Extrinsic Geometric Flows on Codimension-One Foliations

Abstract

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae for deformations of geometric quantities as the Riemannian metric varies along the leaves of a foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second-order parabolic PDEs, for which the existence/uniqueness and in some cases convergence of a solution are shown. Applications to the problem of prescribing the mean curvature function of a codimension-one foliation, and examples with harmonic and umbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given.

Appendix: PDEs

In Sect. A.1, following [12] we discuss parabolic PDEs and prove Proposition 12 about the quasi-linear heat equation. In Sect. A.2, using the generalized companion matrix, defined in [9], we prove Proposition 14 about an infinite system of parabolic PDEs.

1.1 A.1 Parabolic PDEs in One Space Variable

We recall some facts about parabolic systems of quasi-linear PDEs.

Let A=(a _ij (t,x,u)) be an n×n-matrix, $a=(a_{i}(t,x,u,\partial_{x}\/u))$—n-vector, and t,x∈ℝ. Consider the quasilinear system of PDEs, n equations in n unknown functions u=(u ₁,…,u _n )

$$ \partial_tu= A(t,x,u)\,\partial^2_{xx}\/u+ a(t,x,u,\partial_x\/u).$$

(53)

PDEs (53) are homogeneous if a≡0. When A depends on t and x only, the system is semilinear. When A and a are functions of t and x, the system is linear. The initial value problem for (53) with given smooth data A, a, and u ₀,

$$ u(0,x)=u_0(x),$$

(54)

consists of finding a smooth function u(t,x) satisfying (53)–(54).

The parabolicity condition for (53) says that there is c=const>0 such that

$$ \langle A\,v,v\rangle \ge c\langle v,v\rangle ,\quad\forall\,v\in \mathbb{R}^n.$$

(55)

Theorem A

(12, Ch. 15, Proposition 8.3)

Suppose that A of class C ^∞ satisfies the parabolicity condition (55). Then the system (53)–(54) with u ₀∈H ^s(ℝ) (the Sobolev space) for some s>1, admits a unique solution u∈C([0,T), H ^s(ℝ))∩C ^∞((0,T]×ℝ), which persists as long as $\|u\|_{C^{r}}$ is bounded, given r>0.

One may apply this and the maximum principle to obtain the global existence and uniqueness result for a scalar parabolic equation (i.e., n=1 and A ₁₁=k(u)>0) on S ¹.

The following proposition is standard; for the convenience of the reader, we give its proof.

Proposition 12

Let a function k(u)∈C ^∞(ℝ) (the thermal diffusivity) satisfy

$$0<c_1\le k(u) \le c_2<\infty $$

for some real c ₁≤c ₂. Then the quasi-linear heat equation on a unit circle S ¹

$$ \partial_tu = \partial_x\/\bigl(k(u)\,\partial_x\/u\bigr), \qquad u(0,x)=u_0(x) \in C^\infty\bigl(S^1\bigr)$$

(56)

admits a unique solution u∈C ^∞([0,∞)×S ¹). Moreover, there exists lim _t→∞ u(t,x)=u _∞∈ℝ, and for some real α,K>0 the following inequalities are satisfied:

$$ \|u(t,\cdot) -u_\infty\|_{S^1}\le K\,\frac{c_2}{c_1}\,e^{-\alpha\,t}\| u_0 -u_\infty \|_{S^1}.$$

(57)

Proof

By Proposition 9.11 in [12, Ch. 15], there is a unique solution u∈C ^∞([0,∞)×S ¹). By the maximum principle, $\|u(t,\cdot)\|_{S^{1}}\le\|u_{0}\|_{S^{1}}$ for all t>0. Define the function v=φ(u) of variables (t,x), where φ′(u)=k(u). The monotone function φ (of one variable) satisfies inequalities

$$ c_1\le\varphi'\le c_2.$$

(58)

The PDE (56)₁ reads as $\partial_{t}u= \partial^{2}_{xx}\/v$, hence (in view of derivation ∂ _t v=φ ′(u) ∂ _t u) is equivalent to

$$\partial_tv = k(u)\,\partial^2_{xx}\/v,\qquad v(0,x)=\varphi\bigl(u_0(x)\bigr).$$

Let us compare it with the linear PDE on S ¹,

$$ \partial_t\widetilde{v} =\tilde{k}(t,x)\,\partial^2_{xx}\/\widetilde{v},\qquad\widetilde{v}(0,x)=\varphi\bigl(u_0(x)\bigr),$$

(59)

where $\tilde{k}(t,x)=k(u(t,x))$ is given. Indeed, $c_{1}\le\tilde{k}(t,x)\le c_{2}$ for all t>0 and x∈S ¹. From the existence and uniqueness of a solution to (59) we conclude that $\widetilde{v}=v$. Denote by u _∞∈ℝ the average of u ₀ over S ¹, and set v _∞=φ(u _∞). The function w=v(t,x)−v _∞ solves the linear PDE on S ¹

$$ \partial_t\widetilde{w} =\tilde{k}(t,x)\,\partial^2_{xx}\/\widetilde{w},\qquad \widetilde{w}(0,x)=\varphi\bigl(u_0(x)\bigr)-v_\infty.$$

(60)

By the theory of linear PDEs, (60) possesses a fundamental solution $\widetilde{G}(t,x,y)$ which can be built by the classical parametrix method, and satisfies the inequalities $0\le\widetilde{G}\le K G$ for some real K>0. Here G is the fundamental solution of the heat equation $\partial_{t}u=\alpha\,\partial^{2}_{xx}\/u$ for some constant α>0; see survey [5]. Recall that $G(t,x,y)=\sum_{j\ge0} e^{-\alpha\, j^{2}\,t}\phi_{j}(x)\,\phi_{j}(y)$, where ϕ _j denotes the eigenfunction (of operator $-\alpha\,\partial^{2}_{xx}\/$ on S ¹ with eigenvalue λ _j =αj ²) satisfying $\int_{S^{1}}\phi^{2}_{j}(x)\,dx=1$. Indeed, $\widetilde{w}(t,x)=\int_{S^{1}} \widetilde{G}(t,x,y)\,(\varphi (u_{0}(y))-v_{\infty})\,dy$. In particular,

$$\bigl\|\widetilde{v}(t,\cdot)-v_\infty\bigr\|_{S^1}\le e^{-\alpha\, t}\bigl\|\widetilde{v}(0,\cdot)-v_\infty\bigr\|_{S^1}.$$

Thus, for all t>0 and x∈S ¹ we have the a priori estimate

$$\bigl\|\varphi\bigl(u(t,\cdot)\bigr)-\varphi(u_\infty)\bigr\|_{S^1}\le K\,e^{-\alpha\,t}\bigl\| \varphi(u_0)-\varphi(u_\infty)\bigr\|_{S^1}.$$

From this, using inequalities

see (58), we deduce (57). □

Example 8

(i) Consider the heat equation over infinite (or semi-infinite) spatial interval ℝ,

$$ \partial_tu = \partial^2_{xx}\/u+ a(t,x),\qquad u(0,x)=u_0(x)\in L^2(\mathbb{R}).$$

(61)

Here we assume that u ₀(x) is a bounded function. All the solutions of (61),

$$u(t,x)=\int_\mathbb{R}u_0(\xi) G(t,x,y)\,d\,y +\int _0^t\int_\mathbb{R}a(y,s)G(t-s,x,y)\,d\,y\,ds,$$

include the heat kernel $G(t,x,y)={(4\pi t)^{-1/2}}\,e^{-(x-y)^{2}/(4t)}$ (the fundamental solution of homogeneous (61) for u ₀(x)=δ _x (ξ) – the Dirac delta function). For any function u ₀∈L ₂(ℝ), a unique solution of the homogeneous heat equation, u(t,x)=∫_ℝ u ₀(y)G(t,x,y) d  y, converges uniformly to a linear function, as t→∞. Indeed, if u ₀ is bounded then the linear function is constant.

(ii) The rough Laplacian for tensors is defined by $\Delta=\operatorname{div}\nabla$. For a unit circle S ¹, the eigenvalues of −Δ are the numbers λ _j =j ² with eigenfunctions $\phi_{j}(x)=\frac{1}{\sqrt{2\pi}}e^{-i j x}$, where j∈ℤ∖{0}. The heat equation on S ¹,

$$ \partial_tu = \partial^2_{xx}\/u,\qquad u(0,\cdot)=u_0\in H^2\bigl(S^1\bigr),$$

(62)

has a unique solution for functions u∈C([0,∞), H ²(S ¹))∩C ¹((0,∞], L ²(S ¹)). By the Sobolev embedding theorem, H ²(S ¹)⊂C ¹(S ¹). The solution of (62) has the property that u(t,⋅)∈C ^∞(S ¹) for all t>0. Moreover, $u(t,\cdot)\to\bar{u}_{0}$ as t→∞, where $\bar{u}_{0}=\frac{1}{2\pi}\int_{S^{1}} u_{0}(x)\,dx$ and

$$\bigl\|u(t,\cdot)-\bar{u}_0\bigr\|\le e^{-t}\|u_0-\bar{u}_0\|.$$

Consider the Jacobi theta function

$$\theta(z,\tau)=\sum_{n=1}^\infty e^{\pi i n^2\tau+2\pi i n z} =1+2\sum_{n=1}^\infty \bigl(e^{\pi i\tau}\bigr)^{n^2}\cos(2\pi n z),$$

where i ²=−1, z is a complex number, and τ is confined to the upper half-plane. Taking z=x∈ℝ and τ=4πt  i with real t>0, we write $\theta(x, 4\pi t\,i)=1+2\sum_{n=1}^{\infty}e^{-4 \pi^{2} n^{2} t}\cos(2\pi n x)$, which satisfies the heat equation (62)₁. Since lim _t→0 θ(x,4πt  i)=∑ _n∈ℤ δ(x−n), the solution of (62) can be specified by convolving the periodic boundary condition at t=0 with θ(x,4πt  i).

(iii) Following [3], denote $U(t,x)=\frac{\sin x}{\sqrt{\cos^{2}x+e^{2t}}}\ (t\ge0)$ and $k(u)=\frac{1}{1+u^{2}}$. We have a 3-parameter family u(t,x)=U(ω ²(t+β ₁),ω(x+β ₂)) of exact solutions to the PDE (56), $\partial_{t}u=\partial_{x}\/(k(u)\,\partial_{x}\/u))$ on S ¹. Set ω=1 and β ₁=β ₂=0, hence u(t,x)=U(t,x) and $u_{0}(x)=\frac{\sin x}{\sqrt{\cos^{2} x+1}}$. Indeed, lim _t→∞ u(t,x)=u _∞=0 for all x∈S ¹. Since 0≤u ²≤1, we conclude that $\frac{1}{2}\le k(u)\le1$. Finally, we have $\|u(t,\cdot)\|_{S^{1}}\le e^{-t}$ and $\|u_{0}\|_{S^{1}}=1$, which is consistent with (57).

1.2 A.2 The Generalized Companion Matrix

Let P _n =k ⁿ−p ₁ k ⁿ⁻¹−⋯−p _n−1 k−p _n be a polynomial over ℝ and k ₁≤k ₂≤⋯≤k _n be the roots of P _n for n>0. Hence, p _i =(−1)ⁱ⁻¹ σ _i , where σ _i are elementary symmetric functions of the roots k _i . The following generalized companion matrix (see [9]) plays a key role in this work:

(63)

Lemma 6

(see [9])

The matrix (63) has the following properties:

• (a) The characteristic polynomial of B _n,1 is P _n .

• (b) $v_{j}=(1,\,2\,k_{j},\,3\,k_{j}^{2},\ldots, n\,k_{j}^{n-1})$ is the eigenvector of B _n,1 for the eigenvalue k _j .

• (c) B _n,1 V=VD, where $V=\{\frac{n}{i} k_{j}^{i-1}\}_{1\le i,j\le n}$ is the Vandermonde type matrix, and D=diag(k ₁,…,k _n ) is a diagonal matrix. (If all k _i ’s are distinct, then V ⁻¹ B _n,1  V=D.)

Proposition 13

(see [9])

Let τ _i (t,x) (i∈ℕ) be the power sums of smooth functions k _i (t,x) (1≤i≤n). Given m>0, consider the infinite system of linear PDEs

$$ \partial_t\tau_i=-\frac{i\,m}{2(i+m-1)}\,\partial_x\/\tau_{i+m-1},\quad i\in \mathbb{N}.$$

(64)

Then the n-truncated (64), i.e., τ _n+i ’s are eliminated using suitable polynomials of τ ₁,…,τ _n , is

$$\partial_t\overrightarrow{\!\tau} =- \frac{m}{2}\,(B_{n,1})^{m-1}\partial_x\/\overrightarrow{\!\tau}.$$

Remark that for m=1 the system (64) has diagonal form: $\partial_{t}\tau_{i} =-\frac{1}{2}\,\partial_{x}\/\tau_{i}$, and for m=2 the n-truncated system (64) reads as $\partial_{t}\tau_{i} =-B_{n,1}\,\partial_{x}\/\tau_{i}$.

Proposition 14

Let τ _i (t,x) (i∈ℕ) be as in Proposition 13. Given m>0, consider the infinite system of linear PDEs

$$ \partial_t\tau_i=\frac{i\,m}{2(i+m-1)}\,\partial^2_{xx}\/\tau_{i+m-1},\quad i\in \mathbb{N}.$$

(65)

Then the n-truncated system (65) has the form

$$\partial_t\overrightarrow{\!\tau} = \frac{m}{2}\,(B_{n,1})^{m-1}\partial^2_{xx}\/\overrightarrow{\!\tau }+a_m(\overrightarrow{\!\tau},\partial_x\/\overrightarrow{\!\tau}).$$

Proof

By Proposition 13, we have $\partial_{x}\/\tau_{n+i}=\sum_{j=1}^{n}\tilde{b}_{ij}\,\partial_{x}\/\tau_{j}$, where $\tilde{b}_{ij}$ are the coefficients of the matrix B _n,m−1. Derivation of this leads to $\partial^{2}_{xx}\/\tau_{n+i}=\sum_{j=1}^{n}\tilde{b}_{ij}\,\partial^{2}_{xx}\/\tau_{j}+\partial_{x}\/\tilde{b}_{ij}\,\partial_{x}\/\tau_{j}$. □

Remark 5

For m=1 the system (65) has diagonal form: $\partial_{t}\tau_{i}=\frac{1}{2}\,\partial^{2}_{xx}\/\tau_{i}$, and for m=2 the n-truncated system (65) reads as $\partial_{t}\overrightarrow{\!\tau} =B_{n,1}\,\partial^{2}_{xx}\/\overrightarrow{\!\tau}$.

For small values of m, m=1,2, the terms $\partial^{2}_{xx}\/\tau_{n+m}$ are given by

(66)

(67)

By Proposition 14, the last row of the matrix B _n,1 or $\frac{3}{2}(B_{n,1})^{2}$ consists of the coefficients at $\partial^{2}_{xx}\/\tau_{i}$’s on the right-hand side of (66), respectively, of (67), and so on.

Example 9

For f _j =−2 δ _j1, (65) reduces to the (system of) heat equations

$$\partial_t\tau_i =\partial^2_{xx}\/\tau_{i},\quad i\in \mathbb{N},$$

whose solution is known. Consider more complicated cases.

1. For f _j =−δ _j2, (65) reduces to the system

$$ \partial_t\tau_i =\frac{i}{i+1}\,\partial^2_{xx}\/\tau_{i+1},\quad i\in \mathbb{N},$$

(68)

whose n-truncated version reads as: $\partial_{t}\overrightarrow{\!\!\tau}=B_{n,1}\,\partial^{2}_{xx}\/\overrightarrow{\!\!\tau}+a_{2}(\overrightarrow{\!\tau},\partial_{x}\/\overrightarrow{\!\tau})$. For n=2, we have two PDEs

and the matrix . If the roots k ₁≠k ₂, the eigenvectors of B _2,1 are v _j =(1, 2 k _j ), j=1,2. For n=3, (68) reduces to the quasilinear system of three PDEs with the matrix

$$B_{3,1}=\left ( \begin{array}{c@{\quad }c@{\quad }c}0 & \frac{1}{2} & 0 \\0 & 0 & \frac{2}{3} \\3\,\sigma_{3} & -\frac{3}{2}\,\sigma_{2} & \sigma_{1} \\\end{array} \right ),$$

whose eigenvalues are k _j , and the eigenvectors are $v_{j}=(1, 2\,k_{j}, 3\,k_{j}^{2})$.

2. For f _j =−δ _j3, (65) reduces to the system

For n=3, the system has the following matrix:

$$\frac{3}{2} (B_{3,1})^2 =\left ( \begin{array}{c@{\quad }c@{\quad }c}0 & 0 & \frac{1}{2} \\3\,\sigma_3 &-\frac{3}{2}\sigma_2 & \sigma_1 \\\frac{9}{2}\sigma_1\sigma_3 &\frac{9}{4}(\sigma_3-\sigma_1\sigma_2) &\frac{3}{2}(\sigma_1^2-\sigma_2) \\\end{array}\right ),$$

whose eigenvalues are $\frac{3}{2}\,k_{j}^{2}$. This series of examples can be continued as long as one desires.