On the horizontal Mean Curvature Flow for Axisymmetric surfaces in the Heisenberg Group
aa r X i v : . [ m a t h . A P ] O c t ON THE HORIZONTAL MEAN CURVATURE FLOW FORAXISYMMETRIC SURFACES IN THE HEISENBERG GROUP
FAUSTO FERRARI, QING LIU, AND JUAN J. MANFREDI
Abstract.
We study the horizontal mean curvature flow in the Heisenberg group byusing the level-set method. We prove the uniqueness, existence and stability of axisym-metric viscosity solutions of the level-set equation. An explicit solution is given for themotion starting from a subelliptic sphere. We also give several properties of the level-setmethod and the mean curvature flow in the Heisenberg group. Introduction
We are interested in a family of compact hypersurfaces { Γ t } t ≥ in the Heisenberg group parametrized by time t ≥
0. The motion of the hypersurfaces is governed by the followinglaw: V H = κ H , (1.1)where V H denotes its horizontal normal velocity and κ H stands for the horizontal meancurvature in the Heisenberg group. The geometric motion (1.1) is thus called horizontalmean curvature flow . The objective of this work is to investigate the evolution of thesurface Γ t for t > .We implement a version of the level-set method adapted to the Heisenberg group. Let usassume, for the moment, that Γ t is smooth for any t ≥
0. If there exists u ∈ C ( H× [0 , ∞ ))such that Γ t = { p ∈ H : u ( p, t ) = 0 } for t ≥
0, then one may represent the horizontal normal velocity V H as V H = u t |∇ H u | and the horizontal mean curvature κ H as κ H = div H (cid:18) ∇ H u |∇ H u | (cid:19) = 1 |∇ H u | tr (cid:20)(cid:18) I − ∇ H u ⊗ ∇ H u |∇ H u | (cid:19) ( ∇ H u ) ∗ (cid:21) . Here u t , ∇ H u and ( ∇ H u ) ∗ respectively denote the derivative in t , the horizontal gradient and the (symmetrized) horizontal Hessian of u , and div H is the horizontal divergence Date : May 21, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Mean curvature flow equation, Heisenberg groups, viscosity solutions, level-setmethod.F. F. is supported by MURST, Italy, by University of Bologna,Italy by EC project CG-DICE and by theERC starting grant project 2011 EPSILON (Elliptic PDEs and Symmetry of Interfaces and Layers for OddNonlinearities). F. F. wishes to thank the Department of Mathematics at the University of Pittsburghfor the kind hospitality.Q. L. and J. M. are supported by NSF award DMS-1001179. operator . The horizontal gradient of u is given by ∇ H u = ( X u, X u ), where X = ∂∂p − p ∂∂p ; X = ∂∂p + p ∂∂p . In order to understand the law of motion by curvature (1.1), it therefore suffices to solve(MCF) u t − tr (cid:20)(cid:18) I − ∇ H u ⊗ ∇ H u |∇ H u | (cid:19) ( ∇ H u ) ∗ (cid:21) = 0 in H × (0 , ∞ ), (1.2) u ( p,
0) = u ( p ) in H . (1.3)with a given function u ∈ C ( H ) satisfyingΓ = { p ∈ H : u ( p ) = 0 } . We refer the reader to [CGG, ES, G] for a detailed derivation of (MCF) in the Euclideanspaces and to [CDPT, CC] for the analogue in the Heisenberg group.In this work, we aim to establish the uniqueness, existence and stability of the solutionsof (MCF) that are spatially axisymmetric about the third coordinate axis. Namely, weare interested in the solutions u satisfying u ( p , p , p , t ) = u ( p ′ , p ′ , p , t ) when ( p ′ ) + ( p ′ ) = p + p . (1.4)The symmetric structure of the functions is useful to obtain positive results. We thusconsider our contribution as a first step in order to prove more general results. Consult[AAG, SS] for the results on motion by mean curvature for axisymmetric surfaces in theEuclidean spaces.The symmetry with respect to the third axis in the Heisenberg group is not accidental.Indeed it is well known that this coordinate plays a key role in the Heisenberg group inseveral cases. In particular, we recall, for example, that { (0 , , p ) ∈ H : p ∈ R } is thecenter of the Heisenberg group and moreover the points along the p -axis correspond toconjugate points for the exponential map [Mo]. We warn the reader that, in general, ourresults do not apply to functions with different axes of symmetry.Hereafter the property (1.4) is sometimes referred to as “spatial symmetry about thevertical axis” or simply as “axisymmetric”.Since the general regularity of u is not known a priori , we discuss the problem in theframework of viscosity solutions [CIL]. As it is easily observed from the equation, a keydifficulty lies at the characteristic set of the level set Γ t , i.e., at the points where ∇ H u = 0.1.1. Uniqueness.
Even in the Euclidean case [CGG, ES, S, G], the proof of the com-parison principle and the uniqueness of solutions for this type of degenerate equationsneed special techniques to deal with the characteristic set. The comparison principle weexpect is as follows: for any upper semicontinuous subsolution u and lower semicontinuoussupersolution v defined on H × [0 , ∞ ) satisfying u ( p, ≤ v ( p,
0) for all p ∈ H , we have u ( p, t ) ≤ v ( p, t ) for any t ≥
0. L. Capogna and G. Citti [CC] extended the results of [ES]and proved a comparison principle by excluding the characteristic points. Their compar-ison principle further required that (i) either u or v be uniformly continuous and (ii) theinitial surfaces are completely separated in the horizontal directions, i.e., u ( p, ≤ v ( q, p = ( p , p , p ) , q = ( q , q , q ) ∈ H such that p i = q i for i = 1 ,
2. The generalcomparison principle, as stated above, remains an open question.In this paper, we follow [CGG, G] and give a comparison principle without assumingthose two conditions above but requiring that either u or v be axisymmetric. We alsorestrict ourselves to the case of compact surfaces for simplicity. The comparison theoremwe present is as follows. ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 3
Theorem 1.1 (Comparison theorem) . Let u and v be respectively an upper semicontinuoussubsolution and a lower semicontinuous supersolution of u t − tr (cid:20)(cid:18) I − ∇ H u ⊗ ∇ H u |∇ H u | (cid:19) ( ∇ H u ) ∗ (cid:21) = 0 in H × (0 , T ) for any T > . Assume that there is a compact set K ⊂ H and a, b ∈ R with a ≤ b such that u ( p, t ) = a and v ( p, t ) = b for all p ∈ H \ K and t ∈ [0 , T ] . Assume thateither u or v is spatially axisymmetric about the vertical axis. If u ( p, ≤ v ( p, for all p ∈ H , then u ≤ v for all ( x, t ) ∈ H × [0 , T ] . The uniqueness of the axisymmetric solutions follows immediately from the theoremabove. It is worth remarking that when showing comparison principles involving viscositysolutions, one usually needs to double the variables and maximize u ( p, t ) − v ( q, s ) − φ ( p, q ) + | t − s | ε , where ε > p, q ∈ H , t, s ∈ [0 , ∞ ) and φ is a smooth penalty function on H × H , andargues by contradiction.The typical choice of φ in the Euclidean spaces, as discussed in [CIL] and [G], is aquadratic function φ ( x, y ) = | x − y | usually or a quartic function φ ( x, y ) = | x − y | formean curvature flow equation (for x, y ∈ R n ). The advantages of these choices are:(a) The derivatives of φ with respect to x and y are opposite, i.e., ∇ x φ = −∇ y φ . Wewould plug these derivatives in the viscosity inequalities, since they serve as semi-differentials for the compared functions. This construction enables us to derive acontradiction.(b) When discussing (mild) singular equations such as curvature flow equations, it willbe convenient to have the second derivatives be 0 whenever the first derivativesare 0, as in the case of | x − y | .The analogue of the choice | x − y | is not immediate in the Heisenberg group. Since thegroup multiplication is not commutative, the two natural options f ( p, q ) = | q − · p | and g ( p, q ) = | p · q − | are different. It seems that we have more options but it turns out thatneither of them satisfies both conditions above. By direct calculation, we may find that g fulfills the requirement (a) above but its derivatives do not satisfy (b). The function f is good for our requirement (b) but unfortunately fails to have the property (a). Hence,the main difficulty of the uniqueness argument in the Heisenberg group consists in a wisechoice of the penalty function φ .Our approach combines both choices f and g . On one hand, we use f to derive a relaxeddefinition (Definition 3.2) of solutions of (3.1), facilitating us to overcome the singularity.On the other hand, under the help of axial symmetry, we obtain the property (b) whenemploying g type of penalty functions in the proof of the comparison principle. Thesymmetry plays an important role since it largely simplifies the structure of characteristicpoints; see [FLM2] for some geometric details.Roughly speaking, when a smooth function u ( p, t ) is spatially symmetric about thevertical axis, i.e., u = u ( r, p , t ), where r = ( p + p ) / , we get X u = p r ∂∂r u − p ∂∂p u ; X u = p r ∂∂r u + p ∂∂p u. Then ∇ H u ( p, t ) = 0 implies that either ∂u/∂r = ∂u/∂p = 0 or p + p = 0. Thisobservation enables us to obtain property (b) for a large power of the function g . F. FERRARI, Q. LIU, AND J. J. MANFREDI
Our definition of viscosity solutions is actually an extension of that introduced in [CGG,G] to the Heisenberg group. In Section 3, we discuss the equivalence of this definition andthe others.1.2.
Existence.
Generally speaking, there are at least three possible approaches to getthe existence of solutions of (MCF). One may follow [ES] to use the uniformly parabolictheory by considering a regularized equation u t − tr (cid:20)(cid:18) I − ∇ H u ⊗ ∇ H u |∇ H u | + ε (cid:19) ( ∇ H u ) ∗ (cid:21) = 0 in H × (0 , ∞ ), u ( p,
0) = u ( p ) in H .and take the limit of its solution as ε →
0; see [CC] for results in the
Carnot groups withthis method. Another possible option is to employ
Perron’s method by considering thesupremum of all subsolutions or the infimum of all supersolutions, as is shown in [CGG, G]for the Euclidean case. We refer to [I, CIL] for a general introduction of this method inthe framework of viscosity solutions.A third method for existence is based on the representation theorem involving optimalcontrol or game theory, which recently generated a spur of activity. Consult the works[CSTV, KS1, KS2, MPR1, MPR2, PSSW, PS, ST] for the development of this approach tovarious equations in Euclidean spaces. For the mean curvature flow in the sub-Riemanniangeometry, a stochastic control-based formulation analogous to [ST] is addressed in [DDR],where the authors found a solution via a suitable optimal stochastic control problem.In this work, we adapt the deterministic game-theoretic approach of R. V. Kohn and S.Serfaty [KS1] to the Heisenberg group. For any given axisymmetric continuous function u , we set up a family of games, whose value functions u ε converge to the solution u tothe mean curvature flow equation. We not only get the existence of solutions but alsoobtain a game interpretation of the equation in the Heisenberg group. The proof is basedon the dynamic programming principle , which can be regarded as a (nonlinear) semigroup.Our convergence theorem relies on the comparison principle given in Theorem 1.1. Moreprecisely, taking the half relaxed limits , defined on H × [0 , ∞ ), u ( p, t ) : = limsup ∗ ε → u ε ( p, t )= lim δ → sup { u ε ( q, s ) : s ≥ , < ε < δ, | p − q | + | t − s | < δ } (1.5)and u ( p, t ) : = liminf ∗ ε → u ε ( p, t )= lim δ → inf { u ε ( q, s ) : s ≥ , < ε < δ, | p − q | + | t − s | < δ } , (1.6)we show that u and u are respectively a subsolution and a supersolution of (1.2) using thedynamic programming principle. We also show that u ( p, ≤ u ( p ) ≤ u ( p,
0) and that u ε , u and u are spatially axisymmetric about the vertical axis. Our game approximationthen follows immediately from the comparison theorem. See Section 5 for more details onthe game setting and the existence theorem.We discuss asymptotic mean value properties related to random tug-of-war games for p -harmonic functions on the Heisenberg group in [FLM1].1.3. Stability and uniqueness of the evolution.
We give a stability theorem, which isused to show that the equation (1.2) is invariant under the change of dependent variable.We prove that for any continuous function θ : R → R , the composition θ ◦ u is a solutionprovided that u is a solution. Note that this is clear if θ is smooth and strictly monotone,since the mean curvature flow equation is geometric and orientation-free ; see [G] for moreexplanation. Our stability result is applied so as to weaken the regularity of θ . ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 5
It follows from the invariance property that any axisymmetric evolution Γ t does notdepend on the particular choice of u but depends on Γ only, which is important for thelevel-set method.1.4. Evolution of spheres.
Our uniqueness and existence results enable us to discussmotion by mean curvature with a variety of initial hypersurfaces including spheres, toriand other compact surfaces. We are particularly interested in the motion of a subellipticsphere. It turns out that if u is a defining function of the sphere centered at 0 with radius r , say u ( p ) = min { ( p + p ) + 16 p − r , M } with p = ( p , p , p ) ∈ H and M > u ( p, t ) = min { ( p + p ) + 12 t ( p + p ) + 16 p + 12 t − r , M } for any t ≥
0. We need to truncate the initial function and the solution by a constant M because all of our wellposedness results are for solutions that are constant outside acompact set. It is obvious that the zero level set Γ t of u vanishes after time t = r / √ asymptotic profile at the extinction time , we normalize the evolutionΓ t initialized from the sphere and find that the normalized surface Γ t / √ r − t convergesto an ellipsoid given by the following equation: √ r ( p + p ) + 16 p = 1 . The asymptotic profile above depends on r , the size of the initial surface, which is quitedifferent from the Euclidean case.The paper is organized in the following way. We present an introduction in Section 2.1about the Heisenberg group including calculations of some particular functions we willuse later. In Section 3, we discuss various kinds of definitions of solution to (1.2). Wepropose a new definition and show its equivalence with the others. An explicit solutionrelated to the evolution of a subelliptic sphere is given at the end of this section. Thecomparison principle, Theorem 1.1, is proved in Section 4. We establish the games andshow the existence theorem in Section 5. Section 6 is devoted to the stability results andSection 7 is intended to show further properties of the evolution including the uniquenessand finite extinction with the interesting asymptotic profile.2. Tools from Calculus in H Good references for this section are the course notes [M] and the monograph [CDPT].2.1.
Preliminaries.
Recall the that Heisenberg group H is R endowed with the non-commutative group multiplication( p , p , p ) · ( q , q , q ) = (cid:18) p + q , p + q , p + q + 12 ( p q − q p ) (cid:19) , for all p = ( p , p , p ) and q = ( q , q .q ) in H . The Haar measure if H is the usualLebesgue measure in R . The Kor´anyi gauge is given by | p | = (( p + p ) + 16 p ) / , and the left-invariant Kor´anyi or gauge metric is d ( p, q ) = | q − · p | . The Kor´anyi ball of radius r > p is B r ( p ) := { q ∈ H : d ( p, q ) < r } . F. FERRARI, Q. LIU, AND J. J. MANFREDI
The Lie Algebra of H is generated by the left-invariant vector fields X = ∂∂p − p ∂∂p ; X = ∂∂p + p ∂∂p ; X = ∂∂p . One may easily verify the commuting relation X = [ X , X ] = X X − X X .For any smooth real valued function u defined in an open subset of H , the horizontalgradient of u is ∇ H u = ( X u, X u )while the complete gradient of u is ∇ u = ( X u, X u, X u ) . For further details about the relation between sub-Riemannian metrics in Carnot groupand Riemaniann metrics see [AFM].The symmetrized second horizontal Hessian ( ∇ H u ) ∗ is the 2 × ∇ H u ) ∗ := (cid:18) X u ( X X u + X X u ) / X X u + X X u ) / X u (cid:19) . We will also consider the symmetrized complete Hessian ( ∇ u ) ∗ defined as the 3 × ∇ u ) ∗ := X u ( X X u + X X u ) / X X u + X X u ) / X X u + X X u ) / X u ( X X u + X X u ) / X X u + X X u ) / X X u + X X u ) / X u , Derivatives of auxiliary functions.
Here we include several basic calculations forsome test functions related to the Kor´anyi distance, which will be used in the proof ofcomparison theorem for generalized horizontal mean curvature flow.We are interested in the first and second horizontal derivatives of f ( p, q ) : = d ( p, q ) = (cid:0) ( p − q ) + ( p − q ) (cid:1) + 16 (cid:18) p − q − q p + 12 q p (cid:19) . We use the super index p to denote derivatives with respect to the p variable and followthe same convention for derivatives with respect to q .Let us record the results of our calculation: X p f ( p, q ) = 4 (cid:0) ( p − q ) + ( p − q ) (cid:1) ( p − q ) − p − q ) (cid:18) p − q + 12 ( q p − q p ) (cid:19) ; (2.1) X p f ( p, q ) = 4 (cid:0) ( p − q ) + ( p − q ) (cid:1) ( p − q )+ 16( p − q ) (cid:18) p − q + 12 ( q p − q p ) (cid:19) ; (2.2) X q f ( p, q ) = − (cid:0) ( p − q ) + ( p − q ) (cid:1) ( p − q ) − p − q ) (cid:18) p − q + 12 ( q p − q p ) (cid:19) ; (2.3) ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 7 X q f ( p, q ) = − (cid:0) ( p − q ) + ( p − q ) (cid:1) ( p − q )+ 16( p − q ) (cid:18) p − q + 12 ( q p − q p ) (cid:19) ; (2.4)It is clear that in general ∇ pH f ( p, q ) = −∇ qH f ( p, q ), which is not the case in the Eu-clidean case. But the following Euclidean property still holds here. Proposition 2.1.
If either ∇ pH (cid:0) | q − · p | (cid:1) = 0 or ∇ qH (cid:0) | q − · p | (cid:1) = 0 , then the horizontalcomponents of p and q are equal, i.e., p = q and p = q .Proof. Set A := 4 (cid:0) ( p − q ) + ( p − q ) (cid:1) ,B := 16 (cid:18) p − q + 12 ( q p − q p ) (cid:19) . When ∇ pH (cid:0) | q − · p | (cid:1) = 0, the calculations (2.1) and (2.2) read ( A ( p − q ) − B ( p − q ) = 0; B ( p − q ) + A ( p − q ) = 0 (2.5)with det (cid:18) A − BB A (cid:19) = A + B ≥
0. Since A + B = 0 implies that p i = q i for i = 1 , A + B = 0. If the determinant is not zero, then we alsoobtain q = p and q = p by solving the linear system (2.5). The same argument appliesto the case when ∇ qH (cid:0) | q − · p | (cid:1) = 0. (cid:3) We next calculate the second horizontal derivatives. X ,p f ( p, q ) = X ,q f ( p, q ) = 12( p − q ) + 12( p − q ) ; (2.6) X ,p f ( p, q ) = X ,q f ( p, q ) = 12( p − q ) + 12( p − q ) ; (2.7) X p X p f ( p, q ) = X q X q f ( p, q ) = − (cid:18) p − q + 12 ( q p − q p ) (cid:19) = − B ; (2.8) X p X p f ( p, q ) = X q X q f ( p, q ) = 16 (cid:18) p − q + 12 ( q p − q p ) (cid:19) = B. (2.9)It is clear that 12 ( X p X p f + X p X p f ) = 12 ( X q X q f + X q X q f ) = 0 . For later use, let us investigate the derivatives of another function. Take g ( p, q ) := | p · q − | = (cid:0) ( p − q ) + ( p − q ) (cid:1) + 16 (cid:18) p − q − p q + 12 p q (cid:19) . (2.10)Then X p g ( p, q ) = 4(( p − q ) + ( p − q ) )( p − q ) − p + q ) (cid:18) p − q − p q + 12 p q (cid:19) ; (2.11) F. FERRARI, Q. LIU, AND J. J. MANFREDI X p g ( p, q ) = 4(( p − q ) + ( p − q ) )( p − q )+ 16( p + q ) (cid:18) p − q − p q + 12 p q (cid:19) ; (2.12) X q g ( p, q ) = − p − q ) + ( p − q ) )( p − q )+ 16( p + q ) (cid:18) p − q − p q + 12 p q (cid:19) ; (2.13) X q g ( p, q ) = − p − q ) + ( p − q ) )( p − q ) − p + q ) (cid:18) p − q − p q + 12 p q (cid:19) . (2.14) Remark . In this case, we do have ∇ pH g ( p, q ) = −∇ qH g ( p, q ). But the property inProposition 2.1 does not hold in general.The second derivatives are given below. X ,p g ( p, q ) = X ,q g ( p, q ) = 12( p − q ) + 4( p − q ) + 8( p + q ) ; (2.15) X ,p g ( p, q ) = X ,q g ( p, q ) = 4( p − q ) + 12( p − q ) + 8( p + q ) ; (2.16) X p X p g ( p, q ) = X q X q g ( p, q ) =8( p − q )( p − q ) − p + q )( p + q )+ 16( p − q − p q + 12 p q ); (2.17) X p X p g ( p, q ) = X q X q g ( p, q ) =8( p − q )( p − q ) − p + q )( p + q ) − p − q − p q + 12 p q ); (2.18)12 ( X p X p g + X p X p g ) = 12 ( X q X q g + X q X q g )= 8( p − q )( p − q ) − p + q )( p + q ) . (2.19)2.3. Extrema in the Heisenberg group. As | p | ≈ p + p + | p | in Heisenberg group,the Taylor formula reads u ( p ) = u (ˆ p ) + h ˆ p − · p, ∇ u (ˆ p ) i + 12 h ( ∇ H u ) ∗ (ˆ p ) h, h i + o ( | ˆ p − · p | ) , (2.20)where h = ( p − ˆ p , p − ˆ p ) is the horizontal projection of ˆ p − · p .The following proposition follows easily from the Euclidean analog. Proposition 2.2 (Maxima on Heisenberg group) . Suppose O is an open subset of H . Let u ∈ C ( O ) and ˆ p ∈ O . If u ( p ) ≤ u (ˆ p ) for all p ∈ O , then ∇ u (ˆ p ) = 0 and ( ∇ H u ) ∗ (ˆ p ) ≤ .Analogously, for minima we have that if u ( p ) ≥ u (ˆ p ) for all p ∈ O , then ∇ u (ˆ p ) = 0 and ( ∇ H u ) ∗ (ˆ p ) ≥ . ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 9 Definitions of solutions
General definitions.
For a vector η ∈ R and a 2 × Y ∈ S wedefine F ( η, Y ) = − tr (cid:18)(cid:18) I − η ⊗ η | η | (cid:19) Y (cid:19) . In any open subset
O ⊂ H × (0 , ∞ ) the mean curvature flow equation u t − tr (cid:20)(cid:18) I − ∇ H u ⊗ ∇ H u |∇ H u | (cid:19) ( ∇ H u ) ∗ (cid:21) = 0 in O (3.1)can be written as u t + F ( ∇ H u, ( ∇ H u ) ∗ ) = 0 in O .We next define the semicontinuous envelopes in the following way: for any function h defined on a set O of a metric space M with values in R ∪ {±∞} , we take h ⋆ ( x ) = lim r → sup { h ( y ) : y ∈ O ∩ B r ( x ) } (3.2)and h ⋆ ( x ) = lim r → inf { h ( y ) : y ∈ O ∩ B r ( x ) } (3.3)for any x ∈ O , where B r ( x ) denotes the ball with radius r > x . It is easilyseen that F ⋆ (0 ,
0) = F ⋆ (0 ,
0) = 0; F ⋆ ( η, X ) = F ⋆ ( η, X ) = F ( η, X ) for all ( η, X ) ∈ R \ { } × S .One type of definition of viscosity solutions of (3.1) is as follows. Definition 3.1.
An upper (resp., lower) semicontinuous function u defined on O ⊂ H × (0 , ∞ ) is a subsolution (resp., supersolution) of (3.1) if(i) u < ∞ (resp., u > −∞ ) in O ;(ii) for any smooth function φ such thatmax O u − φ = ( u − φ )(ˆ p, ˆ t ) , (resp., min O u − φ = ( u − φ )(ˆ p, ˆ t ) , )it satisfies φ t + F ⋆ ( ∇ H φ, ( ∇ H φ ) ∗ ) ≤ p, ˆ t ),(resp., φ t + F ⋆ ( ∇ H φ, ( ∇ H φ ) ∗ ) ≥ p, ˆ t )).A function u is called a solution of (3.1) if it is both a subsolution and a supersolution.We now propose another definition for the horizontal mean curvature flow equationfollowing Giga [G]. Definition 3.2.
An upper (resp., lower) semicontinuous function u defined on O ⊂ H × (0 , ∞ ) is a subsolution (resp., supersolution) of (3.1) if(i) u < ∞ (resp., u > −∞ ) in O ;(ii) for any smooth function φ such thatmax O u − φ = ( u − φ )(ˆ p, ˆ t ) , (resp., min O u − φ = ( u − φ )(ˆ p, ˆ t ) , )it satisfies φ t + F ( ∇ H φ, ( ∇ H φ ) ∗ ) ≤ p, ˆ t ),(resp., φ t + F ( ∇ H φ, ( ∇ H φ ) ∗ ) ≥ p, ˆ t ),) when ∇ H φ (ˆ p, ˆ t ) = 0 and φ t (ˆ p, ˆ t ) ≤ , (resp., φ t (ˆ p, ˆ t ) ≥ , )when ∇ H φ (ˆ p, ˆ t ) = 0 and ( ∇ H φ ) ∗ (ˆ p, ˆ t ) = 0.A function u is called a solution of (3.1) if it is both a subsolution and a supersolution. Remark . One may replace the maximum (resp., minimum) in condition (ii) of theabove definitions with a strict maximum by adding a positive (resp., negative) smoothgauge to φ .The definition using subelliptic semijets is as follows. Definition 3.3.
An upper (resp., lower) semicontinuous function u defined on O ⊂ H × (0 , ∞ ) is a subsolution (resp., supersolution) of (3.1) if(1) u < ∞ (resp., u > −∞ ) in O ;(2) for any ( τ, η, X ) ∈ J , + H u (ˆ p, ˆ t ) (resp., ( τ, η, X ) ∈ J , − H u (ˆ p, ˆ t )) with (ˆ p, ˆ t ) ∈ O , wehave φ t + F ⋆ ( ∇ H φ, ( ∇ H φ ) ∗ ) ≤ p, ˆ t ),(resp., φ t + F ⋆ ( ∇ H φ, ( ∇ H φ ) ∗ ) ≥ p, ˆ t ),)A function u is called a solution of (3.1) if it is both a subsolution and a supersolution.It is not hard to see that Definition 3.3 is equivalent to Definition 3.1. Roughly speaking,in Definition 3.2 and Definition 3.3 we restrict the test function space to the following A = { φ ∈ C ∞ ( H ) : ∇ H φ ( p ) = 0 implies ( ∇ H ) ∗ φ ( p ) = 0 } . The next result, which is actually a variant of [G, Proposition 2.2.8] for the Heisenberggroup, indicates the equivalence between this new definition and the known one in spiteof the restriction on the test functions.
Proposition 3.1 (Equivalence of definitions) . An upper (resp., lower) semicontinuousfunction u : O → R is a subsolution (resp., supersolution) of (3.1) defined as in Definition3.2 (in O ) if and only if it is a subsolution (resp., superolution) in O in the sense ofDefinition 3.1.Proof. It is obvious that Definition 3.2 is a relaxation of Definition 3.1. We prove thereverse implication only for subsolutions. The statement for supersolutions can be provedsimilarly. Suppose there are a smooth function φ and (ˆ p, ˆ t ) ∈ O such thatmax O ( u − φ ) = ( u − φ )(ˆ p, ˆ t )By usual modification in the definition of viscosity solutions, we may assume it is a strictmaximum. We constructΨ ε ( p, q, t ) := u ( p, t ) − ε | q − · p | − φ ( q, t ) . It is clear thatΨ ∗ ( p, q, t ) := limsup ∗ ε → Ψ ε ( p, q, t ) = ( u ( p, t ) − φ ( p, t ) if p = q −∞ if p = q attains a strict maximum at (ˆ p, ˆ p, ˆ t ). By the convergence of maximizers ([G, Lemma2.2.5]), we may take p ε , q ε , t ε converging to ˆ p, ˆ p, ˆ t respectively as ε → ε attainsa maximum at ( p ε , q ε , t ε ). It follows that q
7→ − ε | q − · p ε | − φ ( q, t ) has a maximum at q ε , which, by Proposition 2.2, implies that − ε ∇ qH f ( p ε , q ε ) = ∇ H φ ( q ε , t ε ); (3.4) ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 11 − ε ( ∇ ,qH f ) ∗ ( p ε , q ε ) ≤ ( ∇ H φ ) ∗ ( q ε , t ε ) , (3.5)where f ( p, q ) = | q − · p | .We next discuss the following two cases. Case A. ∇ H φ ( q ε , t ε ) = 0 for a subsequence of ε →
0. (We still use ε to denote thesubsequence.)Since the maximality of Ψ at ( p ε , q ε , t ε ) implies that( p, t ) u ( p, t ) − ε f ( p ε , q ε ) − φ ( p · ( p ε ) − · q ε , t )attains a maximum at ( p ε , t ε ) ∈ O . Denote φ ε ( p, t ) = φ ( p · ( p ε ) − · q ε , t ). We applyDefinition 3.2 to get φ t + F ( ∇ H φ ε , ( ∇ H φ ε ) ∗ ) ≤ p ε , t ε ) (3.6)Since the derivative of the right multiplication tends to 0 as ε → ∇ H φ ε ( p ε , t ε ) → ∇ H φ (ˆ p, ˆ t ) and ( ∇ H φ ε ) ∗ ( p ε , t ε ) → ( ∇ H φ ) ∗ (ˆ p, ˆ t ) as ε → . It follows immediately that φ t + F ⋆ ( ∇ H φ, ( ∇ H φ ) ∗ ) ≤ p, ˆ t ). Case B. ∇ H φ ( q ε , t ε ) = 0 for all sufficiently small ε > ∇ qH f ( p ε , q ε ) = 0, which by Proposition 2.1 yields that p εi = q εi for i = 1 ,
2. (3.7)In terms of (2.1)-(2.2) and (2.6)–(2.9), we have ∇ pH f ( p ε , q ε ) = 0 and ∇ ,pH f ( p ε , q ε ) = 0 . (3.8)Since ( p ε , t ε ) is a maximizer of( p, t ) u ( p, t ) − ε f ( p, q ε ) − φ ( q ε , t ) , applying Definition 3.2 and sending the limit, we obtain φ t (ˆ p, ˆ t ) ≤ , (3.9)On the other hand, by passing to the limit in (3.4) and (3.5), we have ∇ H φ (ˆ p, ˆ t ) = 0 (3.10)and ∇ H φ (ˆ p, ˆ t ) ≥ . (3.11)By (3.10), (3.9) is equivalent to φ t (ˆ p, ˆ t ) + F ⋆ ( ∇ H φ (ˆ p, ˆ t ) , ≤ , which, thanks to (3.11) and the ellipticity of F , implies that φ t (ˆ p, ˆ t ) + F ⋆ ( ∇ H φ (ˆ p, ˆ t ) , ∇ H φ (ˆ p, ˆ t )) ≤ . (cid:3) An explicit solution.
We provide an example of solutions of (3.1) when the initialvalue is the fourth power of a smooth gauge of the Heisenberg group. We can actuallyexpress a solution explicitly.
Proposition 3.2.
For any p = ( p , p , p ) ∈ H , let G ( p ) = | p | = ( p + p ) + 16 p . (3.12) Then w ( p, t ) = ( p + p ) + 12 t ( p + p ) + 16 p + 12 t (3.13) is a continuous solution of (1.2) and w ( p,
0) = G ( p ) .Proof. Since w is smooth, the proof is based on a straightforward calculation of the firstderivatives of w w t = 12( p + p ) + 24 t,X w = Kp − p p , X w = Kp + 16 p p , (3.14)where K := 4( p + p ) + 24 t and the second derivatives X w = X w = 12 p + 12 p + 24 t,X X w = 16 p , X X w = − p , ( ∇ H w ) ∗ = (cid:18) p + 12 p + 24 t
00 12 p + 12 p + 24 t (cid:19) . (3.15)Noting that ( ∇ H w ) ∗ is constant multiple of the identity, we easily conclude from ourcalculation that F ⋆ ( ∇ H w, ∇ H w ) = F ⋆ ( ∇ H w, ∇ H w )= tr (cid:20)(cid:18) I − ∇ H w ⊗ ∇ H w |∇ H w | (cid:19) ( ∇ H w ) ∗ (cid:21) = 12( p + p ) + 24 t = w t , which means that w satisfies (3.1) by Definition 3.1. (cid:3) Remark . There is another way to understand that w is a solution of (3.1) by adoptingDefinition 3.2 when ∇ H w = 0 at ( p, t ) ∈ H× (0 , ∞ ). If ∇ H w ( p, t ) = 0, we have p = p = 0by solving a linear system (cid:18) K − p p K (cid:19) (cid:18) p p (cid:19) = (cid:18) (cid:19) with det (cid:18) K − p p K (cid:19) = K + 16 p > . In addition, ( ∇ H w ) ∗ = (cid:18) t
00 24 t (cid:19) . Note that, by Proposition 2.2, it is not possible to take a smooth function φ touching w from above at ( p, t ) with ∇ H φ ( p, t ) = 0 and ( ∇ H φ ) ∗ ( p, t ) = 0 . (3.16)Therefore w is a subsolution of (1.2) at ( p, t ) by Definition 3.2. On the other hand,whenever a test function φ touches w from below at ( p, t ) with (3.16), we get φ t ( p, t ) = w t ( p, t ) = 24 t >
0, which implies that w is also a supersolution due to Definition 3.2. ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 13
Remark . A basic transformation keeps the solution (3.13) being a solution. To be moreprecise, for any fixed c ∈ R , L > p ∈ H , we define ˆ w ( p, t ) = Lw (ˆ p − · p, t ) + c for all( p, t ) ∈ H × [0 , ∞ ). Then we claim that ˆ w is a solution of (1.2). Indeed, our calculationabove extends to X ˆ w = X ˆ w = 12 L ( p − ˆ p ) + 12 L ( p − ˆ p ) + 24 Lt ; X X ˆ w = − X X ˆ w = 16 L ( p − ˆ p −
12 ˆ p p + 12 p ˆ p ) . The conclusion follows immediately as in the proof of Proposition 3.2.A primary and geometric observation for the explicit solution u in (3.13) is as follows.For any fixed µ >
0, the µ -level set,Γ µt = { p ∈ H : w ( p, t ) = µ } describes the position of surface at time t ≥
0. It is obvious that even if Γ µ = ∅ , Γ t will vanish when t is sufficiently large, which agrees with the usual extinction of meancurvature flows. We will revisit this property in Section 7.2.A natural question now is whether the explicit solution we found is the only solution of(MCF) with the initial data (3.12). This is related to the open question on the uniquenessof solutions of (MCF). In the following sections we will give an affirmative answer for thecase when the initial data are cylindrically symmetric about the vertical axis.4. Comparison principle
Cylindrically symmetric solutions.
Before presenting the proof of Theorem 1.1,let us investigate the properties for the solutions of (MCF) that are axisymmetric withrespect to the vertical axis; in other words, we consider solutions of the form u = u ( r, z, t )where r = ( x + y ) / . Lemma 4.1 (Tests for axisymmetric solutions) . Let u be a subsolution (resp., supersolu-tion) of (3.1). Suppose that there exists (ˆ p, ˆ t ) ⊂ H × (0 , ∞ ) and φ ∈ C ( O ) such that max O ( u − φ ) = ( u − φ )(ˆ p, ˆ t ) ( resp., min O ( u − φ ) = ( u − φ )(ˆ p, ˆ t )) . If ˆ p = (ˆ p , ˆ p , ˆ p ) satisfies ˆ p + ˆ p = 0 and u is axisymmetric about the vertical axis, thenthere exists k ∈ R such that ∂∂p φ (ˆ p, ˆ t ) = ˆ p k and ∂∂p φ (ˆ p, ˆ t ) = ˆ p k. (4.1) Remark . It is clear that k = ∂∂r φ ( p ˆ p + ˆ p , ˆ p , ˆ t ) provided that φ = φ ( r, p , t ), i.e., φ is also axisymmetric about the vertical axis. Proof.
Denote ˆ r = p ˆ p + ˆ p . We only prove the situation when u is a subsolution. Bythe symmetry of u , u ( p , p , ˆ p , ˆ t ) = u (ˆ p , ˆ p , ˆ p , ˆ t ) for all p + p = ˆ r . By assumption, wehave ( u − φ )( p , p , ˆ p , ˆ t ) ≤ ( u − φ )(ˆ p , ˆ p , ˆ p , ˆ t ) for all ( p , p , p , t ) ∈ O ,which implies that φ ( p , p , ˆ p , ˆ t ) ≥ φ (ˆ p , ˆ p , ˆ p , ˆ t )for all ( p , p ) close to (ˆ p , ˆ p ) with p + p = r . Applying the method of Lagrange’smultiplier, we get k ∈ R such that ∂∂p (cid:18) φ ( p , p , ˆ p , ˆ t ) − k p + p − ˆ r ) (cid:19) = 0 ,∂∂p (cid:18) φ ( p , p , ˆ p , ˆ t ) − k p + p − ˆ r ) (cid:19) = 0 at (ˆ p , ˆ p ). We conclude (4.1) by straightforward calculations. (cid:3) Proof of the comparison theorem.
We are now in a position to prove Theorem1.1.
Proof of Theorem 1.1.
Let us assume u is axisymmetric about the vertical axis. The sameargument applies to the case when v is axisymmetric. Suppose by contradiction that thereexists (ˆ p, ˆ t ) ∈ H × (0 , T ) such that ( u − v )(ˆ p, ˆ t ) > p, ˆ t ) satisfies u (ˆ p, ˆ t ) − v (ˆ p, ˆ t ) − σT − ˆ t = max H× [0 ,T ) (cid:18) u ( p, t ) − v ( p, t ) − σT − t (cid:19) = µ > , (4.2)when σ > σ , double the variables and set up an auxiliary functionΦ ε ( p, t, q, s ) = u ( p, t ) − v ( q, s ) − ε g ( p, q ) − ε ( t − s ) − σT − t , where g ( p, q ) = | p · q − | . Let ( p ε , t ε , q ε , s ε ) ∈ ( H × [0 , T )) be a maximizer of Φ ε , then itis clear that Φ ε ( p ε , t ε , q ε , s ε ) = sup ( H× [0 ,T )) Φ ε > Φ ε (ˆ p, ˆ t, ˆ p, ˆ t ) , which implies that1 ε g ( p ε , q ε ) + 12 ε ( t ε − s ε ) ≤ u ( p ε , t ε ) − v ( q ε , s ε ) − u (ˆ p, ˆ t ) + v (ˆ p, ˆ t ) + σT − ˆ t − σT − t ε . (4.3)By the boundedness of u and v , we have | p ε · ( q ε ) − | → | t ε − s ε | → ε → u = a and v = b with a ≤ b outside K × [0 , ∞ ), we may take a subsequence of ε ,still indexed by ε , such that p ε , q ε → p ∈ H and t ε , s ε → t ∈ [0 , T ) as ε →
0. Sending thelimit in (4.3) and applying (4.2), we getlim sup ε → (cid:18) ε g ( p ε , q ε ) + 12 ε ( t ε − s ε ) (cid:19) ≤ . In other words, we have1 ε g ( p ε , q ε ) → ε ( t ε − s ε ) → ε →
0. (4.4)We next claim that t = 0. Indeed, if t = 0, then, since u ( p, ≤ v ( p,
0) for all p ∈ H , weare led to Φ ε ( p ε , t ε , q ε , s ε ) → u ( p, − v ( p, − σT < , which contradicts the fact that Φ ε ( p ε , t ε , q ε , s ε ) ≥ µ . We next apply the Crandall-Ishiilemma and get (cid:18) σ ( T − t ε ) + 1 ε ( t ε − s ε ) , ε ∇ p g ( p ε , q ε ) , X ε (cid:19) ∈ J , + H u ( p ε , t ε ); (cid:18) ε ( t ε − s ε ) , − ε ∇ q g ( p ε , q ε ) , Y ε (cid:19) ∈ J , − H v ( q ε , s ε ) , where J , + H and J , − H denote the closure of the semijets in Heisenberg group and X ε , Y ε ∈ S satisfy hX ε ξ, ξ i − hY ε ξ, ξ i ≤ Cε g ( p ε , q ε ) | p ε · ( q ε ) − | | ξ | = Cε g ( p ε , q ε ) | ξ | for some C > ξ ∈ R . See [B, M] for more details on the semijets and theCrandall-Ishii lemma on the Heisenberg group. It follows from (4.4) thatlim sup ε → ( hX ε ξ, ξ i − hY ε ξ, ξ i ) ≤ ξ ∈ R . Moreover, as is derived from Remark 2.1, the followinggradient relation holds: 1 ε ∇ pH g ( p ε , q ε ) = − ε ∇ qH g ( p ε , q ε ) . Let η ε denote ε ∇ pH g ( p ε , q ε ).Finally, we adopt Definition 3.3 to derive a contradiction. Case A. If η ε = 0 for all ε > σ ( T − t ε ) + 1 ε ( t ε − s ε ) + F ( η ε , X ε ) ≤ ε ( t ε − s ε ) + F ( η ε , Y ε ) ≥ . (4.7)Taking the difference of (4.6) and (4.7) yields σ ( T − t ε ) ≤ tr( I − η ε ⊗ η ε | η ε | )( X ε − Y ε ) . Passing to the limit as ε → σ ( T − t ) ≤ , which is clearly a contradiction. Case B. If η ε j = ε ∇ pH g ( p ε , q ε ) = ε ∇ qH g ( p ε , q ε ) = 0 for a subsequence ε j →
0, weobtain, by computation, that2 g ( p ε j , q ε j ) X p g ( p ε j , q ε j ) = 2 g ( p ε j , q ε j ) (cid:18) ∂∂p g ( p ε j , q ε j ) − p ε j ∂∂p g ( p ε j , q ε j ) (cid:19) = 0;2 g ( p ε j , q ε j ) X p g ( p ε j , q ε j ) = 2 g ( p ε j , q ε j ) (cid:18) ∂∂p g ( p ε j , q ε j ) + p ε j ∂∂p g ( p ε j , q ε j ) (cid:19) = 0 (4.8)and2 g ( p ε j , q ε j ) X q g ( p ε j , q ε j ) = 2 g ( p ε j , q ε j ) (cid:18) ∂∂q g ( p ε j , q ε j ) − q ε j ∂∂q g ( p ε j , q ε j ) (cid:19) = 0;2 g ( p ε j , q ε j ) X q g ( p ε j , q ε j ) = 2 g ( p ε j , q ε j ) (cid:18) ∂∂q g ( p ε j , q ε j ) + q ε j ∂∂q g ( p ε j , q ε j ) (cid:19) = 0 . (4.9)We further discuss two sub-cases.Case 1. When g ( p ε j , q ε j ) = 0, we get p ε j = q ε j , which implies that ∇ pH g ( p ε j , q ε j ) = ∇ qH g ( p ε j , q ε j ) = 0 . Since X ,p g = 2( X p g ) + 2 gX ,p g, X ,p g = 2( X p g ) + 2 gX ,p g,X p X p g = 2 X p gX p g + 2 gX p X p g, X p X p g = 2 X p gX p g + 2 gX p X p g, (4.10)We have ( ∇ ,pH g ) ∗ ( p ε j , q ε j ) = 0. Similarly, we can deduce ( ∇ ,qH g ) ∗ ( p ε j , q ε j ) = 0. ByDefinition 3.2, the viscosity inequalities read σ ( T − t ε j ) + 1 ε j ( t ε j − s ε j ) ≤ and 1 ε j ( t ε j − s ε j ) ≥ , (4.12)whose difference implies that σ/ ( T − t ε j ) ≤
0. This is certainly a contradiction.Case 2. When g ( p ε j , q ε j ) = 0, we get X p g ( p ε j , q ε j ) = X p g ( p ε j , q ε j ) = 0. We first claimthat p ε j = p ε j = 0. Suppose by contradiction that ( p ε j ) + ( p ε j ) = 0. In terms of Lemma4.1, there is k ∈ R such that (4.8) reduces to p ε j k − p ε j ∂∂p g ( p ε j , q ε j ) = 0 and p ε j k + p ε j ∂∂p g ( p ε j , q ε j ) = 0 , which yields that k = 0 and ∂∂p g ( p ε j , q ε j ) = p ε j − q ε j − p ε j q ε j + 12 p ε j q ε j = 0 . It follows from (4.8), (2.11) and (2.12) that p ε j = q ε j , which contradicts the assumptionthat g ( p ε j , q ε j ) = 0. This completes the proof of our claim.As p ε j = p ε j = 0, we apply (4.8), (2.11) and (2.12) again and get4(( q ε j ) + ( q ε j ) )( − q ε j ) − q ε j ( p ε j − q ε j ) = 0;4(( q ε j ) + ( q ε j ) )( − q ε j ) + 16 q ε j ( p ε j − q ε j ) = 0 . We are then led to q ε j = q ε j = 0. Now simplifying the second derivatives of g in (4.10)by using (2.15)–(2.19), we obtain ( ∇ ,pH g ) ∗ ( p ε j , q ε j ) = 0. An analog of calculation yieldsthat ( ∇ ,qH g ) ∗ ( p ε j , q ε j ) = 0. The proof is complete since Definition 3.2 can be adoptedonce again to get (4.11)–(4.12) and deduce a contradiction. (cid:3) Existence theorem by games
The game setting is as follows. A marker, representing the game state , is initialized ata state p ∈ H from time 0. The maturity time given is denoted by t . Let the step sizefor space be ε >
0. Time ε is consumed for every step. Then the total number of gamesteps N can be regarded as [ t/ε ]. The game states for all steps are denoted in orderby ζ , ζ , . . . , ζ N with ζ = p . Two players, Player I and Player II participate the game.Player I intends to minimize at the final state an objective function , which in our case is u : H → R , while Player II is to maximize it. At the ( k + 1)-th round ( k < N ),(1) Player I chooses in H a unit horizontal vector v k , i.e., v k = ( v k , v k ,
0) satisfying | v k | = ( v k ) + ( v k ) = 1. We denote by S h the set of all unit horizontal vectors.(2) Carol has the right to reverse Paul’s choice, which determines b k = ± ζ k to ζ k · ( √ εb k v k ).Then the state equation is written inductively as ( ζ k +1 = ζ k · ( √ εb k v k ) , k = 0 , , . . . , N − ζ = p. (5.1)The value function is defined to be u ε ( p, t ) := min v max b . . . min v N max b N u ( ζ N ) , (5.2)By the dynamic programming : u ε ( p, t ) = min v ∈ S h max b = ± u ε (cid:16) p · ( √ εbv ) , t − ε (cid:17) (5.3)with u ε ( p,
0) = u ( p ).Our main result of this section is given below. ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 17
Theorem 5.1 (Existence theorem by games) . Assume that u is uniformly continuousfunction in H and is constant C ∈ R outside a compact set. Assume also that u isspatially axisymmetric about the vertical axis. Let u ε be the value function defined as in(5.2). Then u ε converges, as ε → , to the unique axisymmetric viscosity solution of(MCF) uniformly on compact subsets of H × [0 , ∞ ) . Moreover, u = C in ( H \ K ) × (0 , ∞ ) for some compact set K ⊂ H . Before presenting the proof of Theorem 5.1, we first give bounds for the game trajecto-ries under some particular strategies.
Lemma 5.2 (Lower bound of the game trajectories) . For any p ∈ H and t ≥ with N = [ t/ε ] , let ζ k be defined as in (5.1) for all k = 0 , , . . . , N . Then the followingstatements hold. (i) There exists a strategy of Player I such that ( | ζ N | + | ζ N | ) + 16 | ζ N | ≥ ( | p | + | p | ) + 16 | p | (5.4) under this strategy regardless of Player II’s choices. (ii) There exists a strategy of Player II such that (5.4) holds under this strategy re-gardless of Player I’s choices.Proof. (i) By direct calculation, we have(( p + √ εbv ) + ( p + √ εbv ) ) + 16( p + 12 √ εb ( p v − p v )) =( p + p + 2 ε ) + 8 ε ( p + p ) + 16 p + 4 √ εb (( p + p + 2 ε )( p v + p v ) + 4( p p v − p p v )) (5.5)It is clear that Player I may take v = ( v , v , ∈ S h satisfying v = 1 ρ (( p + p + 2 ε ) p + 4 p p ); v = − ρ (( p + p + 2 ε ) p + 4 p p ) , with ρ = ( p + p ) / (( p + p + 2 ε ) + 16 p ) / so that, no matter which b is picked, we have b (( p + p + 2 ε )( p v + p v ) + 4( p p v − p p v )) = 0and, furthermore by (5.5),(( p + √ εbv ) + ( p + √ εbv ) ) + 16( p + 12 √ εb ( p v − p v )) =( p + p + 2 ε ) + 8 ε ( p + p ) + 16 p ≥ ( p + p ) + 16 p . (5.6)We can iterate (5.6) to get( | ζ k | + | ζ k | ) + 16 | ζ k | ≥ ( | ζ k − | + | ζ k − | ) + 16 | ζ k − | for all k = 1 , , . . . , N and (5.4) follows easily.(ii) The proof of (ii) is similar and even easier. Note that Player II may take a proper b = ± b (( p + p + 2 ε )( p v + p v ) + 4( p p v − p p v )) ≥ (cid:3) Lemma 5.3 (Upper bound of the game trajectories) . For any p ∈ H and t ≥ with N = [ t/ε ] , let ζ k be defined as in (5.1) for all k = 0 , , . . . , N . Then the followingstatements hold. (i) There exists a strategy of Player I such that ( | ζ N | + | ζ N | ) + 16 | ζ N | ≤ ( | p | + | p | + 6 N ε ) + 16 | p | (5.7) under this strategy regardless of Player II’s choices. (ii) There exists a strategy of Player II such that (5.7) holds under this strategy re-gardless of Player I’s choices.Remark . With the notation of the gauge G in (3.12), the inequality (5.7) can besimplified into G ( ζ N ) ≤ ( | p | + | p | + 6 t ) + 16 | p | , which is intuitively natural, since the explicit solution given in (3.13) satisfies w ( p, t ) ≤ ( | p | + | p | + 6 t ) + 16 | p | . Proof.
By iteration, it suffices to show there exist strategies of Player I or Player II suchthat (( p + √ εbv ) + ( p + √ εbv ) + jε ) + 16( p + 12 √ εb ( p v − p v )) ≤ ( p + p + ( j + 6) ε ) + 16 p . (5.8)Indeed, the left hand side is calculated to be( p + p + ( j + 2) ε ) + 8 ε ( p + p ) + 16 p + 4 √ εb (( p + p + 2 ε )( p v + p v ) + 4( p p v − p p v ))As in the proof of Lemma 5.3, either Player I or Player II may let b (( p + p + 2 ε )( p v + p v ) + 4( p p v − p p v )) ≤ p + √ εbv ) +( p + √ εbv ) + jε ) + 16( p + 12 √ εb ( p v − p v )) ≤ ( p + p + ( j + 2) ε ) + 8 ε ( p + p ) + 16 p ≤ ( p + p + ( j + 6) ε ) + 16 p , which proves (5.8). (cid:3) Remark . For any ˆ p ∈ H , c ∈ R and L >
0, letˆ G ( p ) = c + LG (ˆ p − · p ) . (5.9)Our proof above can be directly generalized to show thatˆ G ( ζ N ) ≤ c + L ( | p − ˆ p | + | p − ˆ p | + 6 N ε ) + 16 L | p − ˆ p + 12 ( p ˆ p − p ˆ p ) | with either a strategy of Player I or a strategy of Player II.We now return to the proof of Theorem 5.1, which actually rests on showing that u and u , as defined in (1.5) and (1.6), are respectively a subsolution and a supersolution of(MCF). (Note that our definitions are valid since the game value u ε are bounded uniformlyfor all ε > u ( p, ≤ u ( p,
0) and u and u areconstant outside a compact set. Then it follows immediately from the comparison principle(Theorem 1.1) that u ≤ u and therefore u ε → u locally uniformly as ε → ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 19
Proposition 5.4 (Constant value outside a compact set) . Assume that u is uniformlycontinuous function in H and is a constant C ∈ R outside a compact set. Let u ε be thevalue function defined by (5.2). Then for any T > , u ( p, t ) = u ( p, t ) = C for all p ∈ H outside a compact set and for all t ∈ [0 , T ] .Proof. Suppose there exists B r such that u ( p ) = C for any p ∈ H \ B r . Then for anyˆ p ∈ H \ B r and t ≥
0, we use the strategy of Player I introduced in Lemma 5.2, we get ζ N ∈ H \ B r regardless of Player II’s choices, which implies that u ε ( p, t ) ≤ u ( ζ N ) = C. Similarly, we may use the strategy of Player II to deduce that u ε ( p, t ) ≥ C. Hence, u ε = C and u = u = C in H \ B r . (cid:3) Proposition 5.5.
Assume that u is uniformly continuous function in H and is constantoutside a compact set. Let u ε be the value function defined by (5.2). Then u ( p, ≤ u ( p ) and u ( p, ≥ u ( p ) for all p ∈ H . In order to prove this result, we first need to regularize the initial data with the smoothgauge G in (3.12). We define ψ L ( p ) = sup q ∈H { u ( q ) − LG ( p − · q ) } (5.10)and ψ L ( p ) = inf q ∈H { u ( q ) + LG ( p − · q ) } , (5.11)for any p ∈ H and fixed L >
0. These two functions are called the sup-convolution and inf-convolution of u respectively. Our definitions here are slightly different from those in[W] in that we plug p − · q instead of q · p − in G . However, the properties remain thesame. We present one of the important properties for our use. Lemma 5.6 (Approximation by semi-convolutions) . Assume that u is uniformly contin-uous on H and is constant outside a compact set. Let ψ L and ψ L be respectively definedas in (5.10) and (5.11). Then ψ L and ψ L converge to u uniformly in H as L → ∞ .Proof. We only show the statement for ψ L . The proof for the statement on ψ L is sym-metric.It is easily seen that ψ L ≥ u in H . (5.12)On the other hand, since u is uniformly continuous, for any p ∈ H , we may find q L ∈ H such that ψ L ( p ) = sup q ∈H { u ( p ) − LG ( p − · q ) } = u ( q L ) − LG ( p − · q L ) . By (5.12), we have LG ( p − · q L ) ≤ u ( q L ) − u ( p ) , (5.13)which, by the boundedness of u , implies that | p − · q L | ≤ (2 K /L ) / , where K = sup H | u | . By the uniform continuity of u , for any δ >
0, there exists ε > | u ( p ) − u ( q ) | ≤ δ for any p, q ∈ H satisfying | p − · q | ≤ ε . Then we may let L > K /L ) / ≤ ε and therefore u ( q L ) − u ( p ) ≤ δ, which, combined with (5.12), yields | ψ L ( p ) − u ( p ) | ≤ δ for all p ∈ H . (cid:3) Proof of Proposition 5.5.
We arbitrarily fix ˆ p ∈ H . By Lemma 5.6, for any δ >
0, thereexists
L > ψ L (ˆ p ) ≤ u (ˆ p ) + δ, which implies that u ( p ) ≤ u (ˆ p ) + δ + LG (ˆ p − · p ) . Let us use the right hand side, which is exactly ˆ G in (5.9) with c = u (ˆ p ) + δ , as theobjective function of the games. Suppose the game value is w ε . Then by using the specialstrategy of Player I given in Lemma 5.3 and Remark 5.2, we obtain a game estimate w ε ( p, t ) ≤ u (ˆ p ) + δ + LG (ˆ p − · ζ N ) ≤ u (ˆ p ) + δ + L ( | p − ˆ p | + | p − ˆ p | + 6 N ε ) + 16 L | p − ˆ p + 12 ( p ˆ p − p ˆ p ) | no matter what choices are made by Player II during the game. On the other hand, sinceit is clear that u ε ≤ w ε and N ε ≤ t , we get u ε ( p, t ) ≤ u (ˆ p ) + δ + L ( | p − ˆ p | + | p − ˆ p | + 6 t ) + 16 L | p − ˆ p + 12 ( p ˆ p − p ˆ p ) | . Taking the relaxed limit of u ε at (ˆ p,
0) as ε →
0, we have u (ˆ p, ≤ u (ˆ p ) + δ. We finally send δ → u (ˆ p, ≤ u (ˆ p ) for any ˆ p ∈ H .The proof for the statement that u ( p, ≥ u ( p ) for all p ∈ H is symmetric. In fact,the key is to use the strategy of Player II introduced in Lemma 5.3 and Remark 5.2 todeduce u ε ( p, t ) ≥ u (ˆ p ) − δ − L ( | p − ˆ p | + | p − ˆ p | + 6 t ) − L | p − ˆ p + 12 ( p ˆ p − p ˆ p ) | . (cid:3) Proposition 5.7 (Axial symmetry of the game values) . Suppose that u is uniformlycontinuous on H and is spatially axisymmetric with respect to the vertical axis. Let u ε bethe value function defined as in (5.2) Then u ε , u and u are also spatially axisymmetricabout the vertical axis.Proof. We argue by induction. Assume that u ε ( p, t ) = u ε ( p ′ , t ) for some t ≥ p, p ′ ∈ H such that p + p = ( p ′ ) + ( p ′ ) and p = p ′ . (5.14)We aim to show u ε ( p, t + ε ) = u ε ( p ′ , t + ε ) for all p, p ′ ∈ H satisfying the condition (5.14).Since the dynamic programming principle (5.3) gives u ε ( p, t + ε ) = min v ∈ S h max b = ± u ε (cid:16) p · ( √ εbv ) , t (cid:17) , there exists v ∈ S h such that u ε ( p, t + ε ) = max b = ± u ε (cid:16) p · ( √ εbv ) , t (cid:17) . (5.15) ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 21
We claim that there is v ′ ∈ S h such that the coordinates of p · ( √ εbv ) and p ′ · ( √ εbv ′ )satisfy (5.14) as well. Indeed, as p · ( √ εbv ) = (cid:18) p + √ εbv , p + √ εbv , p + 12 √ εb ( p v − p v ) (cid:19) and p ′ · ( √ εbv ′ ) = (cid:18) p ′ + √ εbv ′ , p ′ + √ εbv ′ , p ′ + 12 √ εb ( p ′ v ′ − p ′ v ′ ) (cid:19) , we are looking for v ′ , v ′ ∈ S h such that ( p ′ + √ εbv ′ ) + ( p ′ + √ εbv ′ ) = ( p + √ εbv ) + ( p + √ εbv ) p ′ + 12 √ εb ( p ′ v ′ − p ′ v ′ ) = p + 12 √ εb ( p v − p v ) . Since p and p ′ satisfy (5.14), it suffices to solve the linear system ( p ′ v ′ + p ′ v ′ = p v + p v , − p ′ v ′ + p ′ v ′ = − p v + p v . The problem is trivial if p + p = ( p ′ ) + ( p ′ ) = 0. When p + p = ( p ′ ) + ( p ′ ) = 0,we get a unique pair of solutions v ′ = 1( p ′ ) + ( p ′ ) (cid:0) ( p p ′ + p p ′ ) v + ( p ′ p − p p ′ ) v (cid:1) ,v ′ = 1( p ′ ) + ( p ′ ) (cid:0) ( p p ′ − p ′ p ) v + ( p p ′ + p p ′ ) v (cid:1) . Thanks to the relation (5.14), it is easy to verify that v ′ = ( v ′ , v ′ , ∈ S h , i.e., ( v ′ ) +( v ′ ) = 1. We complete the proof of the claim.In view of the induction hypothesis, we obtain u ε (cid:16) p ′ · ( √ εbv ′ ) , t (cid:17) = u ε (cid:16) p · ( √ εbv ) , t (cid:17) for both b = ± u ε ( p ′ , t + ε ) ≤ max b = ± u ε (cid:16) p ′ · ( √ εbv ′ ) , t (cid:17) ≤ u ε ( p, t + ε ) . We may similarly prove that u ε ( p ′ , t + ε ) ≥ u ε ( p, t + ε ) and therefore u ε ( p ′ , t + ε ) = u ε ( p, t + ε ) for all p, p ′ ∈ H satisfying (5.14).It follows from the definitions (1.5)–(1.6) of half relaxed limits that the same results for u and u hold. (cid:3) Proposition 5.8.
Assume that u ε satisfies the dynamic programming principle (5.3). Let u be the upper relaxed limit defined as in (1.5). Then u is a subsolution of (1.2).Proof. Assume that there exists (ˆ p, ˆ t ) ∈ H × (0 , ∞ ) and φ ∈ C ( H × (0 , ∞ )) such that u − φ attains a strict maximum at (ˆ p, ˆ t ). Then by definitions of u , we may take a sequence,still indexed by ε , ( p ε , t ε ) ∈ H × (0 , ∞ ) such that ( p ε , t ε ) → (ˆ p, ˆ t ) and u ε ( p ε , t ε ) → u (ˆ p, ˆ t )as ε → u ε ( p ε , t ε ) − φ ( p ε , t ε ) = max B r (ˆ p, ˆ t ) ( u ε − φ ) (5.16)Applying the dynamic programming principle (5.3) with ( p, t ) = ( p ε , t ε ), we have u ε ( p ε , t ε ) = min v max b u ε (cid:16) p ε · ( √ εbv ) , t ε − ε (cid:17) , which, combined with (5.16), implies that φ ( p ε , t ε ) ≤ min v max b φ (cid:16) p ε · ( √ εbv ) , t ε − ε (cid:17) . We next use the Taylor expansion for the right hand side at ( p ε , t ε ) and obtain ε φ t ( p ε , t ε ) − min v max b ( h√ εbv, ∇ φ ( p ε , t ε ) i + ε h ( ∇ H φ ) ∗ ( p ε , t ε ) v h , v h i ) ≤ o ( ε ) , (5.17)where v h is the horizontal projection of v , i.e., v h = ( v , v ) for any v = ( v , v , v ). Since v = ( v , v , ε φ t ( p ε , t ε ) − min v max b ( h√ εbv h , ∇ H φ ( p ε , t ε ) i + ε h ( ∇ H φ ) ∗ ( p ε , t ε ) v h , v h i ) ≤ o ( ε ) , (5.18)We discuss two cases: Case A: ∇ H φ (ˆ p, ˆ t ) = 0. Then ∇ H φ ( p ε , t ε ) = 0 for all sufficiently small ε >
0. Letting˜ v = 1 |∇ H φ ( p ε , t ε ) | ( X φ ( p ε , t ε ) , − X φ ( p ε , t ε ) , v h = 1 |∇ H φ ( p ε , t ε ) | ( X φ ( p ε , t ε ) , − X φ ( p ε , t ε )) , we have from (5.17) φ t ( p ε , t ε ) − h ( ∇ H φ ) ∗ ( p ε , t ε )˜ v h , ˜ v h i ) ≤ o (1) . (5.19)Noticing that ˜ v h ⊗ ˜ v h = I − ∇ H φ ( p ε , t ε ) ⊗ ∇ H φ ( p ε , t ε ) |∇ H φ ( p ε , t ε ) | , we are thus led from (5.19) to φ t ( p ε , t ε ) − tr (cid:20)(cid:18) I − ∇ H φ ( p ε , t ε ) ⊗ ∇ H φ ( p ε , t ε ) |∇ H φ ( p ε , t ε ) | (cid:19) ( ∇ H φ ) ∗ ( p ε , t ε ) (cid:21) ≤ o (1) . (5.20)Sending ε →
0, we get φ t (ˆ p, ˆ t ) − tr (cid:20)(cid:18) I − ∇ H φ (ˆ p, ˆ t ) ⊗ ∇ H φ (ˆ p, ˆ t ) |∇ H φ (ˆ p, ˆ t ) | (cid:19) ( ∇ H φ ) ∗ (ˆ p, ˆ t ) (cid:21) ≤ . Case B: ∇ H φ (ˆ p, ˆ t ) = 0. In this case, we have by Definition 3.2 that ∇ H φ ∗ (ˆ p, ˆ t ) = 0 . (5.21)If ∇ H φ ( p ε , t ε ) = 0 for all ε >
0, we may follow the same argument as in Case A, passingto the limit for (5.20) as ε → φ t (ˆ p, ˆ t ) ≤ , (5.22)as desired.If there exists a subsequence ε j such that ∇ H φ ( p ε j , t ε j ) = 0 for all j , then it followsfrom (5.17) that φ t ( p ε j , t ε j ) − h ( ∇ H φ ) ∗ ( p ε j , t ε j ) v h , v h i ) ≤ o (1) for some v ,which again implies (5.22) as the limit when ε j → (cid:3) Proposition 5.9.
Assume that u ε satisfies the dynamic programming principle (5.3). Let u be the lower relaxed limit defined as in (1.6). Then u is a supersolution of (1.2). In order to facilitate the proof, let us present an elementary result.
ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 23
Lemma 5.10 (Lemma 4.1 in [GL]) . Suppose ξ is a unit vector in R and X is a realsymmetric × matrix, then there exists a constant M > that depends only on thenorm of X , such that for any unit vector v ∈ R , |h Xξ ⊥ , ξ ⊥ i − h Xv, v i| ≤ M |h ξ, v i| , (5.23) where ξ ⊥ denotes a unit orthonormal vector of ξ .Proof. Let cos θ = h ξ ⊥ , v i and sin θ = h ξ, v i . Then we have |h Xξ ⊥ , ξ ⊥ i − h Xv, v i| = | tr (cid:16) X ( ξ ⊥ ⊗ ξ ⊥ − v ⊗ v ) (cid:17) |≤k X kk ξ ⊥ ⊗ ξ ⊥ − ( ξ sin θ + ξ ⊥ cos θ ) ⊗ ( ξ sin θ + ξ ⊥ cos θ ) k = k X kk sin θξ ⊥ ⊗ ξ ⊥ − sin θ cos θ ( ξ ⊥ ξ ⊥ + ξ ⊥ ⊗ ξ ) k≤ M | sin θ | , where M > k X k . (cid:3) We refer the reader to [L, Lemma 2.3] for a higher dimensional extension of this lemma.
Proof of Proposition 5.9.
Assume that there exists (ˆ p, ˆ t ) ∈ H × (0 , ∞ ) and φ ∈ C ( H × (0 , ∞ )) such that u − φ attains a strict minimum at (ˆ p, ˆ t ). We may again take a sequence( p ε , t ε ) ∈ H × (0 , ∞ ) such that ( p ε , t ε ) → (ˆ p, ˆ t ) and u ε ( p ε , t ε ) → u (ˆ p, ˆ t ) as ε → u ε ( p ε , t ε ) − φ ( p ε , t ε ) = min B r (ˆ p, ˆ t ) ( u ε − φ ) (5.24)Applying the dynamic programming principle (5.3) with ( p, t ) = ( p ε , t ε ), we have u ε ( p ε , t ε ) = min v max b u ε ( p ε · √ εbv, t ε − ε ) . It then follows from (5.24) that φ ( p ε , t ε ) ≥ min v max b φ ( p ε · √ εbv, t ε − ε ) . As an analogue of (5.18), the Taylor expansion at ( p ε , t ε ) yields φ t ( p ε , t ε ) − min v max b ( 1 ε h√ bv h , ∇ H φ ( p ε , t ε ) i + h ( ∇ H φ ) ∗ ( p ε , t ε ) v h , v h i ) ≥ o (1) , (5.25)as ε → Case A: ∇ H φ (ˆ p, ˆ t ) = 0. Then ∇ φ ( p ε , t ε ) = 0 for all sufficiently small ε >
0. We adoptLemma 5.10 and getmax b (cid:18) ε h√ bv h , ∇ H φ ( p ε , t ε ) i + h ( ∇ H φ ) ∗ ( p ε , t ε ) v h , v h i (cid:19) ≤h ( ∇ H φ ) ∗ ( p ε , t ε )˜ v h , ˜ v h i + M + √ ε ! |h v h , φ H ( p ε , t ε ) i| , where ˜ v h = 1 |∇ H φ ( p ε , t ε ) | ( X φ ( p ε , t ε ) , − X φ ( p ε , t ε )) , as given in the proof of Proposition 5.8. It is now clear, by taking v h = ˜ v h , thatmin v max b (cid:18) ε h√ bv h , ∇ H φ ( p ε , t ε ) i + h ( ∇ H φ ) ∗ ( p ε , t ε ) v h , v h i (cid:19) ≤h ( ∇ H φ ) ∗ ( p ε , t ε )˜ v h , ˜ v h i , which implies through (5.25) that φ t ( p ε , t ε ) − h ( ∇ H φ ) ∗ ( p ε , t ε )˜ v h , ˜ v h i ≥ o (1) . Letting ε →
0, we obtain φ t (ˆ p, ˆ t ) − tr (cid:20)(cid:18) I − ∇ H φ (ˆ p, ˆ t ) ⊗ ∇ H φ (ˆ p, ˆ t ) |∇ H φ (ˆ p, ˆ t ) | (cid:19) ( ∇ H φ ) ∗ (ˆ p, ˆ t ) (cid:21) ≥ . Case B: ∇ H φ (ˆ p, ˆ t ) = 0. We may further assume (5.21) again in this case. We may applythe same argument above and get φ t (ˆ p, ˆ t ) ≥ , (5.26)provided that ∇ H φ ( p ε , t ε ) = 0 for all ε >
0. It remains to show (5.26) when there is asubsequence ε j such that ∇ H φ ( p ε j , t ε j ) = 0. By (5.25), we have on this occasion φ t ( p ε j , t ε j ) − h ( ∇ H φ ) ∗ ( p ε j , t ε j ) v h , v h i ) ≥ o (1) for some v .Sending ε →
0, we get (5.26). (cid:3)
We are now in a position to prove Theorem 5.1.
Proof of Theorem 5.1.
In terms of Proposition 5.7, Proposition 5.8 and Proposition 5.9, u and u are respectively a subsolution and a supersolution of (1.2) that are axisymmetricwith respect to the vertical axis. For any T > u ( p, t ) and u ( p, t ) are constant outside acompact set of H for all t ∈ [0 , T ], owing to Proposition 5.4. Also, since u ( p, ≤ u ( p )and u ( p, ≥ u ( p ) for all p ∈ H , we may apply Theorem 1.1 to get u ≤ u in H × [0 , T ].As it is obvious that u ≥ u , we get u = u in H × [0 , T ] with u ( · ,
0) = u ( · ). In conclusion, u = u = u is the unique continuous solution of (MCF) and the locally uniform convergence u ε → u follows immediately. (cid:3) Stability
The following stability result is standard in the theory of viscosity solutions.
Theorem 6.1 (Stability under the uniform convergence) . Let u ε be solutions of (1.2) and u ε → u locally uniformly in H × [0 , ∞ ) . Then u is also a solution of (1.2). Lemma 6.2. If u ε is a subsolution (resp., supersoution) of (1.2) for all small ε > , then u = limsup ∗ ε →∞ u ε ( resp., u = liminf ∗ ε →∞ u ε ) is also a subsolution (resp., supersoution) of (1.2).Proof. Suppose there exists φ ∈ C ( H × [0 , ∞ )) and (ˆ p, ˆ t ) ∈ H × (0 , ∞ ) such that u − φ attains a strict maximum at (ˆ p, ˆ t ). Then by the convergence of maximizers as shown in[G, Lemma 2.2.5], we can take subsequences of p ε , t ε and u ε , still indexed by ε , satisfying( p ε , t ε ) → (ˆ p, ˆ t ) as ε → u ε − φ )( p ǫ , t ǫ ) = max H× [0 , ∞ ) ( u ε − φ ) . We discuss two cases.
Case 1: ∇ H φ (ˆ p, ˆ t ) = 0. Then ∇ H φ ( p ε , t ε ) = 0 for all ε > φ t − tr (cid:20)(cid:18) I − ∇ H φ ⊗ ∇ H φ |∇ H φ | (cid:19) ( ∇ H φ ) ∗ (cid:21) ≤ p ε , t ε ) . Sending ε →
0, we get the desired inequality φ t − tr (cid:20)(cid:18) I − ∇ H φ ⊗ ∇ H φ |∇ H φ | (cid:19) ( ∇ H φ ) ∗ (cid:21) ≤ p, ˆ t ) . ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 25
Case 2: ∇ H φ (ˆ p, ˆ t ) = 0. Then by Definition 3.2 we only need to discuss the situationwhen ( ∇ H φ ) ∗ (ˆ p, ˆ t ) = 0 also holds. Hence, ∇ H φ ( p ε , t ε ) → ∇ H φ ) ∗ ( p ε , t ε ) → ε →
0. If there exists a subsequence ε j such that ∇ H φ ( p ε j , t ε j ) = 0, then we have φ t − tr (cid:20)(cid:18) I − ∇ H φ ⊗ ∇ H φ |∇ H φ | (cid:19) ( ∇ H φ ) ∗ (cid:21) ≤ p ε j , t ε j ) . Passing to the limit j → ∞ , we obtain φ t (ˆ p, ˆ t ) ≤ ∇ H φ ( p ε , t ε ) = 0 for all ε > φ t ( p ε , t ε ) ≤ φ t (ˆ p, ˆ t ) ≤
0, which completes our proof.One may similarly prove that u = liminf ∗ ε → u ε is a supersolution provided that u ε isa supersolution for all ε > (cid:3) Proof of Theorem 6.1.
Let u = limsup ∗ ε → u ε and u = liminf ∗ ε → u ε . Then in virtue of Lemma 6.2, u is a subsolution of (1.2) and u is a supersolution of (1.2).Noting that u ε → u locally uniformly, we must have u = u = u and therefore u is asolution of (1.2). (cid:3) Properties of the evolution
We have shown that there is a unique solution u of (MCF) for any given continuousfunction u which is axisymmetric with respect to the vertical axis and attains constantvalue outside a compact set. Let us turn to discuss the surface evolution described by thelevel-set equation (MCF). More precisely, given an axisymmetric compact surface Γ ⊂ H ,we choose u ∈ C ( H ) such that it is axisymmetric constant outside a compact set andsatisfies Γ = { p : H : u ( p ) = 0 } . (7.1)We then solve (MCF) for the unique solution u and get the surfaceΓ t = { p ∈ H : u ( p, t ) = 0 } for any t ≥
0. (7.2)In what follows, we first show that the surface represented by the level-set Γ t of u doesnot depend on the particular choice of u .7.1. Uniqueness of the surface evolution.Theorem 7.1 (Invariance) . Assume that θ : R → R is continuous. If u is a solution of(1.2). Then w = θ ◦ u is also a solution of (1.2).Proof. We prove the theorem in several steps.
Step 1.
We first give the proof in the case that θ ∈ C ( R ) and θ ′ >
0. Suppose that thereis φ ∈ C ( H × [0 , ∞ )) and (ˆ p, ˆ t ) ∈ H × (0 , ∞ ) such that θ ◦ u − φ attains a maximum at(ˆ p, ˆ t ). Then it is clear that u − h ( φ ) attains a maximum at (ˆ p, ˆ t ), where h = θ − ∈ C ( R )with h ′ >
0. Denote ψ = h ( φ ). Since u is a subsolution of (1.2), we have ψ t − tr (cid:20)(cid:18) I − ∇ H ψ ⊗ ∇ H ψ |∇ H ψ | (cid:19) ( ∇ H ψ ) ∗ (cid:21) ≤ p, ˆ t ) . Note that ψ t = h ′ φ t , ∇ H ψ = h ′ ∇ H φ and( ∇ H ψ ) ∗ = h ′′ ∇ H φ ⊗ ∇ H φ + h ′ ( ∇ H φ ) ∗ . It follows that φ t − tr (cid:20)(cid:18) I − ∇ H φ ⊗ ∇ H φ |∇ H φ | (cid:19) ( ∇ H φ ) ∗ (cid:21) ≤ p, ˆ t ) , which shows that θ ◦ u is a subsolution of (1.2). An analogue of this argument yields that θ ◦ u is also a supersolution.We also claim that θ ◦ u remains being a solution when θ ∈ C ( R ) and θ ′ <
0. Indeed,when θ is a decreasing function, − θ is increasing. We obtain that − θ ◦ u is a solutionof (1.2). Thanks to the fact that the mean curvature flow is orientation-free or (1.2) ishomogeneous in all of the derivatives, we easily see that θ ◦ u is a solution as well. Inparticular, we note that − u is a solution when u is a solution. Step 2.
We generalize the consequence obtained in Step 1 for a continuous nondecreasingor nonincreasing function. Indeed, for any continuous nondecreasing function θ , we maytake θ n ∈ C ( R ) with θ ′ n > n = 1 , , . . . such thatlimsup ∗ n →∞ θ n ◦ u = θ ◦ u. We refer the reader to [G, Lemma 4.2.3] for details about the construction of θ n . Since θ n ◦ u is a solution of (1.2) for all n , as shown in Step 1, θ ◦ u is a subsolution, due toLemma 6.2.To show that θ ◦ u is a supersolution, we define ˜ θ ( x ) = θ ( − x ) for any x ∈ R andobserve that θ ( u ) = ˜ θ ( − u ). Since ˜ θ is nonincreasing and − u is a solution, we may applya symmetric version of [G, Lemma 4.2.3] to get θ ( u ) = ˜ θ ( − u ) is a supersoluiton.When θ is a continuous nonincreasing function, − θ is nondecreasing. We apply againthe homogeneity of (1.2) to obtain that θ ◦ u is a solution. Since the verification ofdefinition of (1.2) is pointwise, one can further relax the monotonicity condition on θ to alocal monotonicity condition.To conclude this step, we notice that max { min { u, C } − C } is a solution for any C > u is a solution. Step 3.
We finally discuss the situation when θ is assumed to be continuous only. ByTheorem 6.1, it suffices to discuss the bounded function max { min { u, C } − C } insteadof u for arbitrarily large C >
0. We approximate θ uniformly by polynomials θ m in[ − C − , C + 1]. Since polynomials only have finitely many maximizers and minimizers,we may also assume each θ m is constant near all of its local maximizers and minimizers.In fact, if, for instance, θ m attains a local maximum at x ∈ R , we take min { θ m ( x ) , θ ( x ) − ε m } , where ε m > ε m → m → ∞ ) such that θ m is continuous.Now θ m is locally nonincreasing or nondecreasing. We apply the result in Step 2 andfind that θ m ◦ u is a solution of (1.2). Since θ m → θ uniformly, by the stability result givenin Theorem 6.1, we see that θ ◦ u is a solution by sending m → ∞ . (cid:3) An immediate consequence of the theorem above is that our generalized surface evolu-tion does not depend on the choice of the initial level-set function u . Corollary 7.2 (Independence of the choice of the initial function) . Suppose that u and ˜ u are continuous functions in H axisymmetric about the vertical axis and are constantoutside a compact set K ⊂ H . Let Γ = { p ∈ H : u ( p ) = 0 } = { p ∈ H : ˜ u ( p ) = 0 } bebounded. Let u and ˜ u be the unique continuous solutions of (1.2) with the initial conditions u and ˜ u respectively. For any t ≥ , set Γ t = { p ∈ H : u ( p, t ) = 0 } and ˜Γ t = { p ∈ H : ˜ u ( p, t ) = 0 } . Then Γ t = ˜Γ t for all t ≥ .Proof. We follow the proof of [ES, Theorem 5.1]. It is obvious, from Theorem 1.1 andTheorem 5.1, that u and ˜ u are axisymmetric about the vertical axis.We may assume u ≥ u , since | u | is a solutionof (1.2) with the initial condition u ( p,
0) = | u | by Theorem 7.1. Similarly, let us alsoassume that ˜ u ≥ u ≥ ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 27
For any k = 1 , , . . . let E = ∅ and E k = { p ∈ H : u ( p ) > l/k } such that E k isnondecreasing and H ⊂ Γ = ∪ k E k . Define a k = max H\ E k − ˜ u ( k = 1 , , ... ) . Then we have lim k →∞ a k = 0. We then construct a continuous function θ satisfying θ (0) = 0, θ (1 /k ) = a k for all k and θ = a in [1 , ∞ ).Now it is clear that θ ◦ u is an axisymmetric solution of (1.2) with initial data θ ◦ u ,again due to Theorem 7.1. By our construction of θ , we easily see that θ ◦ u ≥ ˜ u .Applying Theorem 1.1 for all T >
0, we get θ ◦ u ≥ ˜ u . This means that Γ t ⊂ ˜Γ t for any t ≥
0. Indeed, for any p ∈ Γ t , we have u ( p, t ) = 0, which implies that θ ◦ u ( p, t ) = 0 andtherefore ˜ u ( p, t ) = 0.We conclude the proof by similarly showing the inclusion ˜Γ t ⊂ Γ t for any t ≥ (cid:3) Finite time extinction.
We give a simple geometric property of the mean curvatureflow. The following result shows that an axisymmetric compact surface evolving by itsmean curvature shrinks and disappears in finite time.
Theorem 7.3 (Finite time extinction for bounded evolution) . Suppose that { Γ t } t ≥ de-notes an axisymmetric surface evolution of the mean curvature flow. If Γ ⊂ B r , for r > , then Γ t = ∅ when t > r / √ .Proof. We may take an axisymmetric u ∈ C ( H ) with a constant value C > B r satisfying (7.1) and u ≥ min {| p | − r , C } . Taking w ( p, t ) = ( p + p ) + 12 t ( p + p ) + 16 p + 12 t as in (3.13), we easily see that w C ( p, t ) := min { w ( p, t ) − r , C } is a solution of (1.2) withinitial data w C ( p,
0) = min {| p | − r , C } , by Theorem 7.1 with θ ( x ) = min { x, C } . We aretherefore led to u ≥ w − r by Theorem 1.1.It is clear that w C ( p, t ) > t > r / √
12 for all p ∈ H , which implies that u > t > r / √
12. Hence Γ t defined in (7.2) is empty when t > r / √
12. Note thatthe conclusion does not depend on the particular choice of u , as explained in Corollary7.2. (cid:3) Remark . Theorem 7.3 indicates that a bounded axisymmetric mean curvature flowencounters singularities at a certain time
T > Remark . The following result stronger than Theorem 7.3 holds: For any continuoussolution u of (MCF) with zero level set Γ t for any t ≥
0, if Γ ⊂ B r with some r >
0, thenΓ t = ∅ when t > r / √
12. Here we do not need to assume the axial symmetry of Γ butwe must specify the solution u since it is not known in general whether or not Γ t dependson the choice of u . Definition 7.1.
We say T ≥ extinction time of the mean curvature flow Γ t in theHeisenberg group, if Γ t = ∅ when t ≤ T and Γ t = ∅ when t > T .We next proceed to investigate the asymptotic profile after normalization for a spherein the Heisenberg group. It is well-known that in the Euclidean space any normalizedcompact convex surface converges to a sphere as t tends to the extinction time [H]. How-ever, the normalized curvature flow from a sphere of radius r in the Heisenberg grouplooks like an ellipsoid E T := { P ∈ H : 12 T ( P + P ) + 16 P = 1 } (7.3) at the extinction time T = r / √ Proposition 7.4.
Suppose that Γ t ⊂ H ( t ≥ ) is the horizontal mean curvature flowas defined in (7.2) with Γ = { p ∈ H : | p | = r } , where r > is a given radius. Thenthe extinction time T = r / √ and the normalized flow Γ t / √ r − t → E T as t → T ,where E T is given in (7.3).Proof. We take w C ( p, t ) = min { ( p + p ) + 12 t ( p + p ) + 16 p + 12 t − r , C } with C >
0. It is easily seen that w C ( p,
0) = 0 if and only if p ∈ Γ . We have also shownthat w C is a solution of (1.2). We track the evolution by setting Γ t = { p ∈ H : w C ( p, t ) =0 } for all t ≥
0. It is clear that Γ t = ∅ when t > r / √
12 and Γ t = ∅ when t ≤ r / √ p ( t ) = ( p ( t ) , p ( t ) , p ( t )) ∈ Γ t , we have12 t ( p ( t ) + p ( t )) + 16 p ( t ) ≤ r − t . (7.4)We normalize the flow by letting P ( t ) = p ( t ) / √ r − t for any p ( t ) ∈ Γ t . Then (7.4) iswritten as 12 t ( P ( t ) + P ( t )) + 16 P ( t ) ≤ . (7.5)By setting U ( P, t ) = ( P + P ) ( r − t ) + 12 t ( P + P ) + 16 P − r − t w C ( p r − t P ( t ) , t ) = U ( P ( t ) , t )) . Sending the limit as t → T with (7.5) taken into account, we obtain12 T ( P ( T ) + P ( T )) + 16 P ( T ) = 1for the limit P ( T ) of any subsequence of P ( t ) as t → ∞ . The consequence above amountsto saying that the limit of the set Γ t / √ r − t is contained in E T .On the other hand, for any P = ( P , P , P ) ∈ E T , we have w C ( p r − t λP ) = ( r − t ) W ( λ, P, t ) , where λ > W ( λ, P, t ) = ( λ ( P + P ) ( r − t ) + λ − λ ( t − T )( P + P ) . One may take λ ( t ) > w C ( √ r − t λ ( t ) P, t ) = 0; in other words, λ ( t ) P ∈ Γ t / √ r − t . Moreover, λ ( t ) → t → T , which implies that P belongs to the limitof a sequence of elements in Γ t / √ r − t .In conclusion, we obtain Γ t / √ r − t → E T as t → T . (cid:3) We stress that this result is very different from that in the Euclidean space. Thenormalized asymptotic shape of horizontal mean curvature flow in the Heisenberg groupstarting from a ball is an ellipsoid. Moreover, the shape of the ellipsoid depends on theextinction time T and therefore the size of the initial surface. It would be interesting toshow this result for a general compact and convex initial surface. ORIZONTAL MEAN CURVATURE FLOW IN THE HEISENBERG GROUP 29
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Dipartimento di Matematica dell’Universit`a di Bologna, Piazza di Porta S. Donato, 5,40126, Bologna, Italy
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