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BY 4.0 license Open Access Published by De Gruyter April 18, 2022

An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space

Paolo Caldiroli
From the journal Advances in Geometry


We construct families of smooth functions H : ℝn+1 → ℝ such that the Euclidean (n + 1)-space is completely filled by not necessarily round hyperspheres of mean curvature H at every point.

MSC 2010: 53A10; 53A05 (53C42)


This article deals with embedding n-dimensional hypersurfaces of prescribed mean curvature into Euclidean space. More precisely, considering a function H : ℝn+1 → ℝ, one looks for compact, embedded hypersurfaces whose mean curvature at any point p is given by H(p). We focus our attention to hypersurfaces which are diffeomorphic to the sphere 𝕊n and we call them H-bubbles.

When H is a nonzero constant, H-bubbles are exactly round spheres of radius ∣H−1 centred at any point of the space. The case of nonconstant functions H is much more delicate and has been intensively investigated in the last years. In fact the behaviour of H crucially affects the existence of H-bubbles; see e.g. [3; 4; 5; 6; 7; 9]. Moreover, concerning the location of H-bubbles, some constraints occur. For example, if H depends just on one variable in a monotone way in some domain D of the space, then no H-bubble can be found in D; see [4, Proposition 4.1]. In a perturbation setup, one sees that H-bubbles concentrate only at critical points of H or of other functions related to H; see e.g. [4, Proposition 4.3], [2, Theorem 2.1].

In this paper, we are interested in the existence of nonconstant curvature functions H such that the whole Euclidean (n + 1)-space can be filled by H-bubbles, in other words, for every point p ∈ ℝn+1 there exists an H-bubble passing through p. We construct families of nonconstant mappings H : ℝn+1 → ℝ of class C1 and with further optional properties (like periodicity or prescribed asymptotics at infinity), for which the desired filling property holds. In fact, we use a kind of inverse approach, starting with the construction of suitable bubbles before we exhibit the corresponding functions H.

This setup might have some importance in applications to weighted isoperimetric inequalities. For example, looking for area minimizing, disk-type surfaces with prescribed boundary and volume (enclosed within another fixed reference surface with the same boundary), a certain sphere-attaching mechanism is involved. Such construction (due to Wente [10, p. 285 ff.], see also [1]) rests on the existence of spheres passing through an a priori unknown point of the space. The analogous issue in case of prescribed weighted volume would be quite interesting but no result similar to [10] seems to be still available. Since prescribing weighted volume is equivalent to prescribing mean curvature, up to a Lagrange multiplier, the presence (or not) of H-bubbles filling the space might have some impact with respect to the above described issue.

Let us spend a few words on the main idea developed in this paper. We firstly construct a smooth Jordan curve Γ lying in the half-plane of ℝn+1 defined by x1 = … = xn−1 = 0 and xn+1 ≥ 0, passing through the origin, and symmetric about the xn+1-axis. Then we introduce a hypersurface of revolution S obtained by revolving the curve Γ around the last coordinate axis. Letting R = maxpSp∣, we define a radial, positive mapping H = H(r) on the ball BR = {p ∈ ℝn+1 ∣ ∣p∣ ≤ R} taking the value of the mean curvature of S at a point pS with ∣p∣ = r. Since H is radially symmetric, every hypersurface obtained by rotation of S about the origin is an H-bubble. Hence the (n + 1)-dimensional ball BR has the filling property with respect to H, that is, for every pBR there is an H-bubble passing through H.

The main difficulty is to construct Γ in such a way that H is well defined, of class C1, nonconstant and with null derivative at r = 0 and r = R. This is achieved by taking Γ as a perturbation of a circle. Consequently, the curvature function H turns out to be a perturbation of a positive constant. We point out that the smallness of the perturbation can be suitably controlled; see Example 3.4.

Since H has null derivative on the boundary of BR, we can extend it outside BR in a C1 way with the constant value H(R). Then the filling property is automatically satisfied on the whole Euclidean space, because the complement of the ball BR can be filled by round hyperspheres with radius H(R)−1. In fact, we can use the function H on BR, for some R′ > R, as a fundamental block, and arrange infinitely many similar blocks, suitably spaced, to cover the whole space. Also in this case the resulting function H has the property that every point of ℝn+1 is touched by an H-bubble. The arrangement of the blocks or even the form of H at every block can be arbitrarily adjusted in order to fulfil additional requirements on H (like periodicity). Hence, a very huge amount of examples, even with C regularity, can be built.

The main results are Theorems 3.1 and 3.2 and are displayed in the last section, together with an explicit example. The first two sections are devoted to the construction of the profile curve Γ and of the reference hypersurface S.

Lastly, we note that whereas, on one hand, the problem of the existence of H-bubbles under global or perturbative conditions on the prescribed mean curvature function requires rather sophisticated tools and arguments (see e.g. [8], [5] and the references therein), on the other hand, our inverse approach can be carried out by means of quite elementary methods and one needs just a basic knowledge of surface theory.

1 The profile curve

By a Jordan curve we mean a closed, simple curve. Here we aim to construct smooth (i.e. C) Jordan curves Γ in the plane, with the following properties:

Γ is symmetric with respect to the vertical axis(1.1a)
Γ has positive curvature at every point(1.1c)
for every r[0,R] there exists a unique z=(x,y)Γ with x0 and |z|=r(1.1d)

where R = maxzΓz∣. More precisely, we prove the following result.

Lemma 1.1

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


satisfies (1.1). In addition, setting


the mappingγεis strictly decreasing and of classC1in [0, π].


Fix hC(ℝ) even and 2π-periodic and consider the function gε and the parametric curve Γε as in (1.2). For every ε ∈ ℝ with ∣ε∣ < h1 we have that gε := 1 + εh > 0 everywhere and the parametric curve Γε defined in (1.2) is a smooth Jordan curve satisfying (1.1a), with 0 ∈ Γε. Let us show that Γε ⊂ {(x, y) ∈ ℝ2y ≥ 0} if ∣ε∣ is sufficiently small. It is enough to prove that

g~ε(t):=gε(π)+gε(t)cost0for all t[0,π].(1.4)

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

g~ε(t)=tπg~ε(s)(ts)ds=1+costεtπ[h(s)coss](ts)ds12εCfor all t[0,2π/3]

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

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


We have

|γε|(t)=gε(t)gε(t)+gε(π)gε(t)costgε(π)gε(t)sintgε(t)2+gε(π)2+2gε(π)gε(t)costfor all t[0,π).(1.6)

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

|γε|(t)=gε(π)2(tπ)+o(tπ)gε(π)|tπ|+o(tπ)as tπ.

Hence ∣γε∣ 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 ∣γε∣ at 0. Writing the Taylor expansion about 0 yields

|γε|(t)=gε(0)gε(0)+gε(π)gε(0)gε(0)gε(π)gε(0)+gε(π)t+o(t)as t0.(1.8)

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


Since gε = 1 + εh, the left-hand side is −1 + O(ε), and then (1.9) holds true taking ∣ε∣ small enough. Hence ∣γε∣′ < 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 ∣γε∣. This immediately implies (1.1d).

Finally, let us evaluate the curvature of Γε. 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 gε = 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 Γε as a function of the distance. More precisely, setting


let us introduce the mapping kε : [0, Rε] → ℝ defined as


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

Lemma 1.2

IfhC(ℝ) is even, 2π-periodic and satisfies


then the mappingkεdefined in(1.12)is of classC1in [0, Rε] andkε′(0) = kε′(Rε) = 0. Moreover, ifhvanishes in neighbourhoods of 0 andπ, thenRε = 2, kεC([0, 2]) andkε = 1 in neighbourhoods of 0 and 2.


Since gεC(ℝ), (1.10) implies that also KγεC(ℝ). Moreover ∣γε−1 is of class C1 in [0, Rε). Hence also kε is so. Moreover, considering that


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


Taking (1.8) into account, we need that

d(Kγε)dt(t)=o(t)as t0.(1.15)

Since gε(t) = O(t), by (1.14), equation (1.15) holds true if

3gε(t)2gε(t)gε(t)gε(t)3gε(t)gε(t)3gε(t)+3gε(t)gε(t)gε(t)2=o(t)as t0.(1.16)

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

2 The reference hypersurface

Now we introduce a family of hypersurfaces of revolution Sε in ℝn+1 obtained by revolving around the last coordinate axis the curves Γε built in Section 1 and placed on the plane x1 = … = xn−1 = 0. The symmetry of Γε with respect to the vertical axis guarantees that Sε is well-defined. We compute the principal curvatures Ki of Sε and we impose conditions on the parameterisation of Γε which guarantee that the functions Ki depend in a C1 way on the distance from the origin, with null derivative at the origin and at the maximum distance.

To carry out this plan, it is convenient to parameterise the n-dimensional unit sphere 𝕊n in the Euclidean (n + 1)-space in hyperspherical coordinates, as follows:

σ(θ1,,θn)=sinθ1sinθ2sinθncosθ1sinθ2sinθncosθ2sinθ3sinθncosθn1sinθncosθnwith θ1[0,2π],θi[0,π] for i=2,,n.

Fixing a mapping g : ℝ → (0, ∞) of class C, even and 2π-periodic, let us introduce the hypersurface Sg parameterised by x(θ1, …, θn) = g(θn)σ(θ1, …, θn), where (θ1, …, θn) ∈ [0, 2π] × [0, π]n−1.

Lemma 2.1

For everyp = x(θ1, …, θn) ∈ Sgthe principal curvatures ofSgatpare given by


where the primes denote derivatives with respect toθn.


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

στi=τiτj=0for all i,j=1,,n,ij.(2.2)

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

Nxθi=0for all i=1,,n.(2.3)

For i = 1, …, n − 1 we have

Nxθi=(α0σ+j=1nαjτj)gτi=αig|τi|2,and Nxθn=(α0σ+j=1nαjτj)(gσ+gτn)=α0g+αng

since ∣σ∣ = ∣τn∣ = 1. Hence the equations (2.3) imply α1==αn1=0,αn=α0gg and then we can write the outward-pointing normal versor in the form


We compute


Since τnθi=cosθnsinθnτi for i = 1, …, n − 1 and τnθn=σ, we obtain


By definition, the principal curvatures at pSg are the eigenvalues of the shape operator Lp : TpSgTpSg given by vLpv=Nv(p) for vTpSg. Since for p = x(θ1, …, θn) and for every i = 1, …, n one has Lpxθi=(Nx)θi, by (2.4) the principal curvatures of S are given by (2.1).□

Fixing hC(ℝ) even and 2π-periodic, for ε ∈ ℝ let gε be as in (1.2) and let Sε be the n-dimensional hypersurface in ℝn+1 parameterised by

xε(θ1,,θn)=gε(θn)σ(θ1,,θn)+gε(π)en+1where (θ1,,θn)[0,2π]×[0,π]n1

and en+1 = σ(0, …, 0). Observe that Sε is obtained from Sgε via translation by an amount of gε(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. Sεis a compact, embeddedn-dimensional hypersurface of revolution around the last coordinate axis ofn+1, diffeomorphic to 𝕊n;

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

  3. maxpSεp∣ = RεwhereRε > 0 is given by(1.11);

  4. the principal curvatures ofSεsatisfyKi(p) > 0 fori = 1, …, n − 1 and for allpSε ∖ {0, Rεen+1}, andKn(p) > 0 for allpSε.


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

gε(t)sintgε(t)cost>0for all t(0,π).(2.5)

One has gε(t) sin tgε(t)cos t = [1 + ε(t)] sin t where h~(t)=h(t)h(t)sint cos t. Since h′(0) = h′(π) = 0,


the mapping 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 Sε as functions of the distance from the origin. To this aim, let us introduce the mapping yε : [0, Rε] → Sε defined by


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

Lemma 2.3



hold, then the mappingsKiyε : [0, Rε] → ℝ (i = 1, …, n) are of classC1in [0, Rε] and with null derivatives at end points. Moreover, ifhvanishes in neighbourhoods of 0 andπ, thenRε = 2, KiyεC([0, 2]) andKiyε = 1 in neighbourhoods of 0 and 2 fori = 1, …, n.


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


we have that Kiyε = ∘ ∣γε−1 (i = 1, …, n − 1). Moreover C1((0, π)). Therefore KiyεC1((0, Rε)), with


We aim to show that d[Kiyε]dr(r)0 as r → 0 and as rRε. Let us study the limit as r → 0. We have




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


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

Now let us study the limit as rRε. We have


We know that


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

gε(t)gε(t)sintcost(sint)2gε(t)gε2(t)+gε(t)2=o(t2)as t0.

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

gε(t)[gε(t)+gε(t)](sint)gε(t)[gε(t)2+gε(t)2]3/2=o(t2)as t0.

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).□

3 Families of curvature functions with the filling property

In this section we construct families of nonconstant mappings H : ℝn+1 → ℝ having the following properties:

  1. HC1(ℝn+1);

  2. there exist C1, C2 > 0 such that C1H(p) ≤ C2 for every p ∈ ℝn+1;

  3. for every p ∈ ℝn+1 there exists an embedded hypersurface S diffeomorphic to 𝕊n with mean curvature H at every point and with pS.

Other optional properties on H like periodicity or some asymptotic behaviour can be added.

As a first result, we exhibit a family of radially symmetric curvature functions which satisfy (H)1–(H)3 and are constant outside a ball.

Theorem 3.1

Leth : ℝ → ℝ be a 2π-periodic, even function of classCsatisfying(1.13)and(2.7). Forε ∈ ℝ letgε, Rε, Ki, andyεas in(1.2), (1.11), (2.1)withg = gε, and(2.6), respectively. Then there existsε0 > 0 such that for everyε ∈ (−ε0, ε0) the functionHε : ℝn+1 → ℝ defined by


satisfies (H)1–(H)3and is radially symmetric. In addition, ifε → 0 thenRε → 2 andHε → 1 inC1(ℝn+1). Moreover, ifhvanishes in neighbourhoods of 0 andπthenHεis of classC.


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

H~ε(r):=(Myε)(r)for all r[0,Rε]

with yε as in (2.6). Thus ε(∣p∣) equals the mean curvature of Sε at p. By radial symmetry, the same holds true for any hypersurface obtained by rotating Sε about the origin of ℝn+1. Hence, for every p ∈ ℝn+1 with ∣p∣ ≤ Rε there exists an embedded hypersurface diffeomorphic to 𝕊n with mean curvature H at every point and with pS. By Lemma 2.3, εC1([0, Rε]) with ε(0) = ε(Rε) = 0 and setting ε(r) := ε(Rε) for r > Rε, we obtain a function of class C1 on [0, ∞). In particular ε(Rε) = 1/gε(0), and by Lemma 2.2 (iv) there exist C1, C2 > 0 such that C1ε(r) ≤ C2 for every r > 0. Moreover, for every point p ∈ ℝn+1 with ∣p∣ ≤ Rε one can take a round hypersphere of radius gε(0) whose mean curvature equals ε. Finally, because of the above discussion, the function Hε(p) = gε(0)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 gε 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 classCsatisfying(1.13)and(2.7). Then there existsε0 > 0 such that for every set of numbers {εj}j∈ℕ ⊂ ℝ withεj∣ < ε0for all j ∈ ℕ and for every set of points {pj}j∈ℕ ⊂ ℝn+1such thatpipj∣ ≥ max{Rεi, Rεj} + 2, the functionH : ℝn+1 → ℝ defined by


withHεas in(3.1)satisfies (H)1–(H)3. In particular, ifεj = εfor alljand the set {pj} is periodic (in all directions), then the mappingHis periodic. Ifεj → 0 asj → ∞, thenH(p) → 1 asp∣ → ∞. Moreover, ifhvanishes in neighbourhoods of 0 andπthenHis of classC.


This follows from Theorem 3.1 and the fact that the gluing of the blocks defined by H(x) = Hεj(xpj) on BRεj(pj) outside the region ⋃iBRεi(pi) is nice because this function takes the common value 1 outside BRεj(pj). The filling property is satisfied because it holds in each ball BRεj+2(pj) and these balls are pairwise disjoint by the lower bound for ∣pipj∣. Hence the region ℝn+1 ∖ ⋃iBRεi(pi) 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 hjC(ℝ → ℝ) (j ∈ ℕ), considering a corresponding sequence of maps Hεj,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∣ < h1 such that

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

the mapping ∣γε∣ 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 gε = 1 + εh with ε ∈ (−1/7, 2/5).


I wish to thank the anonymous reviewer for her/his careful reading of the manuscript and her/his meaningful and helpful remarks.

  1. Funding: The author is member of the Gruppo Nazionale per ľAnalisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

  2. Communicated by: R. Löwen


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Received: 2021-03-22
Published Online: 2022-04-18
Published in Print: 2022-07-26

© 2022 Paolo Caldiroli, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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