Evolution of hypersurfaces in central force fields

Transcription

1 Evolution of hypersurfaces in central force fiels Oliver C. Schnürer an Knut Smoczyk November 000, revise June 00 Abstract We consier flows of hypersurfaces in R n+1 ecreasing the energy inuce by raially symmetric potentials. These flows are similar to the mean curvature flow but ifferent phenomena occur. We show for a natural class of potentials that hypersurfaces converge smoothly to a uniquely etermine sphere if they satisfy a strengthene starshapeness conition at the beginning. 1 Introuction Consier a smooth family F t : M R n+1 \ {0} of oriente hypersurfaces M t := F t (M) that evolve accoring to (1) F = (H φ(s) F,ν )ν fν, t where we have omitte the inex t. Here, H enotes the mean curvature of M t w. r. t. the outer unit normal ν (i. e. H > 0 for the stanar sphere) Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany Mathematics Subject Classification 000: Primary 53C44, Seconary 35K15 1

2 an φ is a smooth, raially symmetric function (reflecting the presence of a central force) epening on s := F. More precisely φ is relate to the potential v on R n+1 \{0} by v(s) := exp n s 1 η(σ) σ σ, φ(s) = σ v(σ) v(σ) = nη(s) s σ=s where η : R + R is a smooth function. The total potential energy V(F) of the hypersurface F(M) will then be efine as V(F) := M v(s(f))µ. A physical interpretation of our flow is as follows. Suppose we have an electrically charge membrane with a constant charge per area. If this surface is in a raially symmetric potential, then the energy of the system is given by the energy in the exterior potential plus other terms, for example are there forces between ifferent parts of the surface, which we neglect. The negative graient flow for this energy is given by F = v(h φ(s) F,ν )ν, t so we see that the stationary solutions of (1) coincie with the stationary solutions of the negative graient flow. We will prove the following main theorem Theorem 1.1 Let F : S n R n+1 be a smooth embeing of a strictly starshape hypersurface such that 0 < s s = F s + <. Further assume that η satisfies (6) (7) s 0 [s,s + ] with η(s 0 ) = 1, η (s) < n c η < 0 s [s,s + ]

3 for a constant c η an that the strengthene starshapeness conition (8) f F,ν < c η hols for F. Then (1) amits an embee solution F t for all t 0 with F 0 = F an S t := F t (S n ) converges exponentially in the C -topology to a stable sphere centere at the origin with raius r 0 = s 0. Remark: Conition (8) can be easily satisfie. Let α > 0, β > 0 an assume that φ has the form φ(s) = α. Let F be s 1+β as in the main theorem, r 0 = max F. At the point where this maximum is attaine we have with s 0 = 1 r 0 c η > f F,ν 1 s 0 ( n α s β 0 ). On the other han η (s 0 ) < n c η implies so we obtain an see that r 0 R(n,α,β). s 0 < αβ s 1+β 0 > c η ( ) 1 n α(1+β) β We have chosen the special ansatz for the potential v as the stability of a stationary sphere of raius r 0 = s 0 is equivalent to the simple conition η (s 0 ) < 0. We wish to explain the conition (8) assume for the embeing F. First we show, that the flow oes not preserve convexity nor starshapeness uring the evolution (so a somehow ifferent initial conition is neee for the smooth long-time existence), then we give a geometric interpretation of (8). We assume a potential v that tens to infinity for s 0 an ecays for s. Then a convex hypersurface oes not nee to stay convex uring 3

4 its evolution. We take a sphere of large raius but assume that the origin is very close to the surface. At the beginning of the evolution the surface is repelle apart from the origin where it is very close to the origin but moves relatively slowly at a large istance from the origin. As a sphere of large raius is nearly flat near a fixe point this estroys convexity. A more elaborate example will show that starshapeness with respect to the origin is not preserve, too. We escribe our surface in polar coorinates as follows. The height above the unit sphere is given by a positive constant times the characteristic function of a small geoesic ball aroun a fixe point. The example is obtaine if we smooth out this situation slightly. Figure 1 shows a cross-section of this hypersurface. Figure 1: Starshape example Rotating it aroun the horizontal axis gives the whole hypersurface. If the neck is thin, n 1 principal curvatures become very large there. The remaining negative principal curvature there is small compare to the others when the neck becomes small. Of course the non-spherical part has to be mae longer simultaneously so that the curvature near the tip remains boune. This construction yiels for a small potential compare to the mean curvature that the motion of the surface is especially large at the neck where we assume that it shrinks so fast that not only the starshapeness is lost but also the evolution will become singular in finite time. 4

5 If we represent the hypersurface asgraph u S n (this is possible for starshape hypersurfaces with respect to the origin), then the time erivative of log u equals f. The change of coorinates u logu, however, is natural as F,ν the inuce metric of the hypersurface in this new coorinates is very similar to the inuce metric for graphs in Eucliean space, in fact, this shows the conformal equivalence of S n R an R n+1 \{0}. More etails can be foun in the appenix. Anotherinterestingfeatureofourflowcanbeobtaineforφ = 1,acasewhich is not consiere in our main theorem. Then the stationary solutions are characterize by H = F, ν an these hypersurfaces shrink homothetically uner the evolution of the mean curvature flow t F = Hν. We consier a potential with appropriate asymptotics as for example the potential inuce by η(s) = 1. Then there are two possibilities to reuce n s the energy; the surface may move apart from the source of the potential or it may contract anthus reuce itsarea anits energy asthe charge isassume to be proportional to the area. When the surface encloses the origin, i. e. the source of the potential, a charge point, these two effects are opposite to each other an so it is reasonable to conjecture that the surface tens to a sphere for which both forces compensate each other when the topology allows this. We wish to mention some further papers on relate problems. In [1], [4], [6] an[7] the evolution of starshape hypersurfaces for various curvature riven flow equations has been consiere. In contrast to outwar irecte flows, where starshapeness usually is preserve, this fails for inwar irecte flows like the mean curvature flow. For these flows convexity is naturally preserve []. But even for inwar irecte flows we have nice properties for starshape hypersurfaces [6]. The paper is organize as follows: In section we introuce notations from ifferential geometry an compute the Euler-Lagrange equations, in section 3 we show how spheres can be use as barriers for our flow. In section 4 we erive evolution equations for geometric quantities, euce a priori estimates an prove the smooth convergence to a stable sphere. Finally, we state aitional properties of our flow in the appenix. We wish to thank Jürgen Jost an the Max Planck Institute for Mathematics in the Sciences (MPI MIS), Leipzig, for their hospitality. 5

6 Euler-Lagrange equations We are intereste in the first an secon variation of V. To this en assume that F t : ( ǫ,ǫ) M R n+1 is a smooth family of immersions of orientable hypersurfaces such that t F t = fν, with a smooth function f epening on t ( ǫ,ǫ) an ν enoting the outwar unit normal. We also recall the Gauß formula, the equations of Gauß, Weingarten, Coazzi an Simons Proposition.1 (13) (14) (15) (16) (17) (18) R ijkl = h ik h jl h il h jk, i j F = h ij ν, i ν = hi l l F, k h ij = j h ik, i j H = h ij Hhi l h lj + A h ij, h ij i j H = A A Z. Here h ij is the secon funamental form an H = g ij h ij, A = g ij g kl h ik h jl, C = g ij g kl g st h ik h js h lt, Z = HC A 4. g ij is the inverse of the inuce metric g ij := F x i, F x j Double inices are always summe from 1 to n, inices are raise an lowere with respect to the inuce metric an enotes the Levi-Civita connection on M w. r. t. g ij. In the sequel we won t istinguish between vectors V T x M an DF(V) T F(x) R n+1 an we also use, both for the scalar prouct on R n+1 an on M. The calculations in [5] give 6.

7 Proposition. (0) t g ij = fh ij, (1) µ = fhµ, t () t ν = DF( f) = gij i f j F, (3) t h ij = i j f fhi k h kj, (4) t H = f +f A, (5) t A = h ij i j f +fc. In aition we get (6) t v = nη s v t s = fφ F,ν v. Lemma.3 The first an secon variation for V an compactly supporte f C0 (M) are given by t V = fv(h φ F,ν )µ, M t V = (H φ F,ν ) M t (fvµ) + v ( f f ( A +φ+φ F,ν ) ) µ. M Proof: The first variation formula is a irect consequence of (1) an (6). For the secon variation we use (), (4) an compute vf t (H φ F,ν ) = vf( f +f A +fφ F,ν +fφ φ s, f ) = iv(vf f)+v ( f f ( A +φ+φ F,ν ) ) an the secon variation formula follows from partial integration. 7

8 Corollary.4 The Euler-Lagrange equation for stationary hypersurfaces w. r. t. V is given by H φ F,ν = 0. A stationary hypersurface is stable (resp. strictly stable) if an only if ( f f ( A +φ+φ F,ν ) ) vµ 0 (resp. > 0) M for all smooth, compactly supporte f 0. Corollary.5 Assume that s 0 satisfies η(s 0 ) = 1 an that η (s 0 ) < 0. Then the sphere with center at the origin an raius r 0 = s 0 is strictly stable. Proof: Left to the reaer. The negative graient flow for V is given by F = v(h φ F,ν )ν. Since t this is stationary if an only if H φ F,ν = 0 we can as well consier the moifie graient flow (30) with t F = fν f := H φ F,ν. We will set M t := F t (M). Then the results in [3] imply Proposition.6 (30) is a system of quasilinear parabolic equations an there exists a maximal time 0 < T such that (30) amits a smooth solution on [0,T). 3 Inclusion principle Lemma 3.1 Assume that the initial value F 0 of a smooth solution F t, 0 t < T, of the flow equation (30) is embee. Then F t, 0 < t < T, is also embee. 8

9 Proof: Lett 0 bethefirsttimewhentheimmersionfailstobeanembeing, F t0 (x 0 ) = F t0 (y 0 ), x 0 y 0. We choose a local coorinate system such that F t0 (x 0 ) = F t0 (y 0 ) = 0 an the unit normals at the origin are ±e n+1. So we may write F t0 aroun x 0 an y 0 as graph(u 1 ) an graph(u ), respectively. Therepresentationasagraphispossibleforasmalltimeinterval[t 0 ε,t 0 +ε], ε > 0, too. We may assume that u 1 > u for t < t 0. If we replace ν by ν in the flow equation, H, F,ν an ν change sign, so we may assume that the normals at (x 0,t 0 ) an (y 0,t 0 ) both equal e n+1. We rewrite the geometric evolution equation as a parabolicevolution equation for u 1 an u an obtain (3) t (u1 u ) = a ij (u 1 ij u ij )+bi (u 1 i u i )+c(u1 u ). We remark that this equation is parabolic as H is elliptic for every smooth solution, i. e. especially for any function τu 1 +(1 τ)u, 0 τ 1. From the strong maximum principle we euce, that u 1 an u coincie locally an furthermore, that these two functions have locally to be equal before t = t 0. This contraicts our assumption about t 0. Corollary 3. Let F be a smooth immerse solution of (30) an F be an immerse solution of this evolution equation. If F is containe in a connecte component of R n+1 \F or in the closure of such a component at the beginning of the evolution, then this remains true uring the evolution. Proof: As the geometric situation at the first point of contact is similar to the situation in the proof of Lemma 3.1, we can argue as we have one there. If F an F touch for t = 0 then the strong maximum principle yiels that they are isjoint at least for small positive times unless there are connecte components of F an F which coincie for t = 0. The following barrier argument will become important in the sequel. Lemma 3.3 Assume s,s 0,s + are positive numbers such that s < s 0 < s + an that η satisfies η(s 0 ) = 1, η (s) < 0 s [s,s + ]. 9

10 Then uner our flow all centere spheres of raius r = s with s [s,s + ] converge to the stable sphere with raius r 0 = s 0. Moreover, if M t,t 0, is a smooth evolution of hypersurfaces such that M 0 is containe in then M t will be containe in {y R n+1 r := s < y < r + := s + }, {y R n+1 r (t) < y < r + (t)}, where r (t) an r + (t) are the raii for the evolutions of the inner an outer sphere. If M t exist for all t 0, then M t converges pointwise to the stable sphere of raius r 0. Proof: The first part is an easy consequence of (35) s = n(η 1) t which hols for centere spheres, see the inepenently proven Lemma 4.. The remainer then follows from the inclusion principle 3.. Remark: From (35), η(s 0 ) = 1 an η (s 0 ) < 0 we euce that the convergence in Lemma 3.3 is even exponentially. 4 Convergence to a stable sphere Lemma 4.1 The evolution equation for f is t f = f φ s, f +f( A +φ+φ F,ν ). Proof: This has been calculate in the proof of Lemma.3. Lemma 4. s satisfies t s = s φ s +sφ n, 10

11 Proof: First we compute i s = F,F i with F i := F x i. Then the Gauß formula implies Therefore i j s = g ij F,ν h ij. s = n H F,ν = n f F,ν φ F,ν. On the other han t s = F, F = f F,ν t an by writing F = DF( s)+ F,ν ν The combination gives φ s = φ(s F,ν ). t s = f F,ν + s n+f F,ν +φ F,ν φ s +φ(s F,ν ) = s φ s +sφ n. Next we want to compute the evolution of F,ν. Lemma 4.3 F, ν satisfies t F,ν = F,ν φ s, F,ν + F,ν ( A +φ+φ F,ν ) H sφ F,ν Proof: We compute an i F,ν = h l i ls i j F,ν = l h ij l s+h ij F,ν h l i h lj, 11

12 where we use the Coazzi equation (16). This gives F,ν = s, H +H A F,ν. On the other han with () an (30) we conclue F,ν = f + F,DF( f) = f + s, f t so that Since t F,ν = F,ν s, (H f) H f + A F,ν. (48) (H f) = F,ν φ s+φ F,ν an we obtain the result. s = s F,ν We want to rewrite equations (4) an (5). To this en we nee an expression for i j (f H). From (48) we obtain i j (f H) = i ( F,ν φ j s+φ j F,ν ) ( = i ( F,ν φ δ l j +φh l j ) ls ) (50) Then = (h k i ksφ δ l j + F,ν φ i sδ l j +φ i sh l j +φ l h ij ) l s ( F,ν φ δ l j +φh l j )(g il F,ν h il ) = φ (h l i js+h l j is) l s φ F,ν i s j s φ l h ij l s (φ φ F,ν )h ij +φ F,ν h l i h lj φ F,ν g ij. (f H) = φ h ij i s j s φ F,ν (s F,ν ) φ s, H Now (4) implies (φ φ F,ν )H +φ F,ν A nφ F,ν. 1

13 Lemma 4.4 t H = H φ s, H +H( A φ+φ F,ν ) φ h ij i s j s+ F,ν (φ F,ν sφ nφ ). (50) also gives h ij i j (f H) = 4φ h il h j l is j s φ F,ν h ij i s j s φ s, A A (φ φ F,ν ) + φ F,ν C φ F,ν H. From (5) an Simons ientity (18) we then erive t A = h ij i j f +fc = h ij i j H +h ij i j (f H)+fC = A A Z 4φ h il h j l is j s φ F,ν h ij i s j s φ s, A A (φ φ F,ν )+φ F,ν C φ F,ν H +fc. Cancellation gives Lemma 4.5 t A = A φ s, A A 4φ h il h j l is j s φ F,ν h ij i s j s + A ( A φ+φ F,ν ) φ F,ν H. In the next steps we want to prove that our spheres remain starshape an that (8) remains true uner the evolution. Therefore we efine the quantity q := f F,ν. A geometric motivation for q will be given in the appenix. Then the evolution equations for f an F,ν imply t q = 1 ( f φ s, f +f( A +φ+φ F,ν ) ) F,ν f ( F,ν φ s, F,ν + F,ν ( A +φ+φ F,ν ) F,ν (5) H sφ F,ν ). 13

14 On the other han we compute q = f F,ν +f( 1 F,ν F,ν + Inserting this in (5) yiels Lemma 4.6 (53) F,ν f, F,ν F,ν 3 F,ν ) = f F,ν f F,ν F,ν F,ν F,ν, q t q = q + F,ν F,ν, q φ s, q +q(q +sφ +φ). Corollary 4.7 Uner the assumptions of Theorem 1 there exists a positive constant ǫ > 0 inepenent of t such that (54) (55) q +sφ +φ ǫ, q (max t=0 q )e 4ǫt as long as a smooth starshape solution of (1) exists. Proof: From Lemma 4.6 we obtain (56) t q = q q + F,ν F,ν, q φ s, q + 4q (q+sφ +φ). Since sφ + φ = n η, the barrier argument, Lemma 3.3, an inequality (7) imply that sφ +φ < c η as long as S t remains smooth. Consequently (58) t q q q + F,ν F,ν, q φ s, q + 4q (q c η ). 14

15 Hencethemaximumprinciplean(8)implythatq cannotevelopapositive maximum with t q > 0 an q c η remains boune above by its initial (negative) maximum. This proves (54) with ǫ := c η max t=0 q. But then the evolution equation for q an (54) even imply that q e 4ǫt remains boune by its initial maximum. Lemma 4.8 Uner the assumptions of Theorem 1 the starshapeness of S t remains true as long as a smooth solution of (1) exists. Proof: We compute the evolution for the quantity 1 F,ν which remains boune above if an only if S t remains starshape. From Lemma 4.3 we get (61) 1 = ( F,ν φ s, F,ν t F,ν F,ν 3 + F,ν ( A +φ+φ F,ν ) H sφ F,ν ) = 1 F,ν 6 φ s, 1 F,ν 4 F,ν F,ν + φ φ F,ν +(q+sφ +φ)). F,ν ( A Now as long as F,ν remains positive we can use (54) an estimate 1 1 t F,ν + F,ν φ s, 1 F,ν F,ν ( A φ ǫ) φ F,ν, 1 F,ν F,ν

16 The barrier argument, Lemma 3.3, implies (6) (63) φ(s(x, t)) φ (s(x,t)) min [ { } { F σ min min t=t,s F 0,max max 0 t=t,s 0 0 min [ { } { F σ min min t=t,s F 0,max max 0 t=t,s 0 0 }]φ(σ) =: c φ,t0, }]φ (σ) =: c φ,t 0 for all x S n an all t t 0, where t 0 0 is a fixe time. For the proof of this lemma we will actually only use t 0 = 0 but we nee the same estimates for a ifferent t 0 later. Hence for all t t 0 the following inequality is vali for starshape S t 1 1 t F,ν (64) F,ν φ s, 1 F,ν ( A +ǫ+c φ,t0 ) F,ν c φ,t 0. + F,ν, 1 F,ν F,ν The last term in the first line is nonpositive but we keep it for later purposes. Let p(t) be the solution of i. e. for ǫ+c φ,t0 0 p(t) = t p = (ǫ+c φ,t 0 )p c φ,t 0, 1 p(t 0 ) = max t=t 0 F,ν, ( p(t 0 )+ c φ,t 0 ǫ+c φ,t0 Then for all t t 0 an starshape S t ( ) ( ) 1 1 t F,ν p F,ν p (ǫ+c φ,t0 ) The maximum principle then implies (66) 1 F,ν ( 1 max t=t 0 F,ν + cφ,t 0 ǫ+c φ,t0 ) e (ǫ+c φ,t 0 )(t t 0 ) c φ,t 0 ǫ+c φ,t0. 16 φ ( 1 F,ν p s, ). ( ) 1 F,ν p ) e (ǫ+c φ,t 0 )(t t 0 ) c φ,t 0 ǫ+c φ,t0

17 for all t t 0. Consequently the quantity F,ν cannot ten to zero in finite time an S t remains starshape. Theorem 4.9 With the same assumptions as in Theorem 1 a smooth solution exists for all t > 0. Proof: We nee bouns for k A for all k 0. We begin with a boun for A F,ν. From Lemma 4.5 an the evolution equation (61) we obtain A = A ( 1 t F,ν F,ν 6 φ s, 1 F,ν 4 F,ν F,ν + φ φ F,ν +(q+sφ +φ)) ) F,ν ( A 1 ( + A φ s, A A F,ν To procee we efine This implies 4φ h il h j l is j s φ F,ν h ij i s j s + A ( A φ+φ F,ν ) φ F,ν H ) = A F,ν A, 1 F,ν φ s, A F,ν A F,ν F,ν 6 A 4φ F,ν F,ν 4 F,ν φ F,ν hij i s j s φ H A +4 F,ν F,ν (q +sφ ). Q := F,ν i h jk i F,ν h jk = F,ν A + A F,ν F,ν F,ν, A. A F,ν F,ν 6 A F,ν 4 = Q F,ν 4 + F,ν A, 1 F,ν F,ν, A 17 F,ν

18 an finally A = s, A t F,ν F,ν φ A F,ν (68) Q F,ν 4 4φ F,ν φ F,ν F,ν hij i s j s φ H A +4 F,ν F,ν (q +sφ ). To procee we nee the estimate F,ν = h i l hlj i s j s A s s A. + F,ν, A F,ν F,ν Moreover H F,ν H F,ν + 1 n A F,ν + 1. Then we use the barrier argument, Lemma 3.3, inequality (54) an estimate A s, A t F,ν F,ν φ A + F,ν, A F,ν F,ν F,ν A (69) + c 1 F,ν +c A F,ν can for two positive constants c 1,c inepenent of t. So we see that increase at most exponentially in time an since by Lemma 4.8 S t remains starshape an F,ν const, this also means that A remains boune on any finite time interval. We can then procee as in [] to erive uniform upper bouns for all higher covariant erivatives of A on any finite time interval [0, T) for which a smooth solution of the flow exists. Consequently T =. Proof of the main theorem: We are now able to prove uniform upper bouns in t also. First we nee a uniform upper boun for 1 F,ν. Therefore 18

19 we go back to inequality (66). From Theorem 4.9 we know that the flow exists for all t > 0 an since by assumption (6) φ(s 0 ) = nη(s 0) s 0 = n s 0 > 0, the barrier argument, Lemma 3.3, implies the existence of t 0 0 estimate from above such that c φ,t0 > 0. But then inequality (66) implies that with a constant c 3 > 0 an for all t > 0 (70) 1 F,ν c 3. Our iea is to a enough of 1 F,ν to A F,ν to obtain a uniform boun for A. Therefore, let B := A +k F,ν for a large constant k to be etermine. Then (64) an (69) give (7) B B φ s, B + t F,ν F,ν, B ( ) + (c 1 k) A F,ν k (ǫ+cφ,t0 ) +c F,ν φ,t 0 +c for all t t 0, with a fixe t 0, without loss of generality t 0 = 0. We choose k = c 1. Then (73) B B φ s, B + t F,ν F,ν, B c 1 B +c 4, where c 4 epens only on c 1,c,c 3,ǫ,c φ,0 an c φ,0 but not on t. Consequently a maximum of B with B > 0 must be smaller than c 4 t c 1 an furthermore B an A are uniformly boune. Once we ve obtaine a uniform boun for A we use the technique in [] to obtain uniform upper bouns for all quantities k A. This shows that F t is uniformly boune in C for all t 0. From the barrier argument, Lemma 3.3, we conclue that s s 0 ecays exponentially. Then we use the elementary interpolation inequality ψ const ( M, g ) ψ ( ψ + ψ ), for C -functions ψ on a compact Riemannian manifol ( M, g), where enotes the covariant erivative w. r. t. g an the constant epens only on 19

20 M an g. This inequality, the C k -bouns for s s 0 (which follow from those for the secon funamental form) an the uniform equivalence of the inuce metrics on M = S n for any t imply an exponential ecay of k (s s 0 ) for all k N. We thus conclue exponential convergence in C to the stable sphere of raius r 0 = s 0. 5 Appenix Corollary 5.1 Assume f > 0 for t = 0. If the absolute value of A + φ+φ F,ν remains boune from above by some positive constant c on the time interval [0,t 0 ),t 0 T, then for t [0,t 0 ) min f(x) min f(x)e ct > 0. x M t x M 0 Proof: This is a irect consequence of the parabolic maximum principle applie to Lemma 4.1. A geometric motivation for f F,ν We wish to represent our evolution problem via graphs over the sphere S n as in [1]. Therefore we use an embeing of the form F = x ( ξ i,t ) u ( x ( ξ i,t ),t ), 0

21 where x S n an ξ i enotes local coorinates of S n. We use covariant erivatives with respect to the metric of S n an compute F i = x i u+xu i, F ij = x ij u+x i u j +x j u i +xu ij, g ij = F i,f j = u σ ij +u i u j = u (σ ij +ϕ i ϕ j ), where ϕ = logu, ( ) g ij = u σ ij ϕi ϕ j, ϕ k = σ kl ϕ l, w := 1+ϕ i ϕ i, w where σ ij enotes the metric of the sphere S n. The outer unit normal is given by ν = 1 ( ) x ϕ k x k. w The Gauß formula relates the secon covariant erivatives of the embeing vector to the secon funamental form an the normal, where the secon erivatives are taken with respect to the inuce metric. At the moment, however, F ij enotes the secon covariant erivatives with respect to the metric σ ij. However, these erivatives are relate by F g ij = Fσ ij + ( Γ k ij(σ) Γ k ij(g) ) F k, so we euce from the Gauß formulae for M an S n Fij,ν σ = h ij ν,ν = h ij, h ij = 1 x 1 w u uk x k,u ij x+u i x j +u j x i uσ ij x = u ( σ ij + 1 w u u iu j 1 ) u u ij = u w (σ ij +ϕ i ϕ j ϕ ij ) = 1 uw g ij u w ϕ ij. 1

22 Next, we will erive an evolution equation for u (79) In view of t (x u) = t x u+x t u+x t F = fν = f 1 w we obtain by multiplying (79) with x ( x ϕ i x i ). 0 = t x,x = t x,x, f 1 w = u t + From the remaining part of (79) we get u, x. t f 1 w ϕi x i = t x u, u, x, t thus t u = f 1 ( ) 1+ ϕ,ϕ i x i w = fw an in view of F,ν = u w ϕ := t ϕ = f F,ν. We now state the parabolic ifferential equations for ϕ, ϕ an W = 1 (1 + ϕ k ϕ k ) to give the reaer the opportunity to compare the evolution equations for ϕ an q, Lemma 4.6, as well as those for W an 1 F,ν, equation (61), ϕ = g ij ϕ ij ne ϕ +φ, ϕ = g ij ϕ ij e ϕ 1 w ( ϕi ϕ j +ϕ i ϕ j )ϕ ij +e ϕ 1 w 4ϕi ϕ j ϕ ij ϕ k ϕ k + ϕ( ϕ+φ+φ e ϕ ), Ẇ = g ij W ij e ϕ ϕ j i ϕi j 1 w e ϕ W i W i + 1 w 4e ϕ (W i ϕ i ) +(W 1) ( e ϕ (n 1) ϕ+φ+φ e ϕ).

23 References [1] C. Gerhart: Flow of nonconvex hypersurfaces into spheres. J. Differential Geom. 3 (1990), [] G. Huisken: Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 0 (1984), [3] G. Huisken, A. Polen: Geometric evolution equations for hypersurfaces. Calculus of variations an geometric evolution problems (Cetraro, 1996), 45 84, Lecture Notes in Math., 1713, Springer, Berlin, [4] N. M. Ivochkina, Th. Nehring, F. Tomi: Evolution of starshape hypersurfaces by nonhomogeneous curvature functions. Algebra i Analiz 1 (000), [5] K. Smoczyk: Symmetric hypersurfaces in Riemannian manifols contracting to Lie-groups by their mean curvature. Calc. Var. Partial Differential Equations 4 (1996), [6] K. Smoczyk: Starshape hypersurfaces an the mean curvature flow. Manuscripta Math. 95 (1998), [7] J. I. E. Urbas: On the expansion of starshape hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 05 (1990), Aress: Max Planck Institute for Mathematics in the Sciences (MPI MIS) Inselstr. -6, D Leipzig, Germany Oliver.Schnuerer@mis.mpg.e, Knut.Smoczyk@mis.mpg.e 3