Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

The Proofs

The proofs in the Euclidean and manifold case are similar. In this section, we present the main steps of the proof. A full pre-sentation is given in ref. 2. Some remarks about the manifold case:

• (a) As mentioned in Remark 3, we will often restrict to working on a single (fixed) chart in local coordinates. When we discuss moving in a direction p, we mean in the local coordinates.

• (b) Let us say a few words about how the dependence on ∥g∥_α⋀1 comes into play. Generally speaking, in all places except one (which we will mention momentarily), the α ⋀ 1-Hölder condition is used to get local bi-Lipschitz bounds on the perturbation of the metric (resp. the ellipticity constants) from the Euclidean metric (resp. the Laplacian). The place where one really uses the Hölder condition is an estimate on how much the gradient of a (global) eigenfunction changes in a ball.

• (c) We will use Brownian motion arguments (on the manifold). To have existence and uniqueness, one needs smoothness assumptions on the metric (say, 𝒞²). Therefore, we will first prove the theorem in the manifold case in the 𝒞² metric category, and then use perturbation estimates to obtain the result for g ^ij ∈ 𝒞^α. To this end, we will often have dependence on the 𝒞^α norm of the coordinates of the g^ij even though we will be (for a specific lemma or proposition) assuming the g has 𝒞² entries.

• (d) We will use estimates from ref. 18. The theorems in ref. 18 are stated only for the case of d ≥ 3. Our theorems are true also for the case d = 2 (and trivially, d = 1). This can beseen indirectly by considering ℳ̃: = ℳ × 𝕋 and noting that the eigenfunctions of ℳ̃ and the heat kernel of ℳ̃ both factor.

The idea of the proof of Theorem 1 and 2 is as follows. We start by fixing a direction p ₁ at z. We would like to find an eigenfunction ϕ_i1 such that |∂_p1ϕ_i1≳ R _z ⁻¹ on B _c1Rz (z). To achieve this, we start by showing that the heat kernel has large gradient in an annulus of inner and outer radius ∼ R _z ⁻¹ around y ₁ (y ₁ chosen such that z is in this annulus, in direction p ₁). We then show that the heat kernel and its gradient can be approximated on this annulus by as the partial sum of 0.1 over eigenfunctions ϕ_λ which satisfy both λ ∼ R _z ⁻² and R _z ^−d/2 ∥ϕ_λ∥_{L²(Bc1Rz (z))} ≳ 1. By the pigeon-hole principle, at least one such eigenfunction, let it be ϕ_i1, has a large partial derivative in the direction p ₁. We then consider ∇ϕ_i1 and pick p ₂ ⊥∇ϕ_i1 and by induction we select ϕ_i1, …, ϕ_id, making sure that at each stage we can find ϕ_ik, not previously chosen, satisfying |∂_pk ϕ_ik| ∼ R _z ⁻¹ on B _c1Rz (z). We finally show that the Φ : = (ϕ_i1, …, ϕ_id) satisfies the desired properties.

Step 1. Estimates on the heat kernel and its gradient.

Let K be the Dirichlet or Neumann heat kernel on Ω or ℳ, corresponding to one of the Laplacian operators considered above associated with g. We have the spectral expansion

When working on a manifold, we can assume in what follows that we fix a local chart containing B _Rz (z).

Proposition 4.

Under Assumption A.1, let g ∈ 𝒞^α, δ₀ sufficiently small, and δ₁ is sufficiently small depending on δ₀. Then there are constants C ₁, C ₂, C′ ₁, C′ ₂, C ₉ > 0, that depend on d, δ₀, δ₁, c _min, c _max, ∥g∥_α⋀1, α ⋀ 1} and C′ ₁, C′ ₂, C ₉ dependent also on C _Weyl, such that the following hold:

• (i) the heat kernel satisfies

• (ii) if (1/2)δ₀ R _z < |z − w|, p is a unit vector in the direction of z − w, and q is arbitrary unit vector, then where C ₉ → 0 as δ₁ → 0 (with δ₀ fixed);

• (iii) if in addition g^ij ∈ 𝒞², (1/2)δ₀ R _z < |z − w|, and q is as above, then for s ≤ t,

• (iv) C ₁, C ₂ both tend to a single function of {d, c _min, c _max, δ₀, C _Weyl}, as δ₁ tends to 0 with δ₀ fixed;

• (v) if g ∈ 𝒞², then also C′ ₁, C′ ₂, C ₉ can be chosen independently of C _Weyl. Furthermore, the above estimates also hold for |ℳ| ≤ + ∞.

At this point we can side track and choose heat kernels {K _t (·, y _i)}_i=1,…,d, with t ∼ R _z ², that provide a local coordinate chart with the properties claimed in Theorem 3.

Proof of Theorem 3.

We start with the case g ∈ 𝒞². Let us consider the Jacobian J̃ (x), for x ∈ B _c1Rz (z), of the map By 0.14 we have |J̃_ij (x) − C′ ₂〈p _i, (x−y _j)/∥x−y _j∥〉 R _z ⁻¹| ≤ C ₉ R _z ⁻¹, with C′ ₂ independent of C _Weyl. As dictated by Proposition 4, by choosing δ₀, δ₁ appropriately (and, correspondingly, c ₁ and c ₆), we can make the constant C ₉ smaller than any chosen ϵ, for all entries, and for all x at distance no greater than c ₁ R _z from z, where we use t = t _z = c ₆ R _z ² for Φ̃. Therefore, for c ₁ small enough compared to c ₄ we can write R _z J̃ (x) = G _d + E(x), where G _d is the Gramian matrix 〈p _i, p _j〉 (indepedent of x), and |E_ij (x)| < ϵ, for x ∈ B _c1Rz (z). This implies that R _z ⁻¹ (σ_min − C _dϵ) ∥υ∥ ≤ ∥ J̃ (x)υ∥ ≤ R _z ⁻¹ (σ_max + C _dϵ) ∥υ∥, with C _d depending linearly on d, where σ_max and σ_min are the largest and, respectively, smallest eigenvalues of G _d. At this point, we choose ϵ small enough, so that the above bounds imply that the Jacobian is essentially constant in B _c1Rz (z), and by integrating along a path from x ₂ to x ₂ in B _c1Rz (z), we obtain the Theorem (Φ and Φ̃ differ only by scalar multiplication). We note that ϵ ∼ 1/d suffices. To get the result when g is only 𝒞^α we use perturbation techniques for the heat kernel (2).

We proceed towards the proof of Theorem 1 and 2. The following steps aim at replacing appropriately chosen heat kernels by a set of eigenfunctions, by extracting the “leading terms”in their spectral expansion.

Proposition 5.

Assume g^ij ∈ 𝒞^α. There exists b ₁ < 1, that depends on d, c _min, c _max, ∥g∥_α⋀1, α ⋀ 1 such that the following holds. For an eigenfunction ϕ_j of Δ_ℳ, corresponding to the eigenvalue λ_j, and R ≤ R _z, the following estimates hold. For w ∈ B _b1R (z) and x, y ∈ B _b1R (z), with constants depending only on d, c _max, c _min, ∥g^ij∥_α⋀1, and P ₁ (x) = (1 + x)^(1/2)+β, P ₂ (x) = (1 + x)^(3/2)+β, P ₃ (x) = (1 + x)^(5/2)+β, with β the smallest integer larger than or equal to (d−2)/4.

We start by restricting our attention to eigenfunctions do not have too high frequency. Let Λ_L (A) = {λ_j : λ_j ≤ At ⁻¹} and Λ_H (A′) = {λ_j : λ_j > A′ t ⁻¹} = Λ_L (A′)^c.

A first connection between the heat kernel and eigenfunc-tions is given by the following truncation Lemma.

Lemma 1.

Assume g ∈ 𝒞². Under Assumption A.1, for A > 1 large enough and A′ < 1 small enough, depending on δ₀, δ₁, C ₁, C ₂, C′ ₁, C′ ₂ (as in Proposition 4), there exist constants C ₃, C ₄ (depending on A, A′ as well as {d, c _min, c _max, ∥g∥_α⋀1, α ⋀ 1}) such that

• (i) The heat kernel is approximated by the truncated expansion

• (ii) If (1/2)δ₀ R _z < |z − w| and p is a unit vector parallel to z − w, then

• (iii) C ₃, C ₄ both tend to 1 as A → ∞ and A′ → 0.

This lemma implies that in the heat kernel expansion, we do not need to consider eigenfunctions corresponding to eigenval-ues larger than At ⁻¹. However, in our search for eigenfunctions with the desired properties, we need to restrict our attention further, by discarding eigenfunctions that have too small a gradient around z. As a proxy for gradient, we use local energy. Recall Ave_R ^z (f) = ( _BR(z) |f|²)^1/2, and let

The truncation Lemma 1 can be strengthened into

Step 3.

Choosing appropriate eigenfunctions.

The set of eigenfunctions with eigenvalues in Λ (as in Lemma 2) is well suited for our purposes, in view of:

Lemma 3.

Assume g ∈ 𝒞². Under Assumption A.1, for δ₀ small enough, there exists a constant C ₇ depending on {C ₁, C ₂, C′ ₁, C′ ₂, C ₅, δ₁} and C ₈ depending on {δ₀, c _min, c _max, ∥g∥_α⋀1, α ⋀ 1} such that the following holds. For any direction p there exist j ∈ Λ := Λ_L (A) ∩Λ_H (A′) ∩Λ_E (z, R _z, δ₀, c ₀) such that and moreover, if ∥z − z′∥ ≤ b ₁ R _z, where b ₁ is a constant that depends on C ₇, C ₈, d, c _min, c _max, ∥g∥_α⋀1, α ⋀ 1, then

This lemma yields an eigenfunction that serves our purpose in a given direction. From this point onwards we let, with abuse of notation, ϕ_j(z′) = (Ave_½ ^zδ₀R_{z z} ϕ_j)⁻¹ · ϕ_j(z′). To complete the proof of the theorems, we need to cover d linearly independent directions. Pick an arbitrary direction p ₁. By Lemma 3 we can find j ₁ ∈ Λ, (in particular j ₁ ∼ t ⁻¹) such that |∂_pl ϕ_jl (z)| ≥ c ₀ R _z ⁻¹. Let p ₂ be a direction orthogonal to ∇ϕ_j1 (z). We apply again Lemma 3, and find j ₂ < At ⁻¹ so that |∂_p2ϕ_j2 (z)| ≥ c ₀ R _z ⁻¹. Note that necessarily j ₂ ≠ j ₁ and p ₂ is linearly independent of p ₁. In fact, by choice of p ₂, ∂_p2ϕ_j1 = 0. We proceed in this fashion. By induction, once we have chosen j ₁, …, j _k (k < d), and the corresponding p ₁, …, p _k, such that |∂_pl ϕ_jl (z)| ≥ c ₀ R _z ⁻¹, for l = 1, …, k, we pick p _k+1 orthogonal to 〈{∇ϕ_jn}_n=1,…,k〉 and apply Lemma 3, that yields j _k+1 such that |∂_pk+1ϕ_jk+1 (z) | ≥ c ₀ R _z ⁻¹.

We claim that the matrix A _k+1 := (∂_pn ϕ_jm)_{m,n=1,…,k+1} is lower triangular and {p ₁, …, p _k+1} is linearly independent. Lower-triangularity of the matrix follows by induction and the choice of p _k+1. Assume a ∈ ℝ^k+1 and Σ_n=1 ^k+1 a _n p _n = 0, then 〈Σ_n=1 ^k+1 a _n p _n, ∇ϕ_jl〉 = 0 for all l = 1, …, k + 1, i.e., a solves the linear system A _k+1 a = 0. But A _k+1 is lower triangular with all diagonal entries non-zero, hence a = 0.

For l ≤ k, we have 〈∇ϕ_jl, p _k+1〉 = 0 and, by Lemma 3, |〈∇ϕ_jl, p _l〉| ≳ R _z ⁻¹. Now let Φ_k = (ϕ_j1, …, ϕ_jk) and Φ = Φ_d. We start by showing that ∥∇Φ|_z (w − z)∥ ≳_d 1/R _z ∥w − z∥. Indeed, suppose that ∥∇Φ_k|_z (w − z)∥ ≤ c/R _z ∥w − z∥, for all k = 1, …, d. For c small enough, this will lead to a contradiction. Let w − z = Σ_l a _l p _l. We have (using say Lemma 3) By induction, |a _k| ≤ Σ_l=1 ^k c ^l ∥w − z∥. For c small enough, |a _i| ≤ ∥w − z∥/d. This is a contradiction since ∥Σ_i a _i p _i∥ = ∥w − z∥ and ∥p _i∥ = 1. We also have, by Proposition 5, Finally, by ensuring ∥z − w _i∥/R _z is smaller than a universal constant for i = 1, 2, we get from Eq. 0.16 which proves the lower bound 0.5. To prove the upper bound of 0.5, we observe that from Proposition 5 we have the upper bound |∂_plϕ_il (z)|≲ Ave_Rz ^z (ϕ_il), and since ϕ_il is L ²-normalized, the right hand side can be as big as ∼ R _z ^−d/2. Therefore, we can choose γ_l as in the statement of the theorem so as to satisfy the upper bound 0.5. This completes the proof for the Euclidean case.

In the manifold case, we consider first the case g ∈ 𝒞², and thus let α ⋀ 1 = 1 in what follows. Let R _z be as in the theorem. We take c ₁ ≤ (1/2)δ₀ chosen so that for all x ∈ B _2c1Rz (z). For this g, the above is carried on in local coordinates. It is then left to prove that the Euclidean distance in the range of the coordinate map is equivalent to the geodesic distance on the manifold. For all x, y ∈ B _{c1 Rz} (z) For the converse, let γ: [0, 1] → ℳ be the geodesic from x to y. γ is contained in B _2dℳ(x,y) (x) on the manifold, whose image in the local chart is contained in B _{2(1+∥g∥α⋀1)dℳ (x,y)} (x). We have

Finally, when g ∈ 𝒞^α we will need:

Examples

Example 2. Mapping with eigenfunctions, non simply-connected domain.

We consider the planar, non-simply connected domain Ω in Fig. 1. We fix a point z ∈ Ω, as in the figure, and display two eigenfunctions whose existence is implied by Theorem 1.

Example 3. Localized eigenfunctions.

In this example, we show that the factors γ₁, …, γ_d in Theorem 1 and 2 may in fact be required to be as small as R _z ^d/2. We consider the “two-drums” domain in Fig. 2, consisting of a unit-size square drum, connected by a small aperture to a small square drum, with size τ/N, where τ is the golden ratio. The with of the connecting aperture is δτ/N, for small δ. For this domain, for small enough δ, and for z in the smaller square, it can be shown that all possible eigenfunctions that may get chosen in the theorem are localized in the smaller square This is essentially a consequence of the fact that the proper frequencies of the two drums, for δ = 0, are all irrational with respect to each other, and therefore eigenfunctions are perfectly localized on each drum. For δ small enough, a perturbation argument shows that the eigenfunctions will be essentially localized on each drum. But the eigenfunctions localized on the small drum, being normalized to L ² norm, will have L ^∞ norm as large as R _z ^−d/2, and therefore the lower bound for the γ_i's is sharp.