Willmore-type variational problem for foliated hypersurfaces

We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(\sigma^{\cal F}_1,\ldots,\sigma^{\cal F}_s)\,{\rm d}V$, where $\sigma^{\cal F}_i$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\cal F$. The first and second variations of such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations.


Introduction
Many authors, e.g., [3,4,8,10,15,16], were looking for an immersion φ : M n → M n+1 of a smooth manifold M n (n ≥ 2) into a Riemannian manifold ( M , ḡ), in particular, Euclidean space R n+1 , which is a critical point of the following functionals for compactly supported variations of φ: Here, h is the scalar second fundamental form of φ(M ), H = 1 n trace g h is the mean curvature, dV is the volume form of the induced metric g on M , and p > 0. These functionals measure how much φ(M ) differs from a minimal hypersurface (H = 0) or a totally geodesic hypersurface (h = 0).The actions (1) are a particular case of functionals W F n = M F (H) dV and JF n = M F ( h ) dV , where F is a C 3 -regular function of one variable, e.g., [2,6,7,9].For a closed oriented smooth hypersurface M n in R n+1 , we get W n,n ≥ C n , where C n = 2π (n+1)/2 Γ((n+1)/2) is the area of the unit n-sphere; the equality W n,n = C n holds if and only if M n is embedded as a hypersphere, see [3].
Variational problems for (1) were first posed by Thomas Willmore in [15] for W 2,2 , which belongs to conformal geometry.The Euler-Lagrange equation for W 2,2 is the well known elliptic PDE where ∆ is the Laplacian and K -the gaussian curvature of M 2 .Solutions of (2) are called Willmore surfaces.An important class of Willmore surfaces in R 3 arise as the stereographic projection of minimal surfaces in the 3-sphere.By Lawson's theorem, any compact orientable surface can be minimally embedded in the 3-sphere.For a closed orientable surface M in R 3 , the inequality W 2,2 ≥ C 2 = 4π holds with the equality for round spheres.If M 2 is a torus in R 3 , then, according the Willmore conjecture proven by F.C. Marques and A. Neves in [10], we have W 2,2 ≥ 2 π 2 ; the equality holds if and only if the generating curve is a circle and the ratio of radii is 1 √ 2 .Willmore surfaces have applications in biophysics, materials science, architecture, etc., e.g., [14].
An interesting problem is the generalization of the Willmore functional to submanifolds with additional structures, such as foliations or almost products.Let M n (n ≥ 2) equipped with an sdimensional (1 ≤ s ≤ n) foliation F be immersed into a Riemannian manifold ( M , ḡ).Let h F be the restriction of the second fundamental form of M on the leaves of F. Denote by For foliation theory we refer to [5], the extrinsic geometry of foliations was developed in [13].We study Reilly-type functionals for compactly supported variations of (M n , F) immersed in R n+1 : which for s = n reduce to (3).For F = (σ F 1 /s) p and F = (τ F 2 ) p/2 , the actions (5) read as which reduce to (1) for s = n.
Remark 1.A foliated hypersurface in R n+1 , whose leaves {L} are minimal submanifolds in R n+1 is an example of a minimizer for (6) 1 with even p.A foliated hypersurface in R n+1 with an asymptotic distribution T F (in particular, a ruled hypersurface) is a minimizer for (6) 2 .It is interesting to find critical hypersurfaces for actions (6) with H F = 0 or h F = 0 on an open dense set of M .
The following special case of ( 5) is invariant under conformal group of ( M , ḡ), see Theorem 1: and reduces to (4) for s = n.Note that s , e.g., for hypersurfaces of revolution in R n+1 foliated by parallels.We hope that foliated hypersurfaces, which are local minima for (5), will be useful for natural sciences and technology related to layered (laminated) non-isotropic materials.
The paper is organized as follows.Section 2 contains some lemmas (proved in Sect.5) that help us calculate variations of Reilly-type functionals on foliated hypersurfaces in R n+1 .In Section 3, conformal invariance of ( 7) is shown, the first variations of the functionals ( 5)- (7) are found, and the corresponding Euler-Lagrange equations for the case of transversally harmonic (for example, Riemannian) foliation are obtained.Then the second variations on critical hypersurfaces of some Willmore type functionals are calculated.In Section 4, applications to hypersurfaces with lowdimensional foliations are given, the critical hypersurfaces of revolution for the actions (6) are presented, and it is shown that they are a local minimum for special variations.

Auxiliary results
Let r : M n → R n+1 be an immersion of a manifold M into R n+1 with Euclidean metric ḡ and the Levi-Civita connection ∇.We identify M with its image r(M ) and restrict our calculations to a relatively compact neighborhood U ⊂ M with induced metric g = • , • and normal coordinates (x 1 , . . ., x n ) centered at a point x ∈ M .Thus, g ij = δ ij (the Kronecker symbol) and Γ k ij = 0 at x. Differentiation of a function f (or a tensor) with respect to the variable x i will be denoted by f i .
Let ∂ i = ∂/∂x i be the coordinate vector fields on U .So, the vectors r i = ∇∂ i r form a local coordinate basis for the tangent bundle T M along U , and we get g = g ij dx i dx j , where g ij = ḡ(r i , r j ) = r i , r j and Einstein summation rule is used.Let N be a unit normal vector field to M on U .The vectors N i = ∇∂ i N belong to the tangent space T x M , i.e., N i , N = 0.
Let h be the scalar second fundamental form of M with respect to unit normal N, A = − ∇ N the Weingarten operator and H = 1 n trace g h the mean curvature.Denote by h j the symmetric tensor dual to A j , i.e., h j (X, Y ) = A j X, Y .For example, h 2 = g kl h li h kj dx i dx j = h k i h kj dx i dx j .Consider a one-parameter family of hypersurfaces r t = r + t u N (|t| < 1).We get a variation δ r = u N, where δ = (d/dt)| t=0 is the variational derivative operator, and u : U → R is a smooth function supported on relatively compact neighborhood U .Obviously, (δ r [11, pp. 48, 69].The Laplacian is ∆u = trace g Hess u = g ij u ij .Note that g, Hess u = ∆ u.The divergence of a vector field Lemma 1 (see [8] for n = 2).The following evolution equations are true: We carry out further calculations for a foliated hypersurface (M, F) and a foliated neighborhood U ⊂ M with normal coordinates (x 1 , . . ., x n ) adapted to F, i.e., (x 1 , . . ., x s ) are variables along the leaves, see [5].Let ∇ F : T M × T F → T F be the induced Levi-Civita connection on the subbundle T F ⊂ T M and on the leaves of F. The leafwise Laplacian on functions is ∆ F = trace g Hess F = div F • ∇ F , where Hess F is the Hessian on the leaves of F. Let P : T M → T F be the orthoprojector, thus P 2 = P and P is self-adjoint.For h F and its dual self-adjoint operator A F we can write h F (X, Y ) = h(P X, P Y ) (X, Y ∈ X M ) and A F = P AP .Let h j F be the symmetric tensor dual to A j F , i.e., h j F (X, Y ) = A j F X, Y .The symmetric tensor h mix is given by We have h F , h mix = 0.The equality h mix = 0 means that P A = AP , i.e., T F is an invariant subbundle for A. Let h F ⊥ be the restriction of h on the normal distribution to The Newton transformations T r (A F ) of A F are defined inductively or explicitly by, e.g.[13], For example, T 1 (A F ) = σ F 1 id T F − A F and T s (A F ) = 0 and the following equalities are true: The "musical" isomorphism ♯ : T * M → T M is used for tensors, e.g., h ♯ = A, and for (0, 2)-tensors B and C we have Lemma 2. The variations of τ F i and σ F r are the following: Proof.By (10), we obtain δA F = Hess F ♯ u + u(A 2 F + A 2 mix ).Using this, ( 14) and the the following variations of τ F i and σ F r , see [13,Sect. 1.5.3]: we get (15).
Lemma 3. The following evolution equations are true: For any smooth function f : M × R → R the following evolution is true: Remark 2. The equations ( 16)-( 17) can be deduced from Lemma 2, but we will prove them in Section 5. To find the second variation of the functionals (5) we need also the variation δ h, Hess F u , but we omit this calculation and consider the second variation of the functionals (6) only.
The following property helps to find the Euler-Lagrange equations using first variation of ( 3)-( 6): Here, (div P )X = n i=1 (∇ e i P )X, e i , where e 1 , . . ., e n is a local orthonormal basis on M .Note that Riemannian foliations (and the leaves of twisted products, e.g., [13]) satisfy (20).Lemma 4. A foliated Riemannian manifold (M, g, F) satisfies (20) if and only if F is transversally harmonic, i.e., the normal distribution has zero mean curvature.
Proof.Using a local orthonormal frame on M such that e i ∈ T F (1 ≤ i ≤ s), we calculate: where (n − s)H ⊥ = P i>s ∇ e i e i is the mean curvature vector of (T F) ⊥ and X ∈ X M .
For any 2-tensor B on M define the adjoint of the covariant derivative The next lemma generalizes (21) and the well known Green's formula, e.g., [11, p. 75].
Lemma 5.If a foliated Riemannian manifold (M, g, F) satisfies (20), then the following formulas are valid for any compactly supported functions u, f and 2-tensor B: 3 Main results In Sect.3.1, we find the Euler-Lagrange equations (and first variations) for the functionals ( 5)- (7), and in Sect.3.2 we find the second variations of ( 5) and ( 6).First we check the conformity of ( 7).
Theorem 1.The functional W conf n,s,r is a conformal invariant of a foliated hypersurface (M, F) in a Riemannian manifold ( M , ḡ).
Proof.Define a new Riemannian metric on M by ḡc = µ 2 ḡ for some positive function µ ∈ C 3 ( M ).Then g c = µ 2 g is the new induced metric on M , thus, the new volume form of M is dV c = µ n dV .If X is a ḡ-unit vector, then X c = X/µ is a g c -unit vector.By the well known formula for the Levi-Civita connection, e.g., [13, p. 14], we get By this with X ∈ T F and Y = N c , the operators A and A c are related by . By the above and A F = P AP , A c F = P A c P , we get

The first variation
We can state our main theorem.
Equations ( 27) and (28) of the next statement follow from Theorem 2, but we will prove them.

The second variation
The following statement generalizes Corollary 1 in [8] when M 2 ⊂ R 3 .
Theorem 3. If (20) is valid, then the Euler-Lagrange equation for (5) At a critical hypersurface satisfying (20), the second variation of (5) Proof.By (26) 1 with F = F (σ F 1 /s), using id T M , h 2 mix = h mix 2 and id T F , Hess F u = ∆ F u, we find the first variation of the functional (5) 1 with F = F (H F ): If (20) is valid, then using (37) and ( 22) we obtain (35).Our next aim is to calculate For the first integral in the last line of (38), using ( 16), ( 19) and δu = 0, we get For the second integral in the last line of (38), using (18), we get By ( 38), ( 39) and ( 40), noting that the first variation vanishes at a critical immersion, we get From ( 41) and ( 35), at a critical hypersurface, we get (36).
Similarly, one can obtain the Euler-Lagrange equation for the functional (5) 2 with F = F ( h F ), we do not present it here.From Theorem 3 with F = H p F we obtain the following.
Corollary 2. At a critical hypersurface satisfying (20), the second variation of the action (6) 1 is Remark 4. By (41) with s = n, the second variation of the action W F n = M F (H) dV is This is compatible with a special case of Eq. ( 7) in [8] for n = 2.As a special case of (43), the second variation of the functional (1) 1 with n = 2 has the following form compatible with [8, Eq. ( 45)]:

Applications
We consider critical hypersurfaces equipped with two-dimensional foliations (i.e., s = 2) in Sect.4.1, and discuss critical hypersurfaces of revolution and their stability in Sect.4.2.

Hypersurfaces with two-dimensional foliations
For s = 2, it is natural to present the functionals (5) in the following form: where is the Gaussian curvature of the leaves.For n = s = 2, (44) reduces to the functional W F 2 = M F (H, K) dV seen in [8].The following equalities are true: From ( 16) and ( 17) with s = 2, we obtain the following evolution equations: Using ( 45) and (46 , we get the evolution equation For n = 2, (45)-( 47) reduce to the equations in [8, Lemma 1].
The next statement for (M n , F 2 ) immersed in R n+1 generalizes Theorem 1 in [8] with n = 2.
Theorem 4. If (20) is valid, then the Euler-Lagrange equation for the action (44) with s = 2 is where F ′ H , F ′ K denote partial derivatives of F (H F , K F ) with respect to H F and K F .At a critical hypersurface foliated by surfaces (s = 2) and satisfying (20), the second variation of the functional (44) with F = F (H F ) is given by Proof.Using (45) and (47 and applying (13), we calculate the first variation of the functional (44) with s = 2: From (50), using (23), we get (48).From Theorem 3 with s = 2 we get (49).
Remark 5. Let F = F (H F ), then from (48) we obtain From this with F = H 2 F , or, from (27), we get the Euler-Lagrange equation for W n,2,2 , see ( ∆ The following particular case of (49), or, Corollary 2 with s = 2, is true.
Corollary 3. At a critical hypersurface satisfying (20), the second variation of W n,p,2 is The following consequence of Theorem 4 was proven for n = 2 in [8].
Corollary 4. The Euler-Lagrange equation for the functional W F 2 with F = F (H, K) is given by

Hypersurfaces of revolution
Hypersurfaces of revolution in Euclidean space R n+1 are naturally foliated into (n − 1)-spheres (parallels) and equipped with rotationally symmetric metrics g = dρ 2 + ρ 2 ds 2 n−1 -a special case of a warped product metric, see [11,Sect. 4.2.3].Such a hypersurface can be represented as a graph x n+1 = f (ρ), where the function f ∈ C 2 is monotonic, ρ = x 2 1 + . . .+ x 2 n and (x i ) are Cartesian coordinates in R n+1 .We obtain the parametrization r = r(φ 1 , . . ., φ n−1 ; f (ρ)), of M , where (φ 1 , . . ., φ n−1 ; ρ) are cylindrical coordinates in R n+1 .The principal curvatures of M (functions of ρ) are k for profile curves -geodesics on M .If the profile curve is a straight line (f ′′ ≡ 0), then k n ≡ 0 and M is a cone, or a cylinder, or a hyperplane.To exclude these cases, we will assume f ′′ = 0. We get Recall that λ j = j(j+n−2) correspond to the solutions with multiplicities N j = C n n+j = (n+j)!n! j! of the eigenvalue problem ∆ u+λ u = 0 on a unit (n−1)-sphere.Any constant function on the round sphere spans the space of λ 0 -eigenfunctions of the Laplacian.Let ⊥ denote the orthogonality of functions with respect to the L 2 inner product.The sphere S 2 (ρ) of radius ρ in R 3 is not a local minimum of W 2,p under volume-preserving deformations for p > 2. For p ≥ 1, S 2 (ρ) is a local minimum of W 2,p under volume-preserving, nonconstant deformations u provided u ∈ {v : ∆ v = (2/ρ 2 )v} ⊥ , see Propositions 2 and 3 in [8].According to (35), a hypersurface of revolution in R n+1 foliated by (n − 1)-spheres-parallels {L ρ } is a critical point of the action (5) In this case, k n and k 1 are functionally related; hence, M is a Weingarten hypersurface.
The following theorem studies the stability of hypersurfaces of revolution critical for (6).
Theorem 5. A hypersurface of revolution M : A critical hypersurface is not a local minimum of W n,p,n−1 for p > n ≥ 2 with respect to general variations, but it is a local minimum for variations u = u(φ 1 , . . ., φ n−1 ) satisfying u| Lρ ⊥ ker ∆ F .

Appendix
Proof.(of Lemma 1).Using δ Thus, since the symmetry h ij = h ji we get the equality (8): From g il g lj = δ i j it follows that (δg il )g lj = −g il (δg lj ) = 2 u g il h lj ; hence, ( 9) is true.We will compute the variation of h.Using N, N i = 0, we find Note that δ N = c i r i for some functions c i .Using N, r j = 0, we get It follows that c i = −g ij u j and δ N = −g ij u j r i = −u i r i .Using the Gauss equation for a hypersurface in R n+1 , we get at x: δ N, r ij = −u i r i , h jl N + Γ k jl r k = 0. Thus, (10) is true: Calculating the variation of the mean curvature, we get (12): The formula δ(dV ) = 1 2 (trace g δ g) dV for variation of dV is valid for any variation δg of a metric.for example, [13].Applying (8) to the above gives (13).Next, we calculate the variation of h 2 : that proves (11).
Proof.(of Lemma 3) First, using ( 8) and (10), we get for 1 ≤ i, j ≤ s and 1 ≤ q ≤ n, that proves (16).Also for 1 ≤ i, j, k, l ≤ s and 1 ≤ q ≤ n we obtain (17): For 1 ≤ i, j, k, l ≤ s, s < α, β, γ ≤ n and 1 ≤ q ≤ n we obtain (18): The proof of ( 19) is similar to the proof of [8, Eq. ( 19)]: instead of M 2 we consider s-dimensional leaves of F. The variation of the Christoffel symbols is the following tensor, e.g., [8]: For the Laplacian ∆ For the first term we get (for 1 ≤ i, j ≤ s) For the second term, using (56) and Γ k ij = 0 at x, we get for 1 ≤ i, j, k ≤ s and 1 ≤ q ≤ n, δ (g ij Γ k ij f k ) = g ij δ (Γ k ij )f k = −g ij g kq u (h jq,i + h iq,j − h ij,q )f k − (u i h jq + u j h iq − u q h ij )f k = −2 u g ij g kq h jq,i f k + u g ij g kq h ij,q f k − 2 g ij g kq u i h jq f k + g ij g kq u q h ij f k . (59) Using the Codazzi-Minardi equation ∇ k h ij − ∇ j h ik = 0, e.g., [11], we get for 1 ≤ i, j, k, l ≤ s, Thus, using normal coordinates and −2 u g ij g kl h jl,i f k + u g ij g kl h ij,l f k = −u g kl (g ij h ij,l )f k , we get s (H F ) l = g ij ∇ l h ij for 1 ≤ i, j, l ≤ s.Therefore, (59) becomes Applying ( 58) and (60) to (57) completes the proof of (19).