. Anciaux. Construction of lagrangian self-similar solutions of the mean curvature flow in C n. Preprint, 000. Sigurd B. Angenent. Shrinking doughnuts. In Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), volume 7 of Progr. Nonlinear Differential Equations Appl., pages Birkhäuser Boston, Boston, MA, 199. Luigi Ambrosio and Halil Mete Soner. Level set approach to mean curvature flow in arbitrary codimension. J. Differential Geom., 43(4): , G. Bellettini and M. Novaga. A result on motion by mean curvature in arbitrary codimension. Differential Geom. Appl., 11(3):05 0, 1999.

19 SELF-SHRINKERS IN ARBITRARY CODIMENSION 19 [CLT0] Jing Yi Chen, Jia Yu Li, and Gang Tian. Two-dimensional graphs moving by mean curvature flow. Acta Math. Sin. (Engl. Ser.), 18():09 4, 00. [GSSZ05] Konrad Groh, Matthias Schwarz, Knut Smoczyk, and Kai Zehmisch. Volume minimization of equivariant Lagrangian immersions. Notes, 005. [HL8] Reese Harvey and H. Blaine Lawson. Calibrated geometries. Acta Math., 148:47 157, 198. [Hui90] Gerhard Huisken. Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom., 31(1):85 99, [Hui93] Gerhard Huisken. Local and global behaviour of hypersurfaces moving by mean curvature. In Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Providence, RI, [SW0] Knut Smoczyk and Mu-Tao Wang. Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differential Geom., 6():43 57, 00. [TW04] Mao-Pei Tsui and Mu-Tao Wang. Mean curvature flows and isotopy of maps between spheres. Comm. Pure Appl. Math., 57(8): , 004. [TY0] R. P. Thomas and S.-T. Yau. Special Lagrangians, stable bundles and mean curvature flow. Comm. Anal. Geom., 10(5): , 00. [Yan71] Kentaro Yano. Submanifolds with parallel mean curvature vector of a euclidean space or a sphere. Kodai Math. Semin. Rep., 3: , Universität Hannover, Fakultät für Mathematik und Physik, Institut für Mathematik, Welfengarten 1, Hannover, Germany address: smoczyk@math.uni-hannover.de