Asymptotic convergence of evolving hypersurfaces
aa r X i v : . [ m a t h . DG ] J a n ASYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES
CARLO MANTEGAZZA AND MARCO POZZETTAA
BSTRACT . If ψ : M n → R n +1 is a smooth immersed closed hypersurface, we considerthe functional F m ( ψ ) = Z M |∇ m ν | dµ, where ν is a local unit normal vector along ψ , ∇ is the Levi–Civita connection of the Rie-mannian manifold ( M, g ) , with g the pull–back metric induced by the immersion and µ the associated volume measure. We prove that if m > ⌊ n/ ⌋ then the unique globallydefined smooth solution to the L –gradient flow of F m , for every initial hypersurface,smoothly converges asymptotically to a critical point of F m , up to diffeomorphisms. Theproof is based on the application of a Łojasiewicz–Simon gradient inequality for the func-tional F m . C ONTENTS
1. Introduction 12. Preliminary computations 53. Analysis of the second variation 84. Convergence 11References 171. I
NTRODUCTION
We consider a closed connected differentiable manifold M n of dimension n and ψ : M n → R n +1 a smooth immersion of M n in the Euclidean space R n +1 . We shall usuallyomit the superscript n denoting the dimension of M . For such an immersion, we alwaysassume that M is endowed with the metric tensor g = ψ ∗ h· , ·i R n +1 , induced by the im-mersion ψ that we also sometimes simply denote as h· , ·i . The Levi–Civita connectionof the Riemannian manifold ( M, g ) is denoted by ∇ and the associated volume measureby µ . Then, for m ∈ N with m ≥ , we consider the functional F m ( ψ ) := Z M |∇ m ν | dµ Date : January 12, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Geometric flows, Łojasiewicz–Simon gradient inequality, smooth convergence. on the smooth immersions ψ : M n → R n +1 , where ν is a (locally defined) unit normalvector field along ψ . Let us specify that in the above definition, if ν = ν α e α and e α is thestandard basis in R n +1 , we mean |∇ m ν | := P n +1 α =1 |∇ m ν α | (notice that F m is independentof the local choice of the unit normal ν ).By the formula for the first variation of F m (see Theorem 2.1), one can prove that theassociated L –gradient flow is defined by an evolution equation ∂ϕ∂t ( p, t ) = − E m ( ϕ t )( p ) ν t ( p ) for a smooth map ϕ : M × [0 , T ) → R n +1 (where ϕ t = ϕ ( · , t ) : M → R n +1 describes themoving hypersurface and ν t is its local unit normal vector field), which turns out to bea (quasilinear and degenerate) parabolic system of PDEs.If m > ⌊ n/ ⌋ , the study carried out in [11] shows that for every initial smooth immersedhypersurface ϕ : M → R n +1 , there exists a unique smooth solution ϕ t with initialdatum ϕ , defined for all positive times; moreover, ϕ t sub–converges to a critical point ϕ ∞ : M → R n +1 of the functional F m , that is, such that E m ( ϕ ∞ ) = 0 (see Theorem 4.3).By sub–convergence we mean that for some sequence of times t j → + ∞ , the sequence ϕ t j smoothly converges to ϕ ∞ , up to diffeomorphisms and translations in R n +1 . More pre-cisely, there exist a sequence of smooth diffeomorphisms σ j : M → M and a sequenceof points p j ∈ R n +1 such that the sequence of immersions ϕ t j ◦ σ j − p j converge to ϕ ∞ in C k ( M ) , for any k ∈ N . From such a sub–convergence result it is anyway not possible toimmediately deduce that the flow fully converges , i.e., that there exists the full limit of ϕ t as t → + ∞ in C k ( M ) for any k (up to diffeomorphisms). Actually, the sub–convergenceof the flow does not even guarantee that the limits of the flow along different divergingsequences of times coincide. Moreover, as the evolution equations involved here are oforder greater or equal than four, it is not even possible to conclude that the flow staysin a compact set of R n +1 for all times by means of comparison arguments, as maximumprinciples are not applicable.In this work we address this issue, that is, we prove that the gradient flow of F m does actually converge, for any initial hypersurface. Our main result is the followingtheorem. Theorem 1.1.
Let ϕ : M n → R n +1 be a smooth immersion of a closed hypersurface and let m > ⌊ n/ ⌋ . Then the unique smooth solution ϕ : M × [0 , + ∞ ) → R n +1 to the evolutionproblem ( ∂ϕ∂t = − E m ( ϕ t ) ν t ϕ (0 , · ) = ϕ converges in C k ( M ) to a smooth critical point ϕ ∞ : M → R n +1 of F m as t → + ∞ , forevery k ∈ N up to diffeomorphisms of M ; more precisely, there exists a one-parameter family ofdiffeomorphisms σ t : M → M such that the flow ϕ t ◦ σ t converges in C k ( M ) to a smooth criticalpoint ϕ ∞ of F m as t → + ∞ , for every k ∈ N .In particular, there exists a compact set K ⊆ R n +1 such that M t = ϕ t ( M ) ⊆ K for any t ≥ . SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 3
A relevant motivation for the study of the gradient flow of the functionals F m goesback to Ennio De Giorgi. In one of his last papers, he conjectured that any compact n –dimensional hypersurface in R n +1 , evolving by the gradient flow of certain functionalsdepending on sufficiently high derivatives of the curvature does not develop singular-ities during the flow (see [6] and [7, Section 5, Conjecture 2] for an English translation,see also [11, Section 9]). This result was central in his program to approximate singulargeometric flows, as the mean curvature flow, with sequences of smooth ones (see [11,Sec. 9] and [2] for a result in this direction). The functionals F m are strictly related tothe ones proposed by De Giorgi since, roughly speaking, the derivative of the normalfield yields the curvature of M (see (1.1)). Though not exactly the same, the energies F m can then play the same role in the approximation process he suggested and the analysisof the asymptotic behavior of their gradient flow is another step in understanding suchprocess.The main tool in the proof of Theorem 1.1 is a Łojasiewicz–Simon gradient inequality forthe functional F m (see Corollary 4.2). Such an estimate bounds a less–than– / powerof the difference in “energy” (the value of the functional) between a critical point and apoint sufficiently close to it in terms of a suitable norm of the first variation of the func-tional. The use of this kind of inequalities in the study of the convergence of parabolicequations of gradient–type goes back to Łojasiewicz [9, 10] and to the seminal paper ofSimon [16]. More recently, useful sufficient hypotheses implying a Łojasiewicz–Simongradient inequality have been derived in [3]. Building on the abstract tools developedin [3], a first recent application of the method is contained in [4], where the authors in-vestigate the Willmore flow of surfaces in the neighborhoods of critical points. In thelast years and in the context of higher order geometric gradient flows, the Łojasiewicz–Simon inequality appeared as a tool for “promoting” the sub–convergence of a flowto its full convergence. As applications of this method we mention [5], in which it isproved the full convergence of the elastic flow of open clamped curves and [14], inwhich the sub–convergence of the p –elastic flow of closed curves on Riemannian man-ifolds is shown to imply the full convergence. The analysis in [14] led to a further sim-plification and deeper understanding of the method, which is exposed in [13]. In thiswork we essentially generalize the strategy employed in [13] to the gradient flows of thefunctionals F m . Moreover we tried to keep most of the arguments as general as possi-ble, in order that the method could be possibly applied also to other geometric gradientflows. Notation and geometry of submanifolds.
In the whole paper we assume that M iscompact and orientable, so that we can choose a smooth global choice of a unit normalvector field ν along any smooth immersion ψ : M → R n +1 . In the case the M is notorientable, given an initial immersion ϕ : M → R n +1 , we can consider the canonicaltwo–fold cover π : f M → M , where f M is orientable and the initial immersion e ϕ = ϕ ◦ π .By uniqueness of the flow ϕ t starting at ϕ (Theorem 4.3), it follows that the flow e ϕ t starting at e ϕ is just e ϕ t = ϕ t ◦ π . Therefore, if we prove that e ϕ t smoothly converges, thenthe same holds for the flow ϕ t . Hence, also in this case Theorem 1.1 holds. SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 4
As the metric g is obtained pulling it back with ψ , in local coordinates, we have g ij ( x ) = (cid:28) ∂ψ ( x ) ∂x i , ∂ψ ( x ) ∂x j (cid:29) and the canonical volume measure induced by the metric g is given in local coordinatesby µ = p det( g ij ) L n where L n is the standard Lebesgue measure on R n .The induced covariant derivative on ( M, g ) of a tangent vector field X is given by ∇ j X i = ∂∂x j X i + Γ ijk X k (in the whole paper we will adopt the Einstein convention of summation over repeatedindices) where the Christoffel symbols Γ ijk are expressed by the formula Γ ijk = 12 g il (cid:18) ∂∂x j g kl + ∂∂x k g jl − ∂∂x l g jk (cid:19) . We will write ∂ i for the coordinates derivatives, opposite to the covariant ones ∇ i . With ∇ k T we will mean the k –th iterated covariant derivative of a tensor T . If f is a smoothfunction on a smooth immersed hypersurface, the symbol ∇ f denotes its gradient and ∇ f its Hessian, whose trace is the Laplacian ∆ f .The second fundamental form A of the immersion ψ is the bilinear symmetric formacting on any pair of tangent vector fields X, Y to the hypersurface as A( X, Y ) = − (cid:10) ∇ R n +1 X Y, ν i , given a (global, since we assumed M orientable) choice of the unit normal vector ν (we will usually identify T M with dψ ( T M ) ⊆ R n +1 and in this formula the field Y isextended locally around ψ ( M ) in R n +1 ). Hence A is defined up to a sign, that is, upto the choice of ν , while A ν is independent of the choice of ν . In local coordinates, thecomponents h ij of A are given by h ij ( x ) = − (cid:28) ν ( x ) , ∂ ψ ( x ) ∂x i ∂x j (cid:29) . We recall that the following Gauss–Weingarten relations hold ∂ ij ψ = Γ kij ∂ k ψ − h ij ν, ∂ i ν = h ij g jk ∂ k ψ. (1.1)The mean curvature H of ψ is the trace of A , that is H( x ) = g ij ( x ) h ij ( x ) . By means of the Gauss equation, the Riemann tensor can be expressed via the secondfundamental form, in local coordinates, as follows R ijkl = h ik h jl − h il h jk . Hence, the formulas for the interchange of covariant derivatives become ∇ i ∇ j X s − ∇ j ∇ i X s = R ijkl g ks X l = R sijl X l = ( h ik h jl − h il h jk ) g ks X l , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 5 ∇ i ∇ j ω k − ∇ j ∇ i ω k = R ijkl g ls ω s = R sijk ω s = ( h ik h jl − h il h jk ) g ls ω s , where we recall that by ∇ i ∇ j X s we mean the s -th component of the field ( ∇ X )( ∂ i , ∂ j ) .Abusing a little the notation, if T , ..., T N is a finite family of tensors, we denote by ⊛ Nk =1 T k := T ∗ . . . ∗ T N a generic contraction of some indices of the tensors T , ..., T N using the coefficients g ij or g ij . We will also denote p s ( T , . . . , T N ) := X i + ... + i N = s C i ,...,i N ∇ i T ∗ . . . ∗ ∇ i N T N , for some constants C i ,...,i N ∈ R . Notice that in every additive term of p s ( T , . . . , T N ) eachtensor appears exactly once (there are no repetitions).We will use instead the symbol q s ( T , . . . , T N ) for “polynomials” of the form q s ( T , . . . , T N ) := X (cid:2) ⊛ M i =1 ∇ i T . . . ⊛ M N i N =1 ∇ i N T N (cid:3) , with M j ≥ for any j = 1 , ..., N and with s = M X i =1 ( i + 1) + . . . + M N X i N =1 ( i N + 1) . Hence, repetitions are allowed in q s and in every additive term there must be presentevery argument of q s .We notice that, by the above relations, the Riemann tensor of the hypersurface can bewritten as R = A ∗ A , exploiting the above notation.2. P RELIMINARY COMPUTATIONS
Let us recall the first variation formula for the functional F m . Theorem 2.1 ([11, Theorem 3.7]) . Let ϕ t : M n → R n +1 be a smooth family of immersionssmoothly depending on t ∈ ( − ε, ε ) and X t = ∂ t ϕ t . Then, for every t ∈ ( − ε, ε ) , there holds ddt F m ( ϕ t ) = Z M E m ( ϕ t ) h ν, X t i dµ t , with E m ( ϕ t ) = 2( − m ∆ m H + q m +1 ( ∇ ν, A) + H , where all the quantities are relative to the hypersurface ϕ t . The next lemma states the evolution formulas for the geometric quantities that weneed in the computation of the second variation of the functional F m . Lemma 2.2.
Let ϕ t : M n → R n +1 be a smooth family of immersions smoothly depending on t ∈ ( − ε, ε ) and ϕ = ϕ . Let X = ∂ t ϕ t | t =0 and assume that X is a normal vector field along ϕ .Then, we have ∂ t g ij | t =0 = 2 h ν, X i h ij ,∂ t g ij | t =0 = − h ν, X i g ik g jl h kl , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 6 ∂ t ν | t =0 = −∇h ν, X i ,∂ t Γ kij (cid:12)(cid:12) t =0 = ∇ A ∗ h ν, X i + A ∗ ∇h ν, X i ,∂ t h ij | t =0 = −∇ ij h ν, X i + h ν, X i h ij , (2.1) ∂ t H | t =0 = − ∆ h ν, X i − h ν, X i| A | , (2.2) ∂ t ∆ m f | t =0 − ∆ m ∂ t f | t =0 = p m ( f , A , h ν, X i ) , (2.3) for any smooth function f ∈ C ∞ ( M × ( − ε, ε )) and m ∈ N with m ≥ , where f = f ( · , and ∂ t q m +1 ( ∇ ν, A) | t =0 = q m +3 ( h ν, X i , ∇ ν, A) + h ν, X i q m +2 ( ∇ ν, A) . (2.4) Proof.
The first four formulas are computed explicitly at page 150 of [11].By means of the Gauss–Weingarten relations (1.1), setting X = βν , hence h ν, X i = β ,we compute ∂ t h ij | t =0 = − ∂ t h ν, ∂ ij ϕ t i| t =0 = − h ν, ∂ ij ( βν ) i + h∇ β, ∂ ij ϕ i = − ∂ ij β − β h ν, ∂ i ( h jl g lk ∂ k ϕ ) i + h ∂ l βg ls ∂ s ϕ, Γ kij ∂ k ϕ − h ij ν i = − ∂ ij β − β h ν, h jl g lk ∂ ik ϕ i + ∂ k β Γ kij = − ∇ ij β + βh ik g kl h lj that is, ∂ t h ij | t =0 = −∇ ij h ν, X i + h ν, X i h ij , hence it follows ∂ t H | t =0 = ∂ t ( g ij h ij ) | t =0 = − h ν, X i| A | − ∆ h ν, X i + h ν, X i| A | = − ∆ h ν, X i − h ν, X i| A | . We now deal with equation (2.3) arguing by induction on m ≥ . Using the previousevolution formulas, for m = 1 we compute ∂ t ∆ f | t =0 = ∂ t ( g ij ( ∂ ij f − Γ kij ∂ k f )) | t =0 = − h ν, X i g ik g jl h kl ∇ ij f + ∆ ∂ t | t =0 f − g ij ( ∇ A ∗ h ν, X i + A ∗ ∇h ν, X i ) ∂ k f , and the claim follows. Now for m + 1 ≥ , by induction we get ∂ t ∆ m +1 f | t =0 = ∆( ∂ t ∆ m f ) | t =0 + p (∆ m f , A , h ν, X i )= ∆ (∆ m ∂ t f | t =0 + p m ( f , A , h ν, X i )) + p m +2 ( f , A , h ν, X i ) . Finally, in order to show equation (2.4), we need to differentiate a generic term of theform ⊛ Nk =1 ∇ i k ∇ ν ⊛ Ml =1 ∇ j l A , with P Nk =1 ( i k + 1) + P Ml =1 ( j l + 1) = 2 m + 1 .For any component ν α of ν we can apply [11, Proposition 3.6] in order to get ∂ t ( ∇ i k ∇ ν α ) | t =0 = −∇ i k +1 ∇ α h ν, X i + p i k ( h ν, X i , ∇ ν, A) , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 7 where ∇ α h ν, X i denotes the α –th component in R n +1 of the gradient ∇h ν, X i . Also,by [11, Lemma 3.5] and formula (2.1), we have ∂ t ( ∇ j l A) | t =0 = ∇ j l ( −∇ h ν, X i + h ν, X i A ∗ A) + p j l (A , A , h ν, X i )= −∇ j l +2 h ν, X i + p j l (A , A , h ν, X i ) . Therefore, using these formulas and the ones above for the derivative of the metric g ij and its inverse g ij , formula (2.4) follows. (cid:3) We can now compute the second variation of F m . Theorem 2.3.
Let ϕ t : M n → R n +1 be a smooth family of immersions smoothly depending on t ∈ ( − ε, ε ) . Denote ϕ = ϕ and assume that ϕ is a critical point for F m , i.e., E m ( ϕ ) = 0 . Let X = ∂ t ϕ t | t =0 and assume that X is normal along ϕ . Then d dt F m ( ϕ t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z M (cid:0) − m +1 ∆ m +1 h ν, X i + Ω( h ν, X i ) (cid:1) h ν, X i dµ, where Ω( h ν, X i ) is linear in h ν, X i and depends on its covariant derivatives of order m at most.Proof. By Theorem 2.1 we have d dt F m ( ϕ t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ddt Z M E m ( ϕ t ) h ν, ∂ t ϕ t i dµ t (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z M (cid:20) ∂∂t E m ( ϕ t ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) t =0 h ν, X i dµ, as E m ( ϕ ) = 0 . Using the explicit expression for E m ( ϕ t ) (Theorem 2.1), applying for-mula (2.3) with f = H and equations (2.2), (2.4), we get ddt E m ( ϕ t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 2( − m +1 ∆ m (∆ h ν, X i + h ν, X i| A | ) + p m (H , A , h ν, X i )+ q m +3 ( h ν, X i , ∇ ν, A) + h ν, X i q m +2 ( ∇ ν, A) − (∆ h ν, X i + h ν, X i| A | )= 2( − m +1 ∆ m +1 h ν, X i + 2( − m +1 ∆ m ( h ν, X i| A | )+ q m +3 ( h ν, X i , ∇ ν, A) + h ν, X i q m +2 ( ∇ ν, A) − (∆ h ν, X i + h ν, X i| A | ) . Hence, the thesis follows by observing that a generic monomial in q m +3 ( h ν, X i , ∇ ν, A) is of the form ⊛ Nk =1 ∇ i k h ν, X i ⊛ Ml =1 ∇ j l ∇ ν ⊛ Ps =1 ∇ r s A , with N X k =1 ( i k + 1) + M X l =1 ( j l + 1) + P X s =1 ( r s + 1) = 2 m + 3 , and N, M, P ≥ and then i k ≤ m for any k . (cid:3) It follows that, by polarization, we can define the bilinear form ( δ F m ) ϕ ( f , f ) := dds ddt F m ( ϕ + sf ν + tf ν ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z M (cid:0) − m +1 ∆ m +1 f + Ω( f ) (cid:1) f dµ , (2.5) SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 8 for any pair of smooth functions f , f : M → R and Ω is as in Theorem 2.3.3. A NALYSIS OF THE SECOND VARIATION
Suppose that ϕ : M → R n +1 is a smooth critical point of F m , i.e., E m ( ϕ ) = 0 . Theformula for the second variation given above shows that ( δ F m ) ϕ ( f , f ) is well definedfor f ∈ W m +2 , ( M, g ) and f ∈ L ( µ ) . This means that ( δ F m ) ϕ ( f, · ) ∈ L ( µ ) ⋆ , for any f ∈ W m +2 , ( M, g ) and it is well defined the map W m +2 , ( M, g ) ∋ f ( δ F m ) ϕ ( f, · ) ∈ L ( µ ) ⋆ . We are going to exploit the theory of Fredholm operators between Banach spaces. Fordefinitions and results on the subject we refer the reader to [8, Section 19.1]. We recallthat if T : V → V is a Fredholm operator between Banach spaces, its index is definedto be the integer number index T := dim ker T − dim coker T. where dim denotes the dimension of a finite dimensional vector space. Proposition 3.1.
Let ϕ : M → R n +1 be a smooth critical point of F m , i.e., E m ( ϕ ) = 0 . Thenthe second variation functional ( δ F m ) ϕ : W m +2 , ( M, g ) → L ( µ ) ⋆ is a Fredholm operator of index zero. In order to prove Proposition 3.1 we need the following commutation rule.
Lemma 3.2.
Let ϕ : M n → R n +1 be a smooth immersion and let T be a tensor defined on M .Assume M is endowed with the pull-back metric g induced by ϕ . Then ∇ ∆ l T − ∆ l ∇ T = p l − (A , A , T ) , for any l ∈ N with l ≥ .Proof. As we need to prove a pointwise identity, we can take a local coordinate frame E , ..., E n which is orthonormal at a given point p (that is, h E i , E j i = δ ij ) and ∇ i E j = 0 at p . In this way we can compute (∆ ∇ T )( E k ) = ( ∇ ( ∇ T )( E i , E i ))( E k )= ( ∇ i ( ∇ i ∇ T ) − ∇ ∇ i E i ∇ T )( E k )= ( ∇ i ( ∇ i ∇ T ))( E k )= ∇ i (( ∇ i ∇ T )( E k )) − ( ∇ i ∇ T )( ∇ i E k )= ∇ i ( ∇ T ( E i , E k )) − ∇ T ( E i , ∇ i E k )= ∇ i ( ∇ T ( E i , E k )) . SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 9 at the point p . On the other hand, using that for any tensor S we have the commutationrule ( ∇ S )( E j , E l ) = ( ∇ S )( E l , E j ) + R ∗ S for any j and l , we obtain ( ∇ ∆ T )( E k ) = ∇ k (trace ∇ T )= trace ∇ k ∇ T = ( ∇ k ( ∇ T ))( E i , E i )= ( ∇ T )( E k , E i , E i )= ( ∇ T )( E i , E k , E i ) + R ∗ ∇ T = ( ∇ i ( ∇ T ))( E k , E i ) + R ∗ ∇ T = ∇ i ( ∇ T ( E k , E i )) − ( ∇ T )( ∇ i E k , E i ) − ( ∇ T )( E k , ∇ i E i ) + R ∗ ∇ T = ∇ i ( ∇ T ( E i , E k ) + R ∗ T ) + R ∗ ∇ T = (∆ ∇ T )( E k ) + ∇ (R ∗ T ) + R ∗ ∇ T = (∆ ∇ T )( E k ) + p (A , A , T ) , where we used that R = A ∗ A , by Gauss equations. Hence, the thesis is proved for l = 1 .Letting now l + 1 ≥ , by induction we obtain ∇ ∆∆ l T = ∆ ∇ ∆ l T + p (A , A , ∆ l T ) = ∆(∆ l ∇ T + p l − (A , A , T )) + p l +1 (A , A , T ) , and the thesis follows. (cid:3) We are now ready to prove Proposition 3.1. A relevant property about Fredholmoperators that we are going to use is the following. If T : V → V is a Fredholmoperator between Banach spaces and K : V → V is a compact operator, then T + K isFredholm and index( T + K ) = index T (see [8, Corollary 19.1.8]). Proof of Proposition 3.1.
For f ∈ W m +2 , ( M, g ) the functional ( δ F m ) ϕ ( f , · ) is given by ( δ F m ) ϕ ( f , f ) = hL ( f ) , f i L ( µ ) , where L : W m +2 , ( M, g ) → L ( µ ) is L ( f ) = 2( − m +1 ∆ m +1 f + Ω( f ) , and Ω is as in Theorem 2.3, hence Ω is a compact operator. Therefore ( δ F m ) ϕ : W m +2 , ( M, g ) → L ( µ ) ⋆ is Fredholm of index zero if and only if the same holds for L : W m +2 , ( M, g ) → L ( µ ) .We then claim that the operator C id + 2( − m +1 ∆ m +1 : W m +2 , ( M, g ) → L ( µ ) is invertible for C > sufficiently large, thus it is Fredholm of index zero. As the inclu-sion id : W m +2 , ( M, g ) → L ( µ ) is compact, this eventually implies that − m +1 ∆ m +1 : W m +2 , ( M, g ) → L ( µ ) is Fredholm of index zero. SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 10
The injectivity of the above operator immediately follows, suppose indeed that we have Cf + 2( − m +1 ∆ m +1 f = 0 , if m = 2 k + 1 , multiplying by f and integrating, we get C Z M f dµ = − Z M f ∆ k +1) f dµ = − Z M (∆ k +1 f ) dµ, then f = 0 . If instead m = 2 k , multiplying by f and integrating we get C Z M f dµ = 2 Z M f ∆ k +1 f dµ = − Z M |∇ ∆ k f | dµ, then f = 0 as well.About the surjectivity, given h ∈ L ( µ ) we aim at finding f ∈ W m +2 , ( M, g ) such that Cf + 2( − m +1 ∆ m +1 f = h . We shall minimize the functional A m : W m +1 , ( M, g ) → R defined by A m ( f ) := (R M (cid:2) C f + (∆ k +1 f ) − f h (cid:3) dµ if m = 2 k + 1 , R M (cid:2) C f + |∇ ∆ k f | − f h (cid:3) dµ if m = 2 k. We can prove that A m is coercive on W m +1 , ( M, g ) , up to choosing C > sufficientlylarge (depending on m and the geometry of ( M, g ) ).We first consider the case m = 2 k + 1 . Integrating by parts in the integral R M (∆ k +1 f ) dµ ,that is, using the divergence theorem and applying the commutation rule of Lemma 3.2we get Z M (∆ k +1 f ) dµ = Z M − h∇ ∆ k f, ∇ ∆ k +1 f i dµ = Z M (cid:2) −h ∆ k ∇ f, ∆ k +1 ∇ f i + ∇ ∆ k f ∗ p k +1) − (A , A , f )+ ∇ ∆ k +1 f ∗ p k − (A , A , f ) (cid:3) dµ = Z M (cid:2) −h ∆ k ∇ f, ∆ k +1 ∇ f i + p k +2 (A , A , f, f ) (cid:3) dµ = Z M (cid:2) ( − k +1 h∇ k +1 f, ∆ k +1 ∇ k +1 f i + p k +2 (A , A , f, f ) (cid:3) dµ = Z M (cid:2) |∇ k +2 f | + p k +2 (A , A , f, f ) (cid:3) dµ = Z M (cid:2) |∇ m +1 f | + p m (A , A , f, f ) (cid:3) dµ. Moreover, by definition of p s , we can apply the divergence theorem on the integral R M p m (A , A , f, f ) dµ in the above expression so that in the polynomial there appearderivatives of f of order m at most. SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 11
We recall that for any covariant tensor T there holds the general inequality (see [1, Chap-ter 3, Section 7.6]) k∇ l T k L ( µ ) ≤ C l,m k∇ m +1 T k lm +1 L ( µ ) k T k m +1 − lm +1 L ( µ ) ≤ ε k∇ m +1 T k L ( µ ) + C l,m ( ε ) k T k L ( µ ) , (3.1)for any l ≤ m and ε > . Therefore we can estimate Z M | p m (A , A , f, f ) | dµ ≤ C m ( k A k ∞ ) X Z M |∇ l f ||∇ l f | dµ, where l , l ≤ m and then Z M | p m (A , A , f, f ) | dµ ≤ εC m ( k A k ∞ ) k∇ m +1 f k L ( µ ) + C m ( k A k ∞ , ε ) k f k L ( µ ) . Therefore, taking εC m ( k A k ∞ ) < / and C = C ( m, k A k ∞ ) sufficiently large, we estimate A m ( f ) ≥ C Z M (cid:2) f + |∇ m +1 f | − h (cid:3) dµ, that by inequality (3.1) implies that A m is coercive on W m +1 , ( M, g ) . Analogously, onecan prove the coercivity of A m also in the case m = 2 k .It follows that there exists a function F ∈ W m +1 , ( M, g ) solving Z M (cid:2) CF f + 2∆ k +1 F ∆ k +1 f (cid:3) dµ = Z M f h dµ ∀ f ∈ W m +1 , ( M, g ) if m = 2 k + 1 , or Z M (cid:2) CF f + 2 h∇ ∆ k F, ∇ ∆ k f i (cid:3) dµ = Z M f h dµ ∀ f ∈ W m +1 , ( M, g ) if m = 2 k . In any case, F is a weak solution to an elliptic equation with constant co-efficients and datum h ∈ L ( µ ) (in the sense of [1, Point (d), Page 85]). Therefore,the standard regularity theory for distributional solutions applies (see [1, Theorem,Page 85]), hence F belongs to W m +2 , ( M, g ) . Integrating by parts, we then get that F solves CF + 2( − m +1 ∆ m +1 F = h , as required. (cid:3)
4. C
ONVERGENCE
Suppose that ϕ : M → R n +1 is a smooth critical point of F m , that is, E m ( ϕ ) = 0 . Thenfor ρ > suitably small, it is well defined the functional E m : B ρ (0) ⊆ W m +2 , ( M, g ) → R given by E m ( f ) := F m ( ϕ + f ν ) . The advantage of the above definition is that the functional E m is now defined on anopen set of a Banach space and we can then look at first and second variation functionalsin the classical sense of functional analysis. More precisely, by Theorem 2.1 we have ( δ E m ) f ( f ) := ddt E m ( f + tf ) (cid:12)(cid:12)(cid:12) t =0 = Z M E m ( ϕ + f ν ) h ν , ν i f dµ , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 12 where ν (resp. ν ) is a unit normal vector along ϕ (resp. ϕ + f ν ) and µ is the volumemeasure induced by ϕ + f ν . In this way we see that δ E m : B ρ (0) ⊆ W m +2 , ( M, g ) → L ( µ ) ⋆ . Analogously, by Theorem 2.3 and formula (2.5) the second variation of E m evaluated at ∈ B ρ (0) is given by ( δ E m ) ( f , f ) = Z M (cid:0) − m +1 ∆ m +1 f + Ω( f ) (cid:1) f dµ, for Ω as in Theorem 2.3, so that ( δ E m ) : W m +2 , ( M, g ) → L ( µ ) ⋆ , and it is a Fredholm operator of index zero by Proposition 3.1.In this setting we can apply the following abstract result stating sufficient conditionsimplying a Łojasiewicz–Simon gradient inequality. Proposition 4.1 ([14, Corollary 2.6]) . Let E : B ρ (0) ⊆ V → R be an analytic map, where V is a Banach space. Suppose that is a critical point for E , i.e., δE = 0 . Assume that thereexists a Banach space Z such that V ֒ → Z , the first variation δE : B ρ (0) → Z ⋆ is Z ⋆ –valuedand analytic and the second variation δ E : V → Z ⋆ evaluated at is Z ⋆ –valued and Fredholmof index zero.Then there exist constants C, θ > and α ∈ (0 , / such that | E ( f ) − E (0) | − α ≤ C k δE f k Z ⋆ , for every f ∈ B θ (0) ⊆ V . The above functional analytic result is a corollary of the useful theory developed in [3]and it has been also proved in [15] independently.Applying Proposition 4.1 to the functional E m we obtain the following corollary. Corollary 4.2.
Let ϕ : M → R n +1 be a smooth critical point of F m , i.e., E m ( ϕ ) = 0 . Let ρ > such that E m : B ρ (0) ⊆ W m +2 , ( M, g ) → R is well defined.Then, there exist constants C > , θ ∈ (0 , ρ ] and α ∈ (0 , / such that |F m ( ϕ + f ν ) − F m ( ϕ ) | − α ≤ C k ( δ E m ) f k L ( µ ) ⋆ , for every f ∈ B θ (0) ⊆ W m +2 , ( M, g ) .Proof. We want to apply Proposition 4.1 with V = W m +2 , ( M, g ) and Z = L ( µ ) . ByProposition 3.1 and the discussion at the beginning of the section, we just need to checkthat E m and δ E m are analytic as maps between Banach spaces.We can rewrite E m ( f ) = Z M n +1 X α =1 h∇ m ν αf , ∇ m ν αf i dµ f where ν f is a unit normal along ϕ + f ν and µ f is the volume measure induced by ϕ + f ν . If ψ : M → R n +1 is any immersion, we have that a unit normal along ψ is ν ψ = ⋆ ∂ ψ ∧ ... ∧ ∂ n ψ | ∂ ψ ∧ ... ∧ ∂ n ψ | , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 13 where ⋆ denotes the Euclidean Hodge star operator. As ψ is an immersion, we see that ψ ν ψ is analytic. It follows that f ν f is analytic as well. As the metric tensorinduced by an immersion ψ : M → R n +1 has components g ij = h ∂ i ψ, ∂ j ψ i , we getthat the metric tensor of ϕ + f ν depends analytically on f and then it is analytic thedependence of µ f and of Christoffel symbols (and thus of the connection) on f . Thenthe integrand in the definition of E m is just a sum of compositions and multiplicationsof functions which are analytic in f . Finally, integration is linear on L ( µ ) , then f m ( f ) ∈ R is analytic for f ∈ B ρ (0) ⊆ W m +2 , ( M, g ) .By the very same arguments, one can check that also f ( δ E m ) f is analytic. Hence,all the hypotheses of Proposition 4.1 are satisfied and the thesis follows. (cid:3) The starting point for proving the smooth convergence of the gradient flow of F m isthe following sub–convergence theorem. Theorem 4.3 ([11, Theorem 7.8, Theorem 8.2]) . Let ϕ : M n → R n +1 be a smooth immersionand let m > ⌊ n/ ⌋ . Then there exists a unique smooth solution ϕ : M × [0 , + ∞ ) → R n +1 tothe evolution equation ( ∂ t ϕ = − E m ( ϕ t ) ν t ,ϕ ( · ,
0) = ϕ , where ν t denotes a unit normal vector field along ϕ t := ϕ ( · , t ) . Moreover, the solution satisfiesthe estimates k∇ k A t k L ∞ ( M,g t ) ≤ C ( k, n, ϕ ) , (4.1) for any t ∈ [0 , + ∞ ) , where A t and g t are the second fundamental form and the metric of ϕ t respectively and there exists a smooth critical point ϕ ∞ : M → R n +1 of F m , a sequence of times t j → + ∞ and a sequence of points p j ∈ R n +1 such that k ϕ t j ◦ σ j − p j − ϕ ∞ k C k ( M ) −−−−→ j → + ∞ , for any k ∈ N , where σ j is a sequence of diffeomorphisms of M . We need a preliminary lemma.
Lemma 4.4.
Let ϕ , ϕ t , ϕ ∞ , σ j , t j , p j be as in Theorem 4.3. Then, for any ε > there is j ε ∈ N such that for any j ≥ j ε there exists δ j > such that the immersion ϕ t − p j coincides with ϕ ∞ + f t ν ∞ up to diffeomorphism, where ν ∞ is a unit normal vector along ϕ ∞ , for some “height”functions f t ∈ C ∞ ( M ) smoothly depending on t ∈ [ t j , t j + δ j ) . Moreover, k f t k W m +2 , ( M,g ∞ ) ≤ ε, for any t ∈ [ t j , t j + δ j ) .Proof. Fixed θ > and k > m + 2 , by Theorem 4.3 there is j θ such that for any j ≥ j θ we have k ϕ t ◦ σ j − p j − ϕ ∞ k C k ( M ) < θ, (4.2)for every t ∈ [ t j , t j + δ j ) , for some δ j > .Let us assume that ϕ ∞ is an embedding. The general statement analogously follows SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 14 by recalling that immersions are local embeddings. So for j θ large enough, ϕ t is anembedding as well for every t ∈ [ t j , t j + δ j ) . Moreover there exists U ⊆ R n +1 open setcontaining N := ϕ ∞ ( M ) such that it is well defined the projection map π : U → N as π ( p ) = p − ∇ R n +1 d N ( p ) , where d N is the distance function from N . The vector ∇ R n +1 d N ( p ) is orthogonal to N at π ( p ) , π is smooth on U and for j θ sufficiently large we have that ( ϕ t ◦ σ j ( M ) − p j ) ⊆ U for every t ∈ [ t j , t j + δ j ) (for a proof of these facts see [12, Proposition 4.2]).Hence, for x ∈ M the “height” function f t ( x ) is uniquely determined by the identity ϕ t ◦ σ j ( x ) − p j = π ( ϕ t ◦ σ j ( x ) − p j ) + f t ( x ) ν ∞ ( ϕ − ∞ ◦ π ◦ ( ϕ t ◦ σ j ( x ) − p j )) , that is, f t ( x ) = h ϕ t ◦ σ j ( x ) − p j − π ( ϕ t ◦ σ j ( x ) − p j ) , ν ∞ ( ϕ − ∞ ◦ π ◦ ( ϕ t ◦ σ j ( x ) − p j )) i . (4.3)Then, the map ( x, t ) f t ( x ) is smooth on M × [ t j , t j + δ j ) and k f t k W m +2 , ( M,g ∞ ) → as θ → , by inequality (4.2) and the fact that k > m + 2 .Hence, for the chosen ε > , taking a suitable θ > we have the estimate in the statementof the lemma. (cid:3) We are now ready for proving our main result. The proof of Theorem 1.1 is essentiallya generalization of the strategy employed in [13] to show the smooth convergence of theelastic flow of closed curves in R n . Proof of Theorem 1.1.
Let ϕ , ϕ t , ϕ ∞ , σ j , t j , p j be as in Theorem 4.3. Fixed k > m + 2 andchosen ε > smaller than the constant θ given by Corollary 4.2, relative to the criticalpoint ϕ ∞ , by Theorem 4.3 and Lemma 4.4, there exists j ε ∈ N such that for every j ≥ j ε we have k ϕ t ◦ σ j − p j − ϕ ∞ k C k ( M ) < ε, for every t ∈ [ t j , t j + δ j ) with some δ j > , moreover, ϕ t ◦ σ j − p j coincides with ϕ ∞ + f t ν ∞ ,up to diffeomorphism, for the functions f t given by Lemma 4.4 (we recall that f t ∈ C ∞ ( M ) depends on j ), satisfying k f t k W m +2 , ( M,g ∞ ) < ε < θ, (4.4)for every t ∈ [ t j , t j + δ j ) .We claim that it is possible to choose ε > small enough such that for any fixed j ≥ j ε ,the hypersurfaces ϕ t ◦ σ j − p j coincide with ϕ ∞ + f t ν ∞ (up to diffeomorphism) for somesmooth functions f t with k f t k W m +2 , ( M,g ∞ ) < θ for any t ∈ [ t j , + ∞ ) .We define H ( t ) := |F m ( ϕ t ) − F m ( ϕ ∞ ) | α , where α ∈ (0 , / is as in Corollary 4.2 applied to the critical point ϕ ∞ and, without lossof generality, we can clearly assume that H ( t ) > for any t . As F m ( ϕ t ) = F m ( ϕ ∞ + f t ν ∞ ) , SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 15 by Corollary 4.2 we have H ( t ) − αα ≤ C k ( δ E m ) f t k L ( µ ∞ ) ⋆ = C (cid:16)Z M | E m ( ϕ ∞ + f t ν ∞ ) h ν t , ν ∞ i| det g t dµ ∞ (cid:17) / ≤ C ( ϕ ∞ , θ ) (cid:16)Z M | E m ( ϕ ∞ + f t ν ∞ ) | dµ t (cid:17) / = C ( ϕ ∞ , θ ) k E m ( ϕ t ) k L ( µ ) , where ν t , g t , µ t = det g t dµ ∞ are the unit normal, metric tensor and volume measure on ϕ ∞ + f t ν ∞ and we estimated √ det g t ≤ C ( ϕ ∞ , θ ) , for any t ≥ t j such that k f t k W m +2 , ( M,g ∞ ) < θ. Differentiating H and using the above inequality, we obtain ∂ t H ( t ) = αH α − α ∂ t F m ( ϕ t )= αH α − α Z M h E m ( ϕ t ) , ∂ ⊥ t ϕ t i dµ = − αH α − α Z M | E m ( ϕ t ) || ∂ ⊥ t ϕ t | dµ = − αH α − α k E m ( ϕ t ) k L ( µ ) k ∂ ⊥ t ϕ t k L ( µ ) ≤ − αC ( ϕ ∞ , θ ) k ∂ ⊥ t ϕ t k L ( µ ) , for any t ≥ t j such that k f t k W m +2 , ( M,g ∞ ) < θ . For such times, possibly choosing asmaller ε , we can assume that | ν t − ν ∞ | < / . Letting e ϕ t := ϕ ∞ + f t ν ∞ we thus get | ∂ ⊥ t e ϕ t | = |h ∂ t e ϕ t , ν t i ν t | = |h ∂ t e ϕ t , ν ∞ i ν t + h ∂ t e ϕ t , ν t − ν ∞ i ν t |≥ |h ∂ t e ϕ t , ν ∞ i| − | ∂ t e ϕ t | = 12 | ∂ t e ϕ t | and the above estimate becomes ∂ t H ( t ) ≤ − αC ( ϕ ∞ , θ ) k ∂ ⊥ t ϕ t k L ( µ ) = − αC ( ϕ ∞ , θ ) k ∂ ⊥ t e ϕ t k L ( µ t ) ≤ − αC ( ϕ ∞ , θ ) k ∂ t e ϕ t k L ( µ t ) SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 16 for any t ≥ t j such that k f t k W m +2 , ( M,g ∞ ) < θ . Integrating the above differential inequal-ity and estimating √ det g t ≥ C ( ϕ ∞ , θ ) > , we obtain k e ϕ τ − e ϕ τ k L ( µ ∞ ) = (cid:13)(cid:13)(cid:13)(cid:13)Z τ τ ∂ t e ϕ t dt (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ∞ ) ≤ C ( ϕ ∞ , θ ) Z τ τ k ∂ t e ϕ t k L ( µ t ) dt ≤ C ( α, ϕ ∞ , θ )( H ( τ ) − H ( τ )) ≤ C ( α, ϕ ∞ , θ )( F m ( ϕ τ ) − F m ( ϕ ∞ )) α then, since possibly choosing a larger j ε we can assume that F m ( ϕ t jε ) − F m ( ϕ ∞ ) ≤ ε /α ,we see that k e ϕ τ − e ϕ τ k L ( µ ∞ ) ≤ C ( α, ϕ ∞ , θ ) ε (4.5)for any τ ≥ τ ≥ t j such that k f t k W m +2 , ( M,g ∞ ) < θ on t ∈ [ t j , τ ] . Finally, since k f t k L ( µ ∞ ) = k e ϕ t − ϕ ∞ k L ( µ ∞ ) , we get k f t k L ( µ ∞ ) ≤ k e ϕ t − e ϕ t j k L ( µ ∞ ) + k e ϕ t j − ϕ ∞ k L ( µ ∞ ) ≤ C ( α, ϕ ∞ , θ ) ε (4.6)for any t ≥ t j such that k f t k W m +2 , ( M,g ∞ ) < θ .Since m > ⌊ n/ ⌋ , estimate (4.4) implies that the hypersurfaces e ϕ t are represented asgraph on ϕ ∞ by means of functions f t with uniformly equibounded gradients (suchbound clearly depends on ε and goes to zero with it). Also, the inequalities (4.1) clearlyhold also for the second fundamental form of the hypersurfaces ϕ t ◦ σ j and e ϕ t , since theycoincide with ϕ t up to diffeomorphism (and translation). These facts imply uniformestimates on the “height” functions f t in W r, ∞ ( M, g ∞ ) ; namely, for any r ∈ N we have k f t k W r, ∞ ( M,g ∞ ) ≤ C ( r, n, ϕ , ϕ ∞ ) , (4.7)for any t ∈ [ t j , t j + δ j ) (a tedious but straightforward way to see this is to differentiateformula (4.3) and use Gauss–Weingarten relations (1.1), taking into account that thecloseness in W m +2 , ( M, g ∞ ) implies that the metric tensor and the Christoffel symbolsof the covariant derivative of e ϕ t are mutually “comparable” with the ones relative to ϕ ∞ ). Hence, if r > m + 2 and ε > is small enough, combining estimates (4.6) and (4.7),the interpolation inequalities (3.1) imply that k f t k W m +2 , ( M,g ∞ ) < θ, for any t ∈ [ t j , t j + δ j ) . By a maximality argument, it clearly follows that we can take δ j = + ∞ , for every j ≥ j ε . Hence, the estimate (4.5), which then holds for any t ≥ t j , im-plies that the flow e ϕ t satisfies the Cauchy criterion for convergence in L ( µ ∞ ) , hence e ϕ t converges in L ( µ ∞ ) , as t → + ∞ . Interpolating as before by means of inequalities (4.7),the same holds for e ϕ t in W r, ( M, g ∞ ) , for any r ∈ N and, by Sobolev embeddings, wethus deduce that there exists the limit lim t → + ∞ e ϕ t in C r ( M ) for any r ∈ N . Therefore, thesame conclusion holds for the original flow ϕ t , up to diffeomorphism. (cid:3) SYMPTOTIC CONVERGENCE OF EVOLVING HYPERSURFACES 17 R EFERENCES
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ARLO M ANTEGAZZA , D
IPARTIMENTO DI M ATEMATICA E A PPLICAZIONI , U
NIVERSIT ` A DI N APOLI ,V IA C INTIA , M
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APOLI , I
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IPARTIMENTO DI M ATEMATICA , U
NIVERSIT ` A DI P ISA , L
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