Lemma 1.1

For every even and 2π-periodic mappingh ∈ C^∞(ℝ) there existsε₀ > 0 such that for everyε ∈ (−ε₀, ε₀) the parametric curve

satisfies (1.1). In addition, setting

the mapping ∣γ_ε∣ is strictly decreasing and of classC¹in [0, π].

Proof

Fix h ∈ C^∞(ℝ) even and 2π-periodic and consider the function g_ε and the parametric curve Γ_ε as in (1.2). For every ε ∈ ℝ with ∣ε∣ < ∥h∥∞−1 we have that g_ε := 1 + εh > 0 everywhere and the parametric curve Γ_ε defined in (1.2) is a smooth Jordan curve satisfying (1.1a), with 0 ∈ Γ_ε. Let us show that Γ_ε ⊂ {(x, y) ∈ ℝ² ∣ y ≥ 0} if ∣ε∣ is sufficiently small. It is enough to prove that

One has g̃″_ε(t) = −cos t + ε [h(t)cos t]″. In particular there exists ε₀ > 0 such that g̃_ε is strictly convex in [2π/3, π]. Since g_ε is even and 2π-periodic, g′_ε(π) = 0 and then also g̃′_ε(π) = 0. Therefore g̃_ε(t) > g̃_ε(π) = 0 for t ∈ [2π/3,π]. In addition

for some constant C independent of ε and t. Hence, taking ∣ε∣ small, we obtain g̃_ε(t) > 0 also for t ∈ [0, 2π/3]. Hence (1.4) holds.

Now let us consider the parameterisation γ_ε defined in (1.3) and let us study the mapping

We have

At t = π one has to be careful since ∣γ_ε∣(π) = 0. Writing the Taylor expansion about π yields

Hence ∣γ_ε∣ is of class C¹ in [0, π]. In particular the left derivative at π is

We also need to control the behaviour of the derivative of ∣γ_ε∣ at 0. Writing the Taylor expansion about 0 yields

In order that the mapping ∣γ_ε∣ is strictly decreasing we need ∣γ_ε∣′₊(0) = 0₋. This occurs if

Since g_ε = 1 + εh, the left-hand side is −1 + O(ε), and then (1.9) holds true taking ∣ε∣ small enough. Hence ∣γ_ε∣′ < 0 in a right neighbourhood of 0 and taking a smaller ε₀ > 0 if necessary, in view of (1.6)–(1.7), we obtain the desired monotonicity property for the mapping ∣γ_ε∣. This immediately implies (1.1d).

Finally, let us evaluate the curvature of Γ_ε. This can be computed by

where, in general, for z ∈ ℝ², iz denotes the anticlockwise rotation of z through an angle π/2. Hence, for g_ε = 1 + εh, (1.1c) is fulfilled by taking ∣ε∣ small, since the leading term is gε2.□

For future convenience, let us study the regularity property of the curvature of Γ_ε as a function of the distance. More precisely, setting

let us introduce the mapping k_ε : [0, R_ε] → ℝ defined as

where ∣γ_ε∣⁻¹ is the inverse of ∣γ_ε∣ : [0, π] → [0, R_ε]. Note that ∣γ_ε∣⁻¹ is well defined thanks to Lemma 1.1.

Lemma 1.2

Ifh ∈ C^∞(ℝ) is even, 2π-periodic and satisfies

then the mappingk_εdefined in(1.12)is of classC¹in [0, R_ε] andk_ε′(0) = k_ε′(R_ε) = 0. Moreover, ifhvanishes in neighbourhoods of 0 andπ, thenR_ε = 2, k_ε ∈ C^∞([0, 2]) andk_ε = 1 in neighbourhoods of 0 and 2.

Proof

Since g_ε ∈ C^∞(ℝ), (1.10) implies that also K ∘ γ_ε ∈ C^∞(ℝ). Moreover ∣γ_ε∣⁻¹ is of class C¹ in [0, R_ε). Hence also k_ε is so. Moreover, considering that

and since ∣γ_ε∣⁻¹(0) = π and g_ε′(π) = g_ε‴(π) = 0 (because g_ε is even and 2π-periodic), one obtains dkεdr(0)=0. Some care is needed at R_ε because ∣γ_ε∣⁻¹(R_ε) = 0, ∣γ_ε∣′(0) = 0, whence ddr|γε|−1(Rε)=∞. One has

Taking (1.8) into account, we need that

Since g′_ε(t) = O(t), by (1.14), equation (1.15) holds true if

For g_ε′(0) = g_ε‴(0) = 0, (1.16) is fulfilled when g_ε″(0) = g_ε⁗(0) = 0, too, which follows from (1.13). Concerning the last statement, if h vanishes in neighbourhoods of 0 and π, then g_ε takes the constant value 1 in the same sets. The same holds for K ∘ γ_ε by (1.10). Hence dkεdr∈Cc∞((0,Rε)),Rε=2 and the assertion follows.□

Lemma 2.1

For everyp = x(θ₁, …, θ_n) ∈ S_gthe principal curvatures ofS_gatpare given by

where the primes denote derivatives with respect toθ_n.

Proof

Let us fix a point p = x(θ₁, …, θ_n) ∈ S_g. For the sake of brevity, we suppress the variables θ₁, …, θ_n in our notation. Let us introduce the vectors τi=∂σ∂θi for i = 1, …, n. One can check that

Thanks to (2.2), the outward-pointing normal versor N at p can be expressed in the form N=α0σ+∑i=1nαiτi where α₀, …, α_n ∈ ℝ satisfy

For i = 1, …, n − 1 we have

since ∣σ∣ = ∣τ_n∣ = 1. Hence the equations (2.3) imply α1=⋯=αn−1=0,αn=−α0g′g and then we can write the outward-pointing normal versor in the form

We compute

Since ∂τn∂θi=cos⁡θnsin⁡θnτi for i = 1, …, n − 1 and ∂τn∂θn=−σ, we obtain

By definition, the principal curvatures at p ∈ S_g are the eigenvalues of the shape operator L_p : T_pS_g → T_pS_g given by v↦Lpv=∂N∂v(p) for v ∈ T_pS_g. Since for p = x(θ₁, …, θ_n) and for every i = 1, …, n one has Lp∂x∂θi=∂(N∘x)∂θi, by (2.4) the principal curvatures of S are given by (2.1).□

Fixing h ∈ C^∞(ℝ) even and 2π-periodic, for ε ∈ ℝ let g_ε be as in (1.2) and let S_ε be the n-dimensional hypersurface in ℝⁿ⁺¹ parameterised by

and e_n+1 = σ(0, …, 0). Observe that S_ε is obtained from S_gε via translation by an amount of g_ε(π) in the direction of the last coordinate axis and satisfies the following properties.

Lemma 2.2

There existsε₀ > 0 such that for everyε ∈ (−ε₀, ε₀):

1. S_εis a compact, embeddedn-dimensional hypersurface of revolution around the last coordinate axis of ℝⁿ⁺¹, diffeomorphic to 𝕊ⁿ;

2. 0 ∈ S_εandS_εis contained in the half-space {(x₁, …, x_n+1) ∈ ℝⁿ⁺¹ ∣ x_n+1 ≥ 0};

3. max_p∈Sε ∣p∣ = R_εwhereR_ε > 0 is given by(1.11);

4. the principal curvatures ofS_εsatisfyK_i(p) > 0 fori = 1, …, n − 1 and for allp ∈ S_ε ∖ {0, R_εe_n+1}, andK_n(p) > 0 for allp ∈ S_ε.

Proof

Properties (i)–(iii) follow from the definition of S_ε and from Lemma 1.1. Let us discuss (iv). According to (1.10) and (2.1), the n-th principal curvature K_n of S_ε equals the curvature of Γ_ε. Therefore K_n > 0 on S_ε, again by Lemma 1.1. By (2.1), for i = 1, …, n − 1, one has K_i > 0 on S_ε ∖ {0, R_εe_n+1} when

One has g_ε(t) sin t − g′_ε(t)cos t = [1 + εh͠(t)] sin t where h~(t)=h(t)−h′(t)sin⁡t cos t. Since h′(0) = h′(π) = 0,

the mapping h͠ is continuous in [0, π]. Therefore, for ∣ε∣ sufficiently small, (2.5) holds true.□

Let us study the regularity property of the principal curvatures of S_ε as functions of the distance from the origin. To this aim, let us introduce the mapping y_ε : [0, R_ε] → S_ε defined by

where γ_ε is defined in (1.3) and ∣γ_ε∣⁻¹ is the inverse of the mapping ∣γ_ε∣ : [0, π] → [0, R_ε]; see Lemma 1.1.

Lemma 2.3

If(1.13)and

hold, then the mappingsK_i ∘ y_ε : [0, R_ε] → ℝ (i = 1, …, n) are of classC¹in [0, R_ε] and with null derivatives at end points. Moreover, ifhvanishes in neighbourhoods of 0 andπ, thenR_ε = 2, K_i ∘ y_ε ∈ C^∞([0, 2]) andK_i ∘ y_ε = 1 in neighbourhoods of 0 and 2 fori = 1, …, n.

Proof

We observe that the mapping K_n ∘ y_ε is the function k_ε defined in (1.12); then the regularity of K_n ∘ y_ε immediately follows from Lemma 1.2. Let us consider K_i ∘ y_ε with i = 1, …, n − 1. Setting

we have that K_i ∘ y_ε = k̄ ∘ ∣γ_ε∣⁻¹ (i = 1, …, n − 1). Moreover k̄ ∈ C¹((0, π)). Therefore K_i ∘ y_ε ∈ C¹((0, R_ε)), with

We aim to show that d[Ki∘yε]dr(r)→0 as r → 0 and as r → R_ε. Let us study the limit as r → 0. We have

Moreover

Since g_ε′(π) = 0, we have that g_ε′(t) − g_ε″(t) sin t cos t = O((t − π)³), and the rule of de ľHôspital implies that gε′(t)sin⁡t→−gε″(π) as t → π. In addition ∣γ_ε∣′(t) → −g_ε(π) < 0 as t → π₋, see (1.7). Therefore

because g_ε″(π) = ε h″(π) = 0 by (2.7).

Now let us study the limit as r → R_ε. We have

We know that

see (1.8). We also know that g_ε′(0) = g_ε″(0) = g_ε‴(0) = g_ε⁗(0) = 0 because of (1.13). Then, by the rule of de ľHôspital, gε′(t)sin⁡t→−gε″(π)=0 as t → 0. Moreover g_ε′(t) − g_ε″(t) sin t cos t = o(t⁴) as t → 0. Then

In addition also g_ε′(t) = o(t⁴) as t → 0. Therefore

In conclusion, also

For the last part of the lemma, one argues exactly as in the last part of the proof of Lemma 1.2, using (2.1).□

Theorem 3.1

Leth : ℝ → ℝ be a 2π-periodic, even function of classC^∞satisfying(1.13)and(2.7). Forε ∈ ℝ letg_ε, R_ε, K_i, andy_εas in(1.2), (1.11), (2.1)withg = g_ε, and(2.6), respectively. Then there existsε₀ > 0 such that for everyε ∈ (−ε₀, ε₀) the functionH_ε : ℝⁿ⁺¹ → ℝ defined by

satisfies (H)₁–(H)₃and is radially symmetric. In addition, ifε → 0 thenR_ε → 2 andH_ε → 1 inC¹(ℝⁿ⁺¹). Moreover, ifhvanishes in neighbourhoods of 0 andπthenH_εis of classC^∞.

Proof

Fixing h as in the statement of the theorem, and taking ε ∈ ℝ with ∣ε∣ small enough, according to Lemma 2.2 the hypersurface S_ε built in Section 2 is diffeomorphic to 𝕊ⁿ and, for every (θ₁, …, θ_n) ∈ [0, 2π] × [0, π]ⁿ⁻¹ the mean curvature of S_ε at p = x_ε(θ₁, …, θ_n) ∈ S_ε is given by M=1n∑i=1nKi with K_i as in (2.1). Let

with y_ε as in (2.6). Thus H͠_ε(∣p∣) equals the mean curvature of S_ε at p. By radial symmetry, the same holds true for any hypersurface obtained by rotating S_ε about the origin of ℝⁿ⁺¹. Hence, for every p ∈ ℝⁿ⁺¹ with ∣p∣ ≤ R_ε there exists an embedded hypersurface diffeomorphic to 𝕊ⁿ with mean curvature H at every point and with p ∈ S. By Lemma 2.3, H͠_ε ∈ C¹([0, R_ε]) with H͠′_ε(0) = H͠′_ε(R_ε) = 0 and setting H͠_ε(r) := H͠_ε(R_ε) for r > R_ε, we obtain a function of class C¹ on [0, ∞). In particular H͠_ε(R_ε) = 1/g_ε(0), and by Lemma 2.2 (iv) there exist C₁, C₂ > 0 such that C₁ ≤ H͠_ε(r) ≤ C₂ for every r > 0. Moreover, for every point p ∈ ℝⁿ⁺¹ with ∣p∣ ≤ R_ε one can take a round hypersphere of radius g_ε(0) whose mean curvature equals H͠_ε. Finally, because of the above discussion, the function H_ε(p) = g_ε(0)H͠_ε(∣p∣), as defined in (3.1), satisfies (H)₁–(H)₃ and is radially symmetric. The last properties plainly follow, because of the definition of g_ε and by arguing as in the last part of the proof of Lemma 1.2.□

Theorem 3.2

Leth : ℝ → ℝ be a 2π-periodic, even function of classC^∞satisfying(1.13)and(2.7). Then there existsε₀ > 0 such that for every set of numbers {ε_j}_j∈ℕ ⊂ ℝ with ∣ε_j∣ < ε₀for all j ∈ ℕ and for every set of points {p_j}_j∈ℕ ⊂ ℝⁿ⁺¹such that ∣p_i − p_j∣ ≥ max{R_εi, R_εj} + 2, the functionH : ℝⁿ⁺¹ → ℝ defined by

withH_εas in(3.1)satisfies (H)₁–(H)₃. In particular, ifε_j = εfor alljand the set {p_j} is periodic (in all directions), then the mappingHis periodic. Ifε_j → 0 asj → ∞, thenH(p) → 1 as ∣p∣ → ∞. Moreover, ifhvanishes in neighbourhoods of 0 andπthenHis of classC^∞.

Proof

This follows from Theorem 3.1 and the fact that the gluing of the blocks defined by H(x) = H_εj(x − p_j) on B_Rεj(p_j) outside the region ⋃_iB_Rεi(p_i) is nice because this function takes the common value 1 outside B_Rεj(p_j). The filling property is satisfied because it holds in each ball B_Rεj+2(p_j) and these balls are pairwise disjoint by the lower bound for ∣p_i − p_j∣. Hence the region ℝⁿ⁺¹ ∖ ⋃_iB_Rεi(p_i) can be filled by round unit spheres.□

Remark 3.3

Even more complicated mappings satisfying (H)₁–(H)₃ can be constructed by taking a sequence of 2π-periodic, even functions h_j ∈ C^∞(ℝ → ℝ) (j ∈ ℕ), considering a corresponding sequence of maps H_εj,hj defined as in (3.1) and then gluing them according to (3.2).

Example 3.4

As a function h which satisfies the assumptions of Theorems 3.1 and 3.2 one can take h(t) = (sin t)⁶. One can also give an estimate on the interval of admissible values for the smallness parameter ε. In fact, considering the proofs of lemmata of Sections 1 and 2, one needs ∣ε∣ < ∥h∥∞−1 such that

g_ε(π) + g_ε(t)cos t > 0 for t ∈ [0, π), see (1.4),

the mapping ∣γ_ε∣ defined in (1.5) is strictly decreasing in [0, π],

and the curvatures written in (1.10) and (2.8) are positive, respectively, in [0, π] and in (0, π).

Taking h(t) = (sin t)⁶, with elementary computations one can check that the previous conditions are fulfilled taking g_ε = 1 + εh with ε ∈ (−1/7, 2/5).