Mean curvature flow in Fuchsian manifolds
aa r X i v : . [ m a t h . DG ] M a y MEAN CURVATURE FLOW IN FUCHSIAN MANIFOLDS
ZHENG HUANG, LONGZHI LIN, AND ZHOU ZHANG
Abstract.
Motivated by the goal of detecting minimal surfaces in hy-perbolic manifolds, we study geometric flows in complete hyperbolic3-manifolds. In general, the flows might develop singularities at somefinite time. In this paper, we investigate the mean curvature flow in aclass of complete hyperbolic 3-manifolds (Fuchsian manifolds) which arewarped products of a closed surface of genus at least two and R . Weshow that for a large class of closed initial surfaces, which are graphsover the totally geodesic surface Σ, the mean curvature flow exists for alltime and converges to Σ. This is among the first examples of convergingmean curvature flows starting from closed hypersurfaces in Riemannianmanifolds. We also provide calculations for the general warped productsetting which will be useful for further works. Introduction
The setting.
The mean curvature flow has been studied extensivelyin various ambient Riemannian manifolds and in most cases the flow ofclosed submanifolds develops singularities in finite time by the avoidanceprinciple for the mean curvature flow. For instance, we know that the fi-nite time singularity has to occur for any compact initial hypersurface inEuclidean space under the mean curvature flow ([
Hui84 ]). The study ofsingularity formation has been a focal point of the field, see for instance[
Hui86, HS09, CM12 ] and many others. Hyperbolic manifolds are knownto possess extremely rich geometric structures. Our motivation is to betterunderstand the mean curvature flow in hyperbolic manifolds, hoping that infurther studies we can detect interesting geometric objects by running themean curvature flow or similar flows to time infinity without developing anysingularity or after handling possible singularities.As a first step, we focus on the Fuchsian manifolds in this paper. Fuchsianmanifolds are probably the most elementary complete, non-simply connectedhyperbolic 3-manifolds. A Fuchsian manifold is obtained as a quotient spaceof H by a Fuchsian group. Let M be a Fuchsian manifold, and we alwaysassume the genus of any incompressible surface of M is at least two so Date : May 21, 2019.2010
Mathematics Subject Classification.
Primary 53C44, 57M10. that it carries its own hyperbolic metric. From differential geometry pointof view, it is a warped product of a hyperbolic surface Σ with R , with themetric ds = dr + cosh ( r ) g , where g is the hyperbolic metric on Σ. Therefore the surface Σ is totallygeodesic in M . Clearly it is the only such surface in M .Our main analytical tool is the mean curvature flow equation, which hasthe following form:(1.1) ∂∂t F ( x, t ) = − H ( x, t ) ν ( x, t ) ,F ( · ,
0) = F , where H ( x, t ) and ν ( x, t ) are the mean curvature and unit normal vectorrespectively at F ( x, t ) of the evolving surface S ( t ), and our convention ofthe mean curvature is the sum of the principal curvatures. Definition 1.1.
A smooth closed surface S in M is a graph over thetotally geodesic surface Σ if there is a constant c > such that the anglefunction Θ = h n , ν i ≥ c , where h· , ·i is the metric over M , n = ∂∂r isthe unit vector field over M which is perpendicular to Σ and ν is the unitnormal vector on S of our choice. Note that Θ ∈ (0 ,
1] if S is a graph, and Θ ≡ S is equidistant from Σ (sometimes called parallel to Σ, or a level surface to Σ).The mean curvature flow in warped product manifolds was also investi-gated by other authors, see e.g. [ BM12 ]. Note that the warped productstructures in [
BM12 ] are completely different from ours. Geometrically,their warped structure can be thought as a real line bundle over a surface,while ours is a surface bundle over the real line. So the evolving hypersur-faces in their case are equidistant graphs over a reference hypersurface, whileour evolving hypersurfaces are the more natural geodesic graphs (i.e. graphsover totally geodesic surfaces). One does not in general expect a geodesicgraph to stay geodesic graphs under the mean curvature flow. In fact, itwas mentioned in [
BM12 ] that in [
Unt98 ] Unterberger gave an exampleof hypersurface which is a geodesic graph but loses this graphical propertywhen it evolves under the mean curvature flow. On the other hand, we willsee in this work that there indeed exists a large class of closed initial surfaces S ’s in Fuchsian manifolds, as geodesic graphs over the totally geodesic sur-face Σ with an explicit lower bound on the angle of S , such that the meancurvature flow starting from S remains as geodesic graphs for all time andconverges smoothly to Σ. CF IN FUCHSIAN MANIFOLDS 3
Main Result.
In this paper, we prove that if the angle function onthe initial surface has a positive lower bound depending only on its maximaldistance to the reference surface Σ, then the mean curvature flow with suchan initial surface exists for all time and converges to the totally geodesicsurface Σ in a Fuchsian manifold( M , g M ) = (cid:0) R × cosh( r ) Σ , dr + cosh ( r ) g (cid:1) . More precisely, we have
Theorem 1.2.
Let M be a Fuchsian manifold and Σ the unique closedtotally geodesic surface in M . Then for any a > , if the initial smoothclosed surface S ⊂ M has hyperbolic distance no larger than a to Σ andthe minimum of the initial angle satisfies (1.2) min p ∈ S Θ ( p ) ≥ tanh( a ) , then the mean curvature flow with initial surface S exists for all time, re-mains as geodesic graph over Σ and converges continuously to Σ . Moreover,the convergence is smooth if the above inequality is strict. Remark 1.3.
Notably, in the special case where min p ∈ S Θ = 1 , namelythe initial surface S is equidistant from the totally geodesic surface Σ , theevolving surface S ( t ) remains equidistant from Σ , i.e. Θ( t ) = h n , ν i ( t ) ≡ for all t ≥ and converges to the totally geodesic surface Σ by the explicitsolution (3.2) , which is the prototype motivating our consideration. Theresult itself illustrates the interaction between geometric data of the flow andthe ambient space. Namely, the lower bound of the angle function Θ of S on the right hand side of (1.2) is nothing but the principle curvatureof the equidistant surface Σ( a ) , where a is the maximal distance of S to Σ . At this moment, it is more like a coincidence as the optimal choiceby our argument. We provide some discussion regarding the formation ofsingularities for the graph case in general in Section 4 and hope to sort outthe underlying connection in future works. Our techniques can be generalized to higher dimensional warped prod-uct manifolds of similar structure with appropriate variations of curvatureconditions.1.3.
Interaction between geometry and analysis.
We want to high-light the interaction between analytical methods and geometric structures.Our setting of Fuchsian manifolds allows us to take advantage of its hyper-bolic geometry in several stages of this work. It is a basic fact that the levelsets { (Σ( r ) , cosh ( r ) g ) } r ∈ R of the totally geodesic surface Σ = Σ(0) forman equidistant foliation of the Fuchsian manifold M . Moreover, each fiber ZHENG HUANG, LONGZHI LIN, AND ZHOU ZHANG of the foliation is umbilic, which enables us to obtain the mean curvatureflow with initial surface Σ( r ) an explicit solution for any fixed r ∈ R (see(3.2)). We use this special mean curvature flow as barriers and the avoid-ance principle for the mean curvature flows (see e.g. [ Hui86 ]) to push flowto the destination Σ (see § V = cosh( r ) ∂∂r (see (2.3)) in a Fuchsian manifold to deriveexplicitly the evolution equation for the angle Θ (see (3.6)).1.4. Outline of the paper.
We provide some preliminary results in § §
3. The scheme is thefollowing. We first show the evolving surface must stay in a bounded regionin M for all time (the Squeeze Lemma 3.1) as long as the flow exists, thenwe derive the evolution equation for the angle function Θ( · , t ) (Theorem 3.6),and then most of the work is devoted to prove that the evolving surfacesstay graphical under the initial condition on distance and angle. Note thatthe uniform positive lower bound of the angle function Θ = h n , ν i locallygives the uniform C -estimate of the graph function which represents theevolving surface. Therefore once we have established uniform bound for Θ,standard parabolic theory ([ LSU68 ]) gives bounds for all higher derivatives.In particular the second fundamental form for the evolving surface S t in M is uniformly bounded. Huisken’s theorem ([ Hui86 ]) then guarantees thatthe mean curvature flow exists for all time. The Squeeze Lemma 3.1 thengives the convergence of the flow. The smoothness will follow.Finally, in § Acknowledgements.
We are grateful to Hengyu Zhou and Biao Wangfor many useful discussions and helpful suggestions. We also thank thereferee for insightful comments which help to clarify several statements inthe paper. Z. Huang acknowledges support from U.S. NSF grants DMS1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representa-tion varieties” (the GEAR Network) and a grant from the Simons Foun-dation (
CF IN FUCHSIAN MANIFOLDS 5 Preliminaries
In this section, we fix our notations, and introduce some preliminary factsthat will be used throughout.2.1.
Fuchsian manifolds.
A Fuchsian manifold M is defined as a warpedproduct space R × Σ. This is a complete hyperbolic 3-manifold of funda-mental importance in hyperbolic geometry and Kleinian group theory. Themetric on a Fuchsian manifold M is explicitly given as:(2.1) ( M , g M ) = ( R × cosh( r ) Σ , dr + cosh ( r ) g ) , where g is the induced metric on the surface Σ which is hyperbolic. Afascinating feature of the geometry of the Fuchsian manifold M is that theequidistant surface Σ( r ) of the totally geodesic surface Σ(0) = Σ forms aglobal foliation of M , and each fiber surface Σ( r ) is umbilic, with constantprincipal curvature of tanh( r ), cf. [ O’N83 ].Another important fact which will be useful for us is the existence of thespecial vector field V in M (see for instance [ Bre13 ]). Namely,(2.2) V = cosh( r ) n satisfies that(2.3) ¯ ∇ X V = sinh( r ) X, for any smooth vector field X , and ¯ ∇ is the Levi-Civita covariant derivativein M . In the paper, notations with a overhead “-” stand for quantitiesfor the ambient manifold M . Making use of this V , one can deduce thefollowing well-known formula. Lemma 2.1.
Let X be any tangent vector field in a Fuchsian manifold M ,then we have (2.4) ¯ ∇ X n = tanh( r )( X − h X, n i n ) . Proof.
Since both sides of the identity are linear in X , we only have to verifythe identity by taking a local frame. Fix any point ¯ q ∈ M and let { ¯ e , ¯ e } bea local normal frame at the corresponding point q ∈ Σ, then { ¯ e , ¯ e , ¯ e = n } forms a local normal frame at ¯ q . Then (2.4) is obviously true for X = n .When X = ¯ e , we have ¯ ∇ ¯ e n = ¯Γ n + ¯Γ j ¯ e j . Since the metric on M is given explicitly as (2.1), its Christoffel symbols¯Γ kij can be computed explicitly. In our case, we have ¯Γ = 0 and ¯Γ j ¯ e j =tanh( r )¯ e j . One sees that (2.4) holds for X = ¯ e . Similarly we can verify thecase of X = ¯ e . ZHENG HUANG, LONGZHI LIN, AND ZHOU ZHANG
Mean curvature flow.
Let F : S → M be the immersion of anincompressible surface S in a Fuchsian manifold M . We assume that S = F ( S ) is a graph over Σ with respect to n , i.e. h n , ν i ≥ c >
0, here c is aconstant to be determined later.We consider a family of immersions of surfaces in M moving under themean curvature flow (1.1): F : S × [0 , T ) → M , ≤ T ≤ ∞ with ∂∂t F ( x, t ) = − H ( x, t ) ν ( x, t ), and F ( · ,
0) = F . For each t ∈ [0 , T ), S ( t ) = { F ( x, t ) ∈ M | x ∈ S } is the evolving surface at time t , and S (0) isthe initial surface S .The short-time existence of the solutions to (1.1) is standard for closedimmersions, see e.g. [ HP99 ]. Initial compact surface develops singularitiesin finite time along the mean curvature flow in Euclidean space, and in factthe norm of the second fundamental form blows up if the singularity occursin finite time, see [
Hui84, Hui86 ].2.3.
Evolution equations.
In this subsection, for completeness we collectand derive a number of evolution equations of some quantities on S ( t ), t ∈ [0 , T ), which are involved in our calculations. We include here standardevolution equations for the mean curvature H ( · , t ), and the square norm ofthe second fundamental form | A ( · , t ) | . Theorem 2.2.
Along the mean curvature flow, one has (cid:18) ∂∂t − ∆ (cid:19) H = H ( | A | − , (2.5) (cid:18) ∂∂t − ∆ (cid:19) | A | = − |∇ A | + 2 | A | + 4( | A | − H ) . (2.6) Proof.
These equations are deduced for general Riemannian manifolds in[
Hui86 ]. In our case of hyperbolic three-manifold, the ambient space M has constant sectional curvature −
1, and the Ricci curvature
Ric ( ν , ν ) = − ν .The lemma then follows from combining these explicit curvature condi-tions and curvature equations ¯ R i j = − g ij , as well as the well-known SimonsIdentity (see e.g. [ Sim68 ] or [
SSY75 ]), satisfied by the second fundamentalform a ij .Our estimates are mainly for the height function u ( x, t ) and the anglefunction , which is the cosine of the geometric angle, Θ( x, t ) on S ( t ): u ( x, t ) = r ( F ( x, t )) , (2.7) Θ( x, t ) = h ν , n i ( F ( x, t )) , (2.8) CF IN FUCHSIAN MANIFOLDS 7 where r ( p ) = ± dist( p, Σ) for all p ∈ M , the signed distance to the fixedreference surface Σ.We have Θ( x, t ) ∈ [0 ,
1] by the choice. It is clear that S ( t ) is a geodesicgraph over Σ if Θ > S ( t ). Θ( · ,
0) = Θ is for the initial surface. Theorem 2.3.
The evolution equations of u and Θ have the following form: ∂∂t u = − H Θ , (2.9) (cid:18) ∂∂t − ∆ (cid:19) Θ = ( | A | − n ( H n ) − H h∇ ν n , ν i (2.10) = ( | A | − ∇ ν L n g )( e i , e i ) − ( ¯ ∇ e i L n g )( ν , e i )(2.11) − a ij L n g ( e i , e j ) , where n ( H n ) is the variation of mean curvature of S ( t ) under the defor-mation vector field n and L n g is the Lie derivative of the metric g in thedirection n .Proof. The first equation is self-evident. The second equation can be foundin ([
Bar84 ], [
EH91 ]) in the Lorentzian setting. We include the proof forthe Riemannian setting in the Appendix for the sake of completeness.
Remark 2.4.
As we will see in the Appendix, the equation for Θ in Theo-rem 2.3 is quite general, and thus very difficult to work with, especially theterm n ( H n ) . One of the key observations in our work is that we can takeadvantage of the explicit nature of the ambient warped product metric andderive more workable equations in our case (see (3.10) ). Proof of Main Theorem
We prove the main theorem in this section. In § S stays in a compactregion in M as long as it exists. This is standard C -estimate using theavoidance principle for the mean curvature flow. In § · , t ). In § § The squeeze.
The following theorem is probably known previously,but we include the proof here for the sake of completeness.
ZHENG HUANG, LONGZHI LIN, AND ZHOU ZHANG
Theorem 3.1.
Let S be as in Theorem 1.2, then as long as the flow existswe have (3.1) − sinh − ( e − t sinh( a )) ≤ u ( · , t ) ≤ sinh − ( e − t sinh( a )) . Proof.
It follows from basic hyperbolic geometry that, if we denote Σ( r )(resp. Σ( − r )) the parallel surface equidistant r to Σ(0) = Σ which staysin the positive (resp. negative) side of Σ, then Σ( r ) (resp. Σ( − r )) is anumbilic surface of constant principal curvature tanh( r ) (resp. − tanh( r )).Now we consider the mean curvature flow with initial surface Σ( a ) suchthat a ≥ dist( x, Σ) for any x ∈ S . By the well known uniqueness of theflow, the mean curvature flow equation is reduced to the following ODE of R ( t ), the r -value of the evolving equidistant surface: dRdt = − R ) , with initial condition R (0) = a , which yields an explicit solution:(3.2) R ( t ) = sinh − ( e − t sinh( a )) . Similar calculations hold for Σ( − a ). One sees that such mean curvatureflow exists for all time, and all evolving surfaces are umbilic and convergeto Σ as t → ∞ .Now by assumption, the initial surface S lies between umbilic slices Σ( a )and Σ( − a ), the conclusion then follows from the avoidance principle for themean curvature flow.Next we derive a general equation for ∆ u . Lemma 3.2.
Let S ⊂ M be a closed surface that is a geodesic graph over Σ , and u ( x ) is the signed distance of x ∈ S to Σ . Then we have: (3.3) ∆ u = tanh( u )(1 + Θ ) − H Θ , where ∆ is the Laplace operator on S with respect to the induced metric.Proof. For any point x ∈ S , choose { e , e } (with e = ν ) to be a localnormal frame of S at x . Then at x we can compute∆ u = X i =1 ∇ e i ∇ e i u = X i =1 ∇ e i h n , e i i = X i =1 ( h ¯ ∇ e i n , e i i + h n , ¯ ∇ e i e i i ) CF IN FUCHSIAN MANIFOLDS 9 = X i =1 h tanh( u )( e i − h n , e i i n ) , e i i + X i =1 h n , ¯ ∇ e i e i i = 2 tanh( u ) − tanh( u )(1 − Θ ) − H Θ= tanh( u )(1 + Θ ) − H Θ , (3.4)where we have used Lemma 2.1. Remark 3.3.
Combining with (2.9) , we have the evolution equation for thehyperbolic distance function u of S ( t ) along the mean curvature flow: (3.5) u t − ∆ u = − tanh( u )(1 + Θ ) , which yields similar decay of u as in Theorem 3.1. Evolution equation for the angle function.
In this subsection, wetake advantage of the presence of a special vector field V = cosh( r ) n (see(2.3)) to derive the evolution equation for the angle function Θ( · , t ). Recallthat, on the evolving surface S ( t ), it is given by Θ( · , t ) = h n , ν i ( · , t ). Wefind the following function more convenient to work with in our hyperbolicsetting:(3.6) η ( · , t ) = cosh( u )Θ( · , t ) = h V, ν i . Lemma 3.4.
The function η ( · , t ) on the evolving surface S ( t ) satisfies thefollowing equation: (3.7) ∆ η = sinh( u ) H − | A | η + h V, ∇ H i . Here ∆ is the Laplace operator on S ( t ) with respect to the induced metric.Proof. For any point p ∈ S ( t ), we choose { e , e } (with e = ν ) to be a localnormal frame at p . Then at p we have∆ η = X i =1 ∇ e i ∇ e i h V, ν i = X i =1 h ¯ ∇ e i ¯ ∇ e i V, ν i + 2 h ¯ ∇ e i V, ¯ ∇ e i ν i + h V, ¯ ∇ e i ¯ ∇ e i ν i = X i =1 h ¯ ∇ e i (sinh( u ) e i ) , ν i + 2 X i,k =1 sinh( u ) h e i , a ik e k i + X i,k =1 h V, ¯ ∇ e i ( a ik e k ) i = − sinh( u ) H + 2 sinh( u ) H + X i,k =1 a ik h V, ¯ ∇ e i e k i + a ik,i h V, e k i = sinh( u ) H − X i,k =1 a ik η + a ii,k h V, e k i = sinh( u ) H − | A | η + h V, ∇ H i , where in the second to the last equality we have used the Codazzi equationand the fact that Fuchsian manifold M is of constant curvature − S ( t ). Lemma 3.5.
With the above notations, we have: ∆Θ = h∇ H, n i − | A | Θ + tanh( u )(1 + Θ ) H − Θ(1 − Θ )cosh ( u ) − u ) h∇ Θ , n i − ( u )Θ . (3.8) Proof.
A direct calculation yields∆ η = ∆(cosh( u )Θ)= cosh( u )∆Θ + 2 sinh( u ) h∇ Θ , n i + Θ(sinh( u )∆ u + cosh( u ) |∇ u | ) . (3.9)Isolating ∆Θ, we have∆Θ = ∆ η cosh( u ) − u ) h∇ Θ , n i − Θ tanh( u )∆ u − Θ(1 − Θ )= tanh( u ) H − | A | Θ + h∇ H, n i − u ) h∇ Θ , n i− tanh ( u )Θ(1 + Θ ) + tanh( u ) H Θ − Θ(1 − Θ ) . Here we have used the fact that |∇ u | = 1 − Θ . Now standard hyperbolictrigonometric identities give − Θ(1 − Θ ) − tanh ( u )Θ(1 + Θ ) = − Θ(1 − Θ )cosh ( u ) − ( u )Θ . This completes the proof.We can now derive the evolution equation for Θ along the flow.
Theorem 3.6.
With the above notations, we have: ∂ Θ ∂t − ∆Θ = | A | Θ − u ) H + 2 tanh( u ) h∇ Θ , n i + Θ(1 − Θ )cosh ( u ) + 2 tanh ( u )Θ . (3.10) Proof.
Recall that for the mean curvature flow we have ∂∂t ν = ∇ H , andgeometrically, one can view ∂∂t as the spatial covariant derivative − H ν here.Therefore ∂ Θ ∂t = ∂∂t h n , ν i = (cid:28) ∂∂t ν , n (cid:29) + h ν , ¯ ∇ − H ν n i = h∇ H, n i − H h ν , ¯ ∇ ν n i . CF IN FUCHSIAN MANIFOLDS 11
Using (2.4) we have ¯ ∇ ν n = tanh( u )( ν − Θ n ) , and thus(3.11) ∂ Θ ∂t = h∇ H, n i − H tanh( u )(1 − Θ ) . Now the conclusion follows from Lemma 3.5.3.3.
Preserving graphical property.
In this subsection, we discuss thepreservation of the graphical property. We formulate the a priori estimateon the angle Θ as follows.
Theorem 3.7.
Let M be a Fuchsian manifold and S be a smooth closedsurface which is a geodesic graph over the unique totally geodesic surface Σ in M , and suppose there is a positive constant a such that S lies entirely be-tween Σ( ± a ) . Then whenever the initial surface S satisfies Θ ≥ tanh( a ) ,the mean curvature flow with initial surface S remains as geodesic graphover Σ , namely Θ( · , t ) > as long as the flow exists.Proof. We have − a ≤ u ( x, ≤ a for any x ∈ S . By Theorem 3.1, wehave:(3.12) | u ( x, t ) | ≤ sinh − ( e − t sinh( a )) . We want to find a positive lower bound for Θ at the initial time to guar-antee a positive lower bound for all time and hence the convergence.It is equivalent to work with the function α = Θ whose evolution equationis slightly more convenient to work with. With the evolution equation (3.10)for Θ, we easily deduce the evolution equation for α : ∂α∂t − ∆ α = 2 | A | α − u ) H Θ + 4 tanh( u )Θ h∇ Θ , n i + 2 α (1 − α )cosh ( u ) + 4 tanh ( u ) α − |∇ Θ | . (3.13)Let φ ( t ) = min S ( t ) α and we only need to consider the case of φ ∈ (0 ,
1) in search of a prioriestimate. At the spatial minimum point of α , we have ∇ Θ = 0 and ∆ α ≥ t > dφdt ≥ ∂α∂t − ∆ α = 2 | A | α − u ) H Θ + 2 α (1 − α )cosh ( u ) + 4 tanh ( u ) α ≥ | A | Θ − √ | u | ) | A | Θ + 2 α (1 − α )cosh ( u ) + 4 tanh ( u ) α = 2( | A | Θ − √ | u | )) − ( u ) + 2 α (1 − α )cosh ( u ) + 4 tanh ( u ) α ≥ − − α ) tanh ( u ) = − − φ ) tanh ( u ) . Combining with | u ( x, t ) | ≤ sinh − ( e − t sinh( a )), we end up with thefollowing ordinary differential inequality(3.14) (cid:18) − φ (cid:19) dφdt ≥ − e − t sinh ( a )1 + e − t sinh ( a ) φ (0) = φ , where by assumption we have(3.15) 1 > φ = min S α ≥ tanh ( a ) = sinh ( a )1 + sinh ( a ) . With a proper choice of ǫ ∈ [0 , φ = ǫ + sinh ( a )1 + sinh ( a ) , where ǫ > φ ( t ) ≥ ǫ + sinh ( a ) e − t ( a ) e − t > ǫ for all t ≥ t ∈ [0 , ∞ ) :(3.18) Θ ( · , t ) > ǫ ≥ , which provides a positive lower bound for the angle Θ in any finite timeinterval, and the evolving surface remains as geodesic graph over Σ as longas it exists. This completes our proof.3.4. Long-time existence and convergence.
Now we can re-assemblethe ingredients and complete the proof of Theorem 1.2.
Proof. (of Theorem 1.2) We have shown in Theorem 3.7 that the mean cur-vature flow (1.1) stays graphical as long as it exists. This provides thegradient estimate for the mean curvature flow for any finite time interval.By the classical theory of parabolic equations in divergent form (for instance[
LSU68 ]), the higher regularity and a priori estimates of the solution followimmediately. This yields the long time existence of the flow by Huisken([
Hui86 ]). Then by Theorem 3.1 and the avoidance principle, the con-tinuous convergence of the flow also follows. When the inequality in theassumption is strict, the proof of Theorem 3.7 gives the uniform estimate of
CF IN FUCHSIAN MANIFOLDS 13 the angle for all time and so the higher order estimates are also uniform forall time, which provides the smooth convergence. The proof of Theorem 1.2is completed. 4.
Remarks on the General Case
In this section, we discuss the general situation after removing the as-sumption on the angle function Θ of the initial graph in Theorem 1.2 andillustrate the relation with possible formation of singularities. In light ofthe evolution equation (3.13), in order to rule out singularities, it will beenough to get a proper bound of H at the point where Θ takes the spatialminimum in (0 , M , we have the following calculation ofits gradient over the surface in general: ∇ Θ = ∇h n , ν i = h ¯ ∇ e i ν , n i e i + h ¯ ∇ e i n , ν i e i = h a ij e j , n i e i + h tanh( u )( e i − h e i , n i n ) , ν i e i = a ij h e j , n i e i − tanh( u ) h e i , n i Θ e i (4.1)where { e , e } is any orthonormal frame at the point of interest on thesurface, and we have used Lemma 2.1 in the second to the last step. Bychoosing e and e to be the two principal directions so that the secondfundamental form is ( a ij ) = diag { a, b } , we have ∇ Θ = ( a − tanh( u )Θ) h e , n i e + ( b − tanh( u )Θ) h e , n i e . (4.2)Notice that the spatial maximum of Θ is clearly 1 and obtained at thepoints with extremal height, where both e i ’s are perpendicular to n . Alsorecall |∇ u | = h e , n i + h e , n i = 1 − Θ . Meanwhile, at the spatialminimal point of Θ, denoted by θ and assumed to be in (0 , e and e is perpendicular to n . Pick a tangent vectorof the surface S , ˆ e perpendicular to n , which is unique up to sign as itis also perpendicular to ν . Then take ˆ e accordingly so that they forman orthonormal frame of the tangent space of S . Use ˆ a ij to denote thecoefficients for second fundamental form with respect to this basis. Applying(4.1), we have ˆ e (Θ) = a h ˆ e , n i , which implies a = 0 since h ˆ e , n i = 1 − θ >
0. So such chosen ˆ e and ˆ e are also principal directions and can be taken as e and e as above. So we can have h e , n i = 0. By (4.2), we have b = tanh( L ) θ where L is the heightof the spatial minimal point under consideration.Since the flow starts with a graph, singularities can occur only when thereis no longer positive lower bound for Θ by the discussion at the end of Section3. Thus by defining T = sup { t ∈ (0 , ∞ ) | θ ≥ C in [0 , t ] for some C > } ∈ (0 , ∞ ] , we know the flow exists as graph exactly in [0 , T ). In the following, we focuson the case of T < ∞ , i.e. the flow fails to be graphical in finite time.In this case, we can choose a time sequence { t i } ∞ i =1 approaching T as i →∞ , such that θ ( t i ) → i → ∞ , i.e. h n , ν i → p i for Θ. Now we analyze p i ∈ S ( t i ) at time t i more carefully. For simplicityof notations, we frequently omit the index i below, and the limit is alwaystaken as i → ∞ . We already know that there is one principal direction e such that h e , n i = 0 at the spatial minimal point. Geometrically, e is thedirection of the curve as the intersection of the evolving graph S ( t ) and theequidistance graph Σ( L ) where L is the height of the spatial minimal point.As h e , n i + h e , n i = 1 − θ , we have h e , n i →
1, i.e. e → n by reversing e if necessary. Consider the geodesics on S ( t i ) starting at p i in the directionof e . Since at p i we have h ¯ ∇ e e , ν i = −h ¯ ∇ e ν , e i = − b = − tanh( L ) θ ,which approaches 0 by the decay of height and θ ( t i ) →
0, we have ¯ ∇ e e → e also stands for the unit tangent vector fieldalong the geodesic. Together with e → n , we know that in the infinitesimalway at p i , this geodesic on S ( t i ) approaches the r -curve which is geodesicof M in the direction n . Intuitively, this is the consequence of the loss ofgraphical property.Meanwhile, the “reason” for the loss of graphical property should be the“relative” blow-up of the principle curvature in the e direction, namely thequantity a in (4.2). We hope to illustrate this point in the following. Using(3.10) and b = tanh( L ) θ , we have(4.3) dθdt ≥ (cid:0) a + tanh ( L ) θ (cid:1) θ − a tanh( L ) + θ (1 − θ )cosh ( L )If | a | ≤ − θ log θ + Cθ for some C > dθdt ≥ Cθ log θ − Cθ. (4.4)Direct calculation yields that θ ≥ Ce − Ce Ct >
0, which rules out the loss ofgraphical property in any finite time and we have the long time existencetogether with the continuous convergence. Motivated by this, for the case
CF IN FUCHSIAN MANIFOLDS 15 of T < ∞ and some C >
0, we set(4.5) I = { t ∈ [0 , T ) | | a | > − θ log θ + Cθ at all spatial minimal points of Θ( · , t ) } . Note that I has the closure in R containing T , since otherwise we can derive apositive lower bound for θ for any finite time interval as above, contradicting T < ∞ . So we can choose the time sequence { t i } ∞ i =1 ⊂ I approaching T as i → ∞ and consider at the point q i where Θ achieves the spatial minimumon S ( t i ) and | a | > − θ log θ + Cθ . We can further make sure that θ → θ will have a uniform positive lowerbound for t ∈ I close to T , and θ will then have a uniform positive lowerbound for t ∈ [0 , T ) \ I by applying the above argument for the interior of[0 , T ) \ I , contradicting the choice of T . Here we make use of the continuityof θ with respect to time.Since θ →
0, the scale of b = tanh( L ) θ is small and way smaller thanthat of a , i.e. “relative” blow-up. In other words, after a proper “blowing-down” (by the scale of θ ( − log θ ) / , for example), we have the violation ofgraphical property modelled as a cylinder with the circle on the equidistantsurface Σ( L ) and pointing in the n direction in the infinitesimal way.By the discussion at the end of Section 3, the blow-up of | A | , i.e. formationof flow singularities as surface, can’t occur before the degeneration of Θ andcertainly might not happen at the same place. This adds to the intriguingfeatures about the singularities. In future works, we hope to provide moreprecise local and global understanding for such singularities, aiming at eitherruling them out or obtaining interesting examples.5. Appendix
In this appendix, we give a detailed proof for the evolution equation forthe angle function in Theorem 2.3, i.e. the equations (2.10) and (2.11)for our Riemannian setting. The calculation is carried out for the meancurvature flow of graphical hypersurfaces of general dimension n , in theambient manifold M n +1 with a general warped product metric.We still use n and ν to denote the unit normal vectors for the warpedproduct foliation and the evolving hypersurface. Also we sum over all re-peated indices in this section.The following computation is done for F ( p, t ) for time t . We choosethe normal frame { e i } ni =1 for the evolving hypersurface. Then we require L n e i = [ n , e i ] = 0 to extend the frame to a neighborhood of F ( p, t ) in M n +1 , i.e. using n to generate a family of hypersurfaces with the initial onebeing the evolving hypersurface at time t . Then the vector field ν belowmeans the normal vector field for this family of hypersurfaces. This won’t affect the result for ∆Θ at F ( p, t ) for time t .∆Θ = e i e i h ν , n i = e i (cid:0) h ν , ¯ ∇ e i n i + h ¯ ∇ e i ν , n i (cid:1) = e i (cid:0) h ν , ¯ ∇ n e i i + h ¯ ∇ e i ν , e j i · h n , e j i (cid:1) = e i (cid:0) h ν , ¯ ∇ n e i i + h ¯ ∇ e j ν , e i i · h n , e j i (cid:1) = h ¯ ∇ e i ν , ¯ ∇ n e i i + h ν , ¯ ∇ e i ¯ ∇ n e i i + h ¯ ∇ e i ¯ ∇ e j ν , e i i · h n , e j i + h ¯ ∇ e j ν , ¯ ∇ e i e i i · h n , e j i + h ¯ ∇ e j ν , e i i · h ¯ ∇ e i n , e j i + h ¯ ∇ e j ν , e i i · h n , ¯ ∇ e i e j i = h ¯ ∇ e i ν , ¯ ∇ n e i i + h ν , ¯ ∇ e i ¯ ∇ n e i i + h ¯ ∇ e i ¯ ∇ e j ν , e i i · h n , e j i + h ¯ ∇ e j ν , e i i · h ¯ ∇ e i n , e j i + h ¯ ∇ e j ν , e i i · h n , ¯ ∇ e i e j i , where we have used ¯ ∇ e i n = ¯ ∇ n e i for the third equality, a ij = h ¯ ∇ e i ν , e j i = h ¯ ∇ e j ν , e i i for the fourth equality and ¯ ∇ e i e i = − a ii ν = − H ν for the lastequality. For these terms, we have h ν , ¯ ∇ e i ¯ ∇ n e i i = h R ( e i , n ) e i , ν i + h ν , ¯ ∇ n ¯ ∇ e i e i i , where R is the Riemannian curvature tensor, and we also have e j ( H ) = e j h ¯ ∇ e i ν , e i i = h ¯ ∇ e j ¯ ∇ e i ν , e i i + h ¯ ∇ e i ν , ¯ ∇ e j e i i = −h R ( e i , e j ) ν , e i i + h ¯ ∇ e i ¯ ∇ e j ν , e i i , using [ n , e i ] = 0, ¯ ∇ e i e j = ¯ ∇ e j e i = − a ij ν and [ e i , e j ] = 0. We also find: h ¯ ∇ e i ν , ¯ ∇ n e i i = h ¯ ∇ e i ν , e j i · h ¯ ∇ n e i , e j i = h ¯ ∇ e j ν , e i i · h ¯ ∇ e i n , e j i = h ¯ ∇ n e i , e j i a ij = a ij n ( h e i , e j i ) . So the previous computation for ∆Θ can be continued as follows:∆Θ = h ¯ ∇ e i ν , ¯ ∇ n e i i + h ν , ¯ ∇ e i ¯ ∇ n e i i + h ¯ ∇ e i ¯ ∇ e j ν , e i i · h n , e j i + h ¯ ∇ e j ν , e i i · h ¯ ∇ e i n , e j i + h ¯ ∇ e j ν , e i i · h n , ¯ ∇ e i e j i = a ij n ( h e i , e j i ) + h R ( e i , n ) e i , ν i + h ν , ¯ ∇ n ¯ ∇ e i e i i + h n , e j i · ( e j ( H ) + h R ( e i , e j ) ν , e i i ) + h ¯ ∇ e j ν , e i i · h n , ¯ ∇ e i e j i . We consider each term separately below: a ij n ( h e i , e j i ) = − a ij n ( g ij ) , h R ( e i , n ) e i , ν i = − Ric( n , ν ) , h ν , ¯ ∇ n ¯ ∇ e i e i i = n (cid:0) h ν , ¯ ∇ e i e i i (cid:1) , h n , e j i · e j ( H ) = h n , ∇ H i , CF IN FUCHSIAN MANIFOLDS 17 h n , e j i · h R ( e i , e j ) ν , e i i = Ric( n l , ν ) , h ¯ ∇ e j ν , e i i · h n , ¯ ∇ e i e j i = a ij · h n , − a ij ν i = − X i,j | a ij | · h n , ν i , where ¯ ∇ e i e i = − a ii ν = − H ν is used for the third one, ( g ij ) is the inversematrix of ( g ij = h e i , e j i ) and n l is the projection of n in the direction of theevolving hypersurface. Remark 5.1.
The equality h ν , ¯ ∇ e i e i i = − H holds only at F ( p, t ) for time t , and so n (cid:0) h ν , ¯ ∇ e i e i i (cid:1) is NOT equal to − n ( H ) . Nevertheless, we still have h ν , ¯ ∇ e i e i i = −h ¯ ∇ e i ν , e i i by the construction of ν at the beginning of thisappendix. Now we can finish the computation for ∆Θ:∆Θ = − a ij n ( g ij ) + n (cid:0) h ν , ¯ ∇ e i e i i (cid:1) − h n , ν i · Ric( ν , ν )+ h n , ∇ H i − X i,j | a ij | · h n , ν i = − n ( H n ) + (Ric( ν , ν ) − X i,j | a ij | )Θ + h n , ∇ H i , where n ( H n ) = a ij n ( g ij ) − n ( h ν , ¯ ∇ e i e i i ).Next we further clarify the term n ( H n ). It stands for the variation ofmean curvature for the family of hypersurfaces starting with the evolvinghypersurface under consideration at time t and flowing out by the vectorfield n . This is the same family of hypersurfaces considered in the previouscalculation, and we are only interested in the initial hypersurface which isthe hypersurface evolving along the mean curvature flow at time t . Clearly,we have n ( H n ) = n ( g ij a ij ) = g ij n ( a ij ) + a ij n ( g ij ) = n ( a ii ) + a ij n ( g ij )at the point since ( g ij ) is the identity matrix at the point under consider-ation, and a ii is not equal to H nearby. The computation for n ( H n ) is as follows, still for just that point. n ( H n ) = n ( a ii ) + a ij n ( g ij )= n (cid:0) h ¯ ∇ e i ν , e i i (cid:1) − a ij n ( h e i , e j i )= h ¯ ∇ n ¯ ∇ e i ν , e i i + h ¯ ∇ e i ν , ¯ ∇ n e i i − a ij L n g ( e i , e j )= − a ij ( L n g )( e i , e j ) + e i (cid:0) h ν , ¯ ∇ n e i i + h e i , ¯ ∇ n ν i (cid:1) − h ¯ ∇ e i e i , ¯ ∇ n ν i− h ν , ¯ ∇ e i ¯ ∇ n e i i − h e i , ¯ ∇ e i ¯ ∇ n ν i + h e i , ¯ ∇ n ¯ ∇ e i ν i = − a ij ( L n g )( e i , e j ) − h ν , ¯ ∇ e i ¯ ∇ n e i i + h R ( n , e i ) ν , e i i = − a ij ( L n g )( e i , e j ) − e i (cid:0) h ν , ¯ ∇ n e i i (cid:1) + h ¯ ∇ e i ν , ¯ ∇ n e i i + h R ( ν , e i ) n , e i i = − a ij ( L n g )( e i , e j ) − e i (cid:0) h ν , ¯ ∇ n e i i (cid:1) + h ¯ ∇ e i ν , ¯ ∇ n e i i + h ¯ ∇ ν ¯ ∇ e i n , e i i − h ¯ ∇ e i ¯ ∇ ν n , e i i − h ¯ ∇ ¯ ∇ ν e i n , e i i + h ¯ ∇ ¯ ∇ ei ν n , e i i , where n ( g ij ) = − g ik n ( g kℓ ) g ℓj and ( g ij ) being identity at the point are usedfor the first equality; [ n , e i ] = 0 is used for the third equality; h e i , ν i = 0, | ν | = 1 and at F ( p, t ), ¯ ∇ e i e i = − a ii ν = − H ν are used for the fifth equality.We have a few more terms to sort out. − a ij L n g ( e i , e j ) = −h e i , ¯ ∇ e j ν i (cid:0) h ¯ ∇ e i n , e j i + h ¯ ∇ e j n , e i i (cid:1) = −h ¯ ∇ ¯ ∇ ej ν n , e j i − h ¯ ∇ e j n , ¯ ∇ e j ν i , ν ( L n g ( e i , e i )) = ν (cid:0) h ¯ ∇ e i n , e i i (cid:1) = h ¯ ∇ ν ¯ ∇ e i n , e i i + h ¯ ∇ e i n , ¯ ∇ ν e i i , − L n g ( ¯ ∇ ν e i , e i ) = −h ¯ ∇ ¯ ∇ ν e i n , e i i − h ¯ ∇ e i n , ¯ ∇ ν e i i , − e i ( L n g ( ν , e i )) = − e i (cid:0) h ¯ ∇ ν n , e i i + h ¯ ∇ e i n , ν i (cid:1) = −h ¯ ∇ e i ¯ ∇ ν n , e i i − h ¯ ∇ ν n , ¯ ∇ e i e i i − e i (cid:0) h ¯ ∇ n e i , ν i (cid:1) . Now it is easy to calculate: n ( H n ) = 12 ν ( L n g ( e i , e i )) − L n g ( ¯ ∇ ν e i , e i ) − e i ( L n g ( ν , e i )) + h ¯ ∇ ν n , ¯ ∇ e i e i i = 12 ( ¯ ∇ ν L n g )( e i , e i ) − e i ( L n g ( ν , e i )) + h ¯ ∇ ν n , ¯ ∇ e i e i i = 12 ( ¯ ∇ ν L n g )( e i , e i ) + h ¯ ∇ ν n , ¯ ∇ e i e i i− ( ¯ ∇ e i L n g )( ν , e i ) − L n g ( ¯ ∇ e i ν , e i ) − L n g ( ν , ¯ ∇ e i e i )= 12 ( ¯ ∇ ν L n g )( e i , e i ) + h ¯ ∇ ν n , ¯ ∇ e i e i i− ( ¯ ∇ e i L n g )( ν , e i ) − L n g ( a ij e j , e i ) − L n g ( ν , − H ν ) . In light of h ¯ ∇ ν n , ¯ ∇ e i e i i = − H h ¯ ∇ ν n , ν i = − HL n g ( ν , ν ), we conclude n ( H n ) = 12 ( ¯ ∇ ν L n g )( e i , e i ) − ( ¯ ∇ e i L n g )( ν , e i ) − a ij L n g ( e i , e j )+ 12 HL n g ( ν , ν ) . CF IN FUCHSIAN MANIFOLDS 19
Noticing h ν , ¯ ∇ ν n i = L n g ( ν , ν ), we have n ( H n ) = 12 ( ¯ ∇ ν L n g )( e i , e i ) − ( ¯ ∇ e i L n g )( ν , e i ) − a ij L n g ( e i , e j ) + H h ν , ¯ ∇ ν n i . We note that the advantage of computing this way is that the terms nowdepend mostly on the evolving hypersurface. The vector field n only appearsin L n g . In the following, we compute ∂ Θ ∂t in detail. We use the coordinatesystem used in [ Hui86 ], i.e. a normal coordinate system { y α } for F ( p, t ) in M with the frame vector for the first coordinate is − ν at time t .Let ν = ν α ∂∂y α and n = n α ∂∂y α . We have ∂ Θ ∂t = ∂ ( g αβ ν α n β ) ∂t . There arethree terms from Leibniz rule. ∂g αβ ∂t = ∂∂t (cid:18)(cid:28) ∂∂y α , ∂∂y β (cid:29)(cid:19) = − H ν (cid:18)(cid:28) ∂∂y α , ∂∂y β (cid:29)(cid:19) = 0 , because the Christoffel symbols vanish at the point. Define ∂ ν ∂t to be ∂ν α ∂t ∂∂y α and ∂ n ∂t to be ∂n α ∂t ∂∂y α , and we see ∂ Θ ∂t = h ∂ ν ∂t , n i + h ν , ∂ n ∂t i . We have ∂ ν ∂t = ∇ H , and ∂ n ∂t = ∂n α ∂t ∂∂y α = − H ν ( n α ) ∂∂y α = − H ¯ ∇ ν n , where the last equality is true again by the choice of { y α } .Finally, we can conclude (2.10) and (2.11). References [Bar84] Robert Bartnik,
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Department of Mathematics, The City University of New York,Staten Island, NY 10314, USAThe Graduate Center, The City University of New York, 365 Fifth Ave.,New York, NY 10016, USA
E-mail address : [email protected] (L. L.) Mathematics Department, University of California, Santa Cruz, 1156High Street, Santa Cruz, CA 95064, USA
E-mail address : [email protected] (Z. Z.) School of Mathematics and Statistics, The University of Sydney,NSW 2006, Australia
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