Convexity and the Dirichlet problem of translating mean curvature flows
aa r X i v : . [ m a t h . DG ] D ec CONVEXITY AND THE DIRICHLET PROBLEM OFTRANSLATING MEAN CURVATURE FLOWS
LI MAA bstract . In this work, we propose a new evolving geometric flow (calledtranslating mean curvature flow) for the translating solitons of hyper-surfaces in R n + . We study the basic properties, such as positivity pre-serving property, of the translating mean curvature flow. The Dirichletproblem for the graphical translating mean curvature flow is studied andthe global existence of the flow and the convergence property are alsoconsidered. Mathematics Subject Classification (2000): 35J60, 53C21, 58J05
Keywords : mean curvature flow, translating solitons, convexity, themaximum principle
1. I ntroduction
In this note, we propose a new evolving flow (called translating meancurvature flow) for the translating solitons of hypersurfaces in R n + . Thisflow is a modification of mean curvature flow with a translation by a fixedvector. We study the basic properties of the translating mean curvature flow.The Dirichlet problem for the graphical translating mean curvature flow isstudied and the global existence of the flow and the convergence propertyare also presented. This work can be considered as a continuation of ourpaper [10].We propose the translating mean curvature flow in the following way.Given a fixed nonzero vector V ∈ R n + . The translating mean curvatureflow for translating soliton is defined as a one parameter family of properlyimmersed hypersurface M t = X ( Σ , t ), where 0 < t < T and X : Σ × [0 , T ) → R n + evolved by the evolution equation(1) X t = ¯ H ( X ) + V N , t > H ( X ) is the mean curvature vector of the hypersurface M t at the po-sition vector X and V N is the normal component of the vector V . We denote V T = V − V N the tangential part of the vector V . Recall that for the outerunit normal ν : = ν ( · , t ) on M t , the mean curvature is defined by H = div ( ν ) Date : May 26th, 2016. ∗ The research is partially supported by the National Natural Science Foundation ofChina (No. 11271111) and SRFDP 20090002110019. and the mean curvature vector is ¯ H = − H ν . Let ( e j ) be a local orthonormalframe on M t . Let a i j = < D ei e j , ν > and let A = ( a i j ) be the second funda-mental form on M t . Then H = − P a j j . Define ¯ X = X − tV . Then we havethe mean curvature ¯ X t = ¯ H ( ¯ X ) , t > X . Therefore, many geometric proper-ties such as convexity, mean convexity, are preserved by the flows. How-ever, the global behaviors of two flows X ( t ) and ¯ X ( t ) are di ff erent. Hencethe flows (1) need to be considered independently.Applying the maximum principle (and Hamilton’s tensor maximum prin-ciple) to derived evolution equations from (1) we obtain the following re-sult. Theorem 1.
Given a translating mean curvature flow M t with bounded sec-ond fundamental form A, t ∈ [0 , T ) with T > .(1). (i). If < V , ν > ≥ on the initial hypersurface M , then < V , ν > ≥ on the hypersurface M t for any t > . Similarly, assume that H ≥ on theinitial hypersurface M . Then H ≥ on the hypersurface M t for any t > .(ii). Assume that A ≥ on the initial hypersurface M . Then A ≥ on thehypersurface M t for any t > .(2). Assume that for some constant β , H − β < V , ν > ≤ on the initialhypersurface M and H − β < V , ν >< at at least point in M . ThenH − β < V , ν >< on M t for t > .(3). If we further assume A + β < V ,ν> n g ≥ at the initial hypersurface M and A + β < V ,ν> n g > at at least one point p ∈ M , we have A + β < V ,ν> n g > on M t for t > . To derive this result, we shall do computations as in [3]. As we havepointed out above, the property (1) can be derived from the mean curvatureflow. For completeness, we give a full proof. Related Harnack inequalitiesfor translating mean curvature flow similar to results in [5] may be the same.One example for hypersurfaces with H − < V , ν >< u ( x ) = λ | x | , where x ∈ R n , n ≥ λ = V = − e n + = (0 , ... , − Du ( x ) = x , v = p + | x | , ν = ( − x , / v , < ν, V > = − / v , and − H = div ( x p + | x | ) = v ( n + ( n − | x | + | x | ) > v . One can compute that for λ > H − < V , ν >> RANSLATING MEAN CURVATURE FLOWS 3 of Monge-Ampere equations. B.White [13] has given a geometric measuretheory argument for the existence of minimizers of the weighted area Z e − λ x n + dA ( x )amongst integral currents over the mean convex domain. Namely, letting W be a bounded domain in R n with piecewise smooth mean convex boundaryand letting Γ be a smooth closed ( n −
1) manifold in ∂ W × R that is a graph-like. Then he has used the global defined radially symmetric solitons y = ϕ ( x ) as barriers for the minimizing process of integral currents which lie inthe region R defined by R = { ( x , y ) ∈ ¯ W × R ; b ≤ y ≤ ϕ ( x ) } where b = inf { y ; ( x , y ) ∈ S } . We remark that his region R (in the proof ofTheorem 10 in [13]) may be replaced by the regionˇ R = { ( x , y ) ∈ ¯ W × R ; ϕ ( x ) − C ≤ y ≤ ϕ ( x ) } for suitable constant C >
0. The choice of the lower barrier ϕ ( x ) − C is nicein the sense that it is a sub-solution to the mean curvature soliton equation.One may get the minimizers by using BV functions. Our approach for theexistence of translating solitons with the Dirichlet boundary condition onconvex domains is the heat flow method. That is, we propose the translatingmean curvature flow to get the solitons as the limits. The uniqueness andconvexity of the translating solitons with convex boundary data φ remain asopen questions.The Dirichlet problem for the graphical mean curvature flow on meanconvex domains has been studied by G.Huisken [6] and Lieberman [9].Their results show that the Dirichlet problem of the graphical mean cur-vature flow on mean convex domains has a global flow and it converges toa minimal surface at time infinity. Their result can not been directly appliedto the following graphical translating mean curvature flow.(2) ∂ t u = p + | Du | div ( Du p + | Du | ) − , in Ω × [0 , ∞ )with the Dirichlet boundary condition u = φ, on ∂ Ω , t ≥ u ( x , = u ( x ) , x ∈ Ω . Here we assume Ω ⊂ R n is a bounded domain with C boundary, φ ∈ C ,α ( Ω ), u ∈ C ,α ( Ω ), and u = φ on ∂ Ω . The flow (2) corresponds to LI MA the negative gradient flow of the weighted area functional F ( u ) = Z Ω p + | Du | e u ( x ) dx . If we let f = u + t , then f satisfies ∂ t u = p + | Du | div ( Du p + | Du | ) , in Ω × [0 , ∞ )with the Dirichlet boundary condition f = φ + t , on ∂ Ω , t ≥ f ( x , = u ( x ) , x ∈ Ω . Observe that the boundary condition now depends on time variable and theknown result [6] can not be applied directly to it.We have the following result.
Theorem 2.
Assume Ω ⊂ R n be a bounded convex domain with C bound-ary. Assume that φ ∈ C ,α ( Ω ) , u ∈ C ,α ( Ω ) , and u = φ on ∂ Ω . Then theDirichlet problem of (2) has a smooth solution and u ( · , t ) converges to thetranslating soliton with boundary data φ as t → ∞ . The plan of this note is below. In section 2 we discuss the positivity pre-serving properties of the general translating mean curvature flow. In section3 we consider the global existence of the Dirichlet problem of graphic meancurvature flows on bounded convex domains in R n .2. positivity preserving property of the translating mean curvature flow We shall use Hamilton’s tensor maximum principle as below (see [2] forfull statement and the proof).
Proposition 3.
Let ( M , g ( t )) be a one parameter family of complete non-compact Riemannian manifolds with bounded curvature. Suppose S = S i j ( x , t ) dx i dx j is a smooth time-dependent symmetric 2-tensor field suchthat ( ∂ t − ∆ g ( t ) ) S ≥ ∇ X S + B ( S , t ) where B ( S , t ) is locally Lipschitz in ( S , t ) and X = X ( t ) is a smooth timedependent vector field on M. Assume that B satisfies the null-eigenvectorassumption in the sense that for some time-parallel vector field v and atsome point x ∈ M such that if S ≥ and T ( v , · ) = , then B ( S , t )( v , v ) ≥ .Assume that S ≥ at the initial time t = . Then S ≥ for all t > . RANSLATING MEAN CURVATURE FLOWS 5
Recall the following formulae for the flow X t = f ν with local coordinates( x j ) on M t , we have for the evolving metric g i j = < ∂ x i X , ∂ x j X > , outer unitnormal ν , and the second fundamental form ( a i j ), we have ∂ t g i j = − f a i j ,∂ t ν = −∇ f , and ∂ t a i j = f i j − f a ik a k j . We shall let f = < V , ν > − H , which is our translating mean curvature flowcase.Let ( g i j ) = ( g i j ) − . As in [3] and [7] we take ( e i ) to be the evolving frameon M t such that ∂ t e i = g jk ∂ t g i j e k = − f g jk a i j e k . Then we have ∂ t g i j = . At a fixed point p ∈ M t we may assume that < e i , e j > = δ i j and ∇ e i e j = ∂ t A ( e i , e j ) = f i j + f a ik a k j . Note that ∇ i < V , ν > = < V , D e i ν > = − < V , e k > a ik , and at p , ∇ j ∇ i < V , ν > = − < V , e k > a ik , j − < V , D e i e j > a ik = a i j , V T − < V , ν > a ik a k j . Then we have ∆ < V , ν > = ∇ V T H − < V , ν > | A | . Since ∂ t < V , ν > = − < V , ∇ f > = −∇ V T f and f + H = < V , ν > , we get( ∂ t − ∆ ) < V , ν > = −∇ V T f − ∇ V T H + < V , ν > | A | = −∇ V T < V , ν > + < V , ν > | A | . That is,(3) ( ∂ t − ∆ ) < V , ν > = −∇ V T < V , ν > + < V , ν > | A | . Recall the well-known formulae that( ∆ A ) i j = −| A | a i j − Ha ik a k j − H i j . Then we have( ∂ t A − ∆ A )( e i , e j ) = ( f + H ) i j + ( f + H ) a ik a k j + | A | a i j , which implies that for the normalized mean curvature flow,( ∂ t A − ∆ A )( e i , e j ) = −∇ V T a i j + | A | a i j , LI MA that is,(4) ( ∂ t A − ∆ A ) = −∇ V T A + | A | A . Applying Hamilton’s tensor maximum principle above (see also Proposition12.31 in [2]) we know that A ≥ ∂ t − ∆ ) H = −∇ V T H + | A | H . We can apply the scalar maximum principle to this equation and to (3) too.This gives the property (1) in Theorem 1.By these formulae for A , H , and < V , ν > we obtain that( ∂ t − ∆ )( A + β < V , ν > n g ) = −∇ V T ( A + β < V , ν > n g ) + | A | ( A + β < V , ν > n g ) , and( ∂ t − ∆ )( H − β < V , ν > ) = −∇ V T ( H − β < V , ν > ) + | A | ( H − β < V , ν > ) . Define the operator L = ∆ − ∇ V T + | A | = L + | A | . Then the above equations can be rewritten as( ∂ t − L )( A + β < V , ν > n g ) = ∂ t − L )( β < V , ν > − H ) = . Applying the maximum principle (and Hamilton’s tensor maximum princi-ple as above) to above two equations we completes the proof of Theorem1. One immediate consequence is the following pinching estimate.
Corollary 4.
Given a translating mean curvature flow M t with boundedsecond fundamental form A, t ∈ [0 , T ) with T > . Assume that for someuniform constants β and β , β < V , ν > ≤ H ≤ β < V , ν > on the initialhypersurface M . Then β < V , ν > ≤ H ≤ β < V , ν > on M t for t > . The proof is the same as (2) in Theorem 1.As in [3] we have for any symmetric 2-tensor and h a positive functionon the manifold M , ( ∂ t − L ) | f | ≤ < f , ( ∂ t − L ) f >, ( ∂ t − L ) | fh | ≤ < fh , ( ∂ t − L fh ) , and ( ∂ t − L ) fh = ( ∂ t − L ) fh − f ( ∂ t − L ) hh + h < ∇ h , ∇ fh > . RANSLATING MEAN CURVATURE FLOWS 7
Then we have ( ∂ t − L ) | fh | ≤ < ∇| fh | , ∇ log h > . Let, for some λ , B = A + λ < V ,ν> n g β < V , ν > − H . Lemma 5.
Let M t ⊂ R n + be a one parameter family of hypersurfacesevolved by the translating mean curvature flow (1). Assume that β < V , ν > − H > on the initial hypersurface for some constant β , and | A | arebounded on each M t . Then ( ∂ t − L ) | B | ≤ < ∇| B | , ∇ log( β < V , ν > − H ) >, on M t . We now point out the geometric meaning of the operator ∆ − ∇ V T + | A | on the hypersurface M . Define the operator L = ∆ − ∇ V T + | A | , which is the Jacobian operator for the weighted volume F ( M ) = Z M e − < V , X > dX . Then, F ′ = − ¯ H − V N = H ν − V N . In fact, for X t = X ′ = f ν and H ′ = ∂ t H ,we have H ′ = − ∆ f − | A | f and ν ′ = −∇ f . Then ( H − < V , ν > ) ′ = − ∆ f − | A | f + < V , ∇ f > = − L f . At the critical point of F where H = < V , ν >, we have F ” = − Z M < f , L f > dm where dm = e − < V , X > dX . LI MA
3. T he D irichlet problem for the translating graphical mean curvatureflow Recall that Ω ⊂ R n is a bounded convex domain with C boundary.Note that the flow (2) corresponds to the negative gradient flow of theweighted area functional F ( u ) = Z Ω p + | Du | e u ( x ) dx . In fact, δ F ( u ) δ u = − Z Ω [ div ( Du p + | Du | ) − v ] δ ue u ( x ) dx , where v = p + | Du | . The functional F ( u ) corresponds to the functional F ( M ) with V = − e n + in the previous section.We point out a similarity between the translating mean curvature flow (2)and the graphical mean curvature flow. Fix any t >
0. Define U = u − t + t .Then U satisfies the following(5) ∂ t u = p + | Du | div ( Du p + | Du | ) , in Ω × [0 , ∞ )with the Dirichlet boundary condition U = φ − t + t , on ∂ Ω , t ≥ U ( x , = u ( x ) − t , x ∈ Ω . Define Q = p + | DU | . Then Q satisfies(6) ∂ t v = D i ( a i j D j v ) + Hd l D l v − a i j a kl D i D k uD j D k u · v , in Ω × (0 , T )where a i = Q − D i U , H = AU , and A i j = ∂ a i /∂ p j . We shall use (6) to get theuniform gradient bound of u . We need to control sup ∂ Ω | Du | first. Becauseof the equation (6) (being the same as the case of mean curvature flow)we believe the result of Theorem 2 should also be true for mean convexdomains. However, we shall not discuss this in this note.We now begin the proof of Theorem 2. Proof.
The existence of short time solution to (2) can be obtained by thestandard method. Let T > u ( x , t ). We claim that T = + ∞ . To obtain this, we need to find a prioriestimates for sup Ω | u | and sup Ω | Du | .Define Au = − div ( Du p + | Du | ) . RANSLATING MEAN CURVATURE FLOWS 9
Let w be the bowl soliton constructed by Altschuler-Wu [1]. Then we have − p + | Dw | Aw = , in R n . Note that w satisfies 2. By adding to w some uniform constant C we mayassume w − C ≤ − sup Ω | u | and w + C ≥ sup Ω | u | . Using w ± C as thebarriers, we obtain that w − C ≤ u ≤ w + C , in Ω × [0 , T ) . This gives us the uniform bound of sup Ω | u | .We now use the fact that the domain Ω is convex. Fix p ∈ ∂ Ω . Recallthat by the result of J.Serrin [11] or applying Cor. 14.3 in [4] to the operator Q ( u ) = − ( p + | Du | ) Au − (1 + | Du | ) , we can construct barriers δ + and δ − such that δ ± ( p ) = φ ( p ), − ( p + | D δ + | ) A δ + ≤ + | D δ + | , δ + ≥ φ and − ( p + | D δ − | ) A δ − ≥ + | D δ − | , δ − ≤ φ in Ω . We may also assume δ − ≤ u ≤ δ + (see [6]).Applying the maximum principle to the evolution equation (2) we knowthat δ − ≤ u ≤ δ + on Ω × [0 , T ). Hence we have at any time t = t we have δ − ( x ) ≤ u ( x , t ) ≤ δ + ( x )on Ω . Since p ∈ ∂ Ω is arbitrary, we know that there is a uniform constant C depending only on ∂ Ω , u , and φ such that | Du | ≤ C , on ∂ Ω × [0 , T ) . Applying the maximum principle the equation (6) for Q , we obtain the uni-form bound for sup Ω | Du | . Once these are done, we then get the existence ofthe unique solution to the Dirichlet problem of (2) for all times 0 < t < ∞ with the uniform gradient bound on sup Ω | Du | . The standard parabolic equa-tion theory [8] guarantees uniform bounds of all higher derivatives of u .Since ∂ t u = ∂ Ω , by the equation we have H + v = ∂ Ω and for dm : = e u dx , ddt Z Ω vdm = − Z Ω ( H + v ) dm . Then Z ∞ Z Ω ( H + v ) dm ≤ Z Ω vdm (0) . Using the uniform bound about v , we can conclude that sup Ω | ∂ t u | and sup Ω | H − v | converges to zero uniformly as t → ∞ . This completes the proof of The-orem 2. (cid:3) R eferences [1] Altschuler,S.J., L F Wu, Translating surfaces of the non-parametric mean curvatureflow with prescribed contact angle , Calc. Var. Partial Di ff erential Equations 2 (1994)101-111, http: // dx.doi.org / / BF01234317[2] Chow, Bennett; Chu, Sun-Chin; Glickenstein, David; Guenther, Christine; Isen- berg,Jim; Ivey, Tom; Knopf, Dan; Lu, Peng; Luo, Feng; Ni, Lei.
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