The isoperimetric profile of a compact Riemannian Manifold for small volumes

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Manifold for small volumes

Stefano Nardulli

To cite this version:

Stefano Nardulli. The isoperimetric profile of a compact Riemannian Manifold for small volumes.

2007. �hal-00177259�

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hal-00177259, version 1 - 6 Oct 2007

Manifold for Small Volumes

Stefano Nardulli 7th October 2007

1. In constant sectional curvature spaces the solutions of the isoperimetric problem are geodesic balls, this fact allow us, in the Euclidean case, to have an explicit formula for the isoperimetric profile and in the case of strictly positive or strictly negative curvature an implicit but satisfactory one.

2. In dimension 2, Frank Morgan, Michael Hutchings et Hugh Howards show in [MHH00] page 4899 (Main isoperimetric theorem) that onR² endowed with a complete rotationally invariant metric whose sectional curvature is a strictly increasing fonction of the distance to the origin, the solutions of the isoperimetric problem are the following:

• a disk centered at the origin,

• the complement of a disk centered at the origin,

• a ring centered at the origin.

This result generalises a previous result of Itai Benjamini and Jianguo Cao [BC96] page 361 (Main Theorem) that contains the case of the isoperimetric profile of a metric of paraboloids of revolution. Finally Manuel Ritor´e in [Rit01] with different methods extends the list of surfaces for which the isoperimetric profile is known including planes of revolution with increasing sectional curvature etc. (see page 1094).

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3. Antonio Ros and Manuel Ritoré determine the isoperimetric profile of RP³ in [RR92], Laurent Hauswirth, Joaquín Pérez, Pascal Romon and Antonio Ros in [HJRR04] do the same for certains flat tori of dimension 3.

4. Per Tomter [Tom93] determines the profile only in a neighborhood of 0, for the Heisenberg group of dimension 3.

5. On certain surfaces of non trivial topology, one can construct metrics with somewhat prescribed isoperimetric profile, see [GP07].

For a more detailed exposition of the state of the art we refer to the P.H.D.

thesis of Vincent Bayle [Bay04] and to the survey by Ros and Ritor´e (on line on the web page of department of mathematics of the University of Granada).

0.2 Results

The problem that we study in this article is the behaviour of the fonctionI_M in a neighborhood of the origin. We compute the first nontrivial coefficient of its asymptotic expansion and open the way to the computation of other ones.

We show that the isoperimetric problem reduces, in a natural manner (in particular, compatible with the symmetry of the ambient manifold) to a variational problem in finite dimension. The geometric content of the following statement will become clear in subsequent paragraphs.

Theorem 1. There exists a smooth functionf :M ×R→ Rand a map β that to an arbitrary point p of M and a real r associates a domain β(p, r), such that, for sufficiently small r > 0, the solutions of the isoperimetric problem at volume v =ωnrⁿ are exactly the images via β of the minima of the function p 7→ f(p, r). f and β are invariant (resp. equivariant) under the group of isometries of M.

As a consequence of theorem 1, in [Tom93], Per Tomter determines the isoperimetric profile of the Heisenberg group of dimension 3 endowed with a left invariant Riemannian metric of maximal symmetry, for small volumes.

Tomter uses an unpublished work of Bruce Kleiner, dating back to 1985, that we can state as follows.

Theorem 0.1 (Kleiner). Let Mbe a Riemannian manifold, G a group of isometries that acts transitively onM, K≤Gthe stabilizer of a point.

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Then for smallv, there exists a solution of the isoperimetric profile in volume v that is K-invariant.

We first learned of this result in may 2005, because there is no written trace and no other reference than [Tom93].

One of the goals of this paper is in particular to recover the proof of theorem 0.1. With some generic hypotheses on the Riemannian metric, we can make more precise the statement of theorem 1 and we can show the differentiability properties of the isoperimetric profile in a neighborhood of 0, for details see section 3.

The basic idea of the proof is to apply the implicit function theorem to the map that to a small hypersurface, seen like a small perturbation of a small geodesic sphere, associates its mean curvature. The linearization of this map is not invertible. Hence we modify this map, introducing a new class of hypersurfaces satisfying a weaker condition than the constancy of mean curvature. We call this new kind of hypersurfaces pseudo-balls. On the other hand it will be nedeed to show that a solution of the isoperimetric problem is a perturbation of a small geodesic sphere.

The proof of theorem 1 has two essentials steps, 1. the construction of pseudo balls,

2. the proof that the solutions of the isoperimetric profile are pseudo balls.

The second step is the subject of a separate article [Nar06].

0.3 Pseudo-balls

Definition 0.2. We call pseudo-ball a hypersurface N embedded in M such that there exists a point p ∈ M and a function u ∈ C^2,α(T_p¹M ⋍ Sⁿ−1,R), such thatN is the graph of u in normal polar coordinates centered atp, i.e. N =

expp(u(θ)θ), θ ∈T_p¹M and Q(H(u)) =const.∈R,

where H is the mean curvature operator, Q= id−P, P is the orthogonal projector of L²(T_p¹M) on the first eigenspace of the Laplacian on the unit sphere Sⁿ⁻¹ of Rⁿ, and T_p¹M is the fiber over p of the unit tangent bundle over the Riemannian manifoldM.

To state a uniqueness theorem for pseudo-balls we need the notion of center of mass.

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Definition 0.3. Let (Ω, µ) be a probability space and f : Ω→ Ma measur- able function. we consider the following functionE :M →[0,+∞[:

E(x) := 1 2

Z

Ω

d²(x, f(y))dµ(y).

We call center of mass of f with respect to the measure µ the unique mini- mum ofE on M, provided that it exists.

Proposition 0.1. We assume that the sectional curvature of M satisfies KM ≤δ. We assume also thatf(Ω)has a diameter less than one half of the injectivity radius and than ^π

2√ δ.

ThenE has a unique minimumc, i.e. the unique nearest point to f(Ω)such that R

Ωexp⁻_c¹f(y)dµ(y) = 0.

In particular, we can speak about the center of mass of a hypersurface of small diameter (we apply proposition 0.1 to the (n−1)-dimensional measure of the boundary).

Definition 0.4. Let F^2,α be the fiber bundle on M with fiber at p equal to C^2,α(T_p¹M,R)

Theorem 2. There exists a C^∞ map, β :M ×R→ F^2,α such that for all p ∈ M, and all sufficiently small r > 0, the hypersurface exp_p(β(p, r)(θ)θ) is the unique pseudo-ball whose center of mass ispenclosing a volume ωnrⁿ where ωn := V ol₍Rⁿ,can)(Bⁿ) is the Euclidean volume of the unit ball Bⁿ of Rⁿ.

Remark: Ifg is an isometry ofM,gsends pseudo-balls to pseudo-balls andg◦β =β◦g (g acts only on the first factorM).

Examples:

1. In constant curvature, pseudo-balls are geodesics spheres.

2. In the 3 dimensional Heisenberg group, pseudo-balls have been calcu- lated by Per Tomter [Tom93].

0.4 Geometric measure theory We use the following result, from [Nar06].

Theorem 3. Let Mⁿ be a compact Riemannian manifold, g_j a sequence of Riemannian metrics of class C^∞ that converges C⁴ to a fixed metric g_∞.

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Let B be a domain in M with smooth boundary ∂B, Tj a solution of the isoperimetric problem in (Mⁿ, g_j) such that

(∗) :Mg∞(B−Tj)→0.

Then∂Tj is the graph of a functionuj in normal exponential coordinates to

∂B. Furthermore for all α ∈]0,1[, uj ∈ C^2,α(∂B), and if we assume that

∂B has constant mean curvature for g_∞, then ||u_j||C^2,α(∂B)→0.

In section 3, we deduce the following theorem, that completes the proof of theorem 1.

Theorem 4. Let T be a current solution of the isoperimetric problem, of sufficiently small volume. Then T is a pseudo-ball.

0.5 Expansion of the isoperimetric profile near 0

The asymptotic expansion of the volume of pseudo-balls and the volume of their boundary can be computed with theorem 1, this yields an expansion for the profile.

The paper [Dru02] of O. Druet provides an alternative approach to theorem 7. But our method gives more information. For instance, when scalar curvature is constant, the next term in the expansion of the profile can be obtained.

0.6 Plan of the article

1. Section 1 describes the construction of pseudo balls via the implicit function theorem in an infinite dimensional context.

2. Section 2 describes why and in what sense approximate solutions, of the isoperimetric problem, in the case of small volumes, are close to Euclidean balls.

3. In section 3 the results of preceding sections and those of [Nar06] are applied to obtain the first two non zero coefficients of the asymptotic expansion of the isoperimetric profile.

0.7 Aknowledgements

I wish to thank Renata Grimaldi of the University of Palermo for the fruitful discussions that we had during my P.H.D. studies, about the subject of my thesis. I thank also my P.H.D. advisor Pierre Pansu. I thank also Guy

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David, S´everine Rigot, Giandomenico Orlandi for their suggestions about the theorem of Jean Taylor. I am grateful to Istituto Nazionale di Alta Matematica ”Francesco Severi” for financial support.

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1 Construction of pseudo-balls

1.1 The normal coordinates context

We place ourselves in TpM, the tangent space of M at p, endowed with the Riemannian metric exp^∗_p(g). This is a smooth Riemannian metric in a neighborhood of the origin, ||x||< ε. Let u be a function on T_p¹M ∼=Sⁿ−1

the unit sphere of TpM, such that ||u||L^∞ < ε, we are interested in the hypersurface

Nu={expp(u(θ)θ)|θ∈T_p¹M}.

Such hypersurfaces will be callednormal graphs. Denote byθthe radial unit vector field onT_p(M)− {0}.

1.2 The mean curvature

LetN be an arbitrary hypersurface. It’s mean curvature is H_ν^N(x) =

n−1

X

1 i

II_x^N(ei, ei) =−

n−1

X

1 i

<∇^eiν, ei >g (x) (1) where (e₁, . . . , en−1) is an orthonormal basis ofTxN,ν a normal unit vector field toN and II_x^N the second fundamental form ofN inx.

We will use in the sequel an extension of ν to the whole Tp(M)− {0} into a vector field that is independent from the distance to the origin, i.e. such that [θ, ν] = 0 where [,] is the Lie bracket of two vector fields.

We set

ν =a+bθ,

where the vector field ais tangent to geodesic spheresS(p, r) centered at p and satisfies obviously [a, θ] = 0. In this case, after calculations analogous to what can be found in paragraph 3.3 of [Nar06] one gets:

H_ν(r, θ) =−div_S(p,r)(a)+<∇aa, a >−bII_θ^r(a, a) +bH_θ^r+b∇ab (2) where div_S(p,r)(a) is the divergence of the vector field a restricted to the geodesic sphere of radius r centered at p, II_θ^r is the second fundamental form of the sphere of radius r centered at p in the outward radial direction andH_θ^r is the trace ofII_θ^r.

The next task is to reformulate (2) in terms of objects that live on the sphere

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T_p¹M. We denote ˜a(r) := (dir)⁻¹(a) if a∈T_exp_p_(rθ)M, ir :

T_p¹M → M θ 7→ exp_p(rθ),

we consider the pull-back gr of the metric g with respect to ir (i.e. gr = i^∗_r(g)).

Hν =−div₍Sⁿ⁻1,gr)(ã)+<∇ãrã,a >˜ gr −br²II_θ^r(ã,ã) +bH_θ^r+b∇rãb (3) We introduce the family of metrics ˜gr := _r¹2gr that have the advantage to have no singularities inr= 0, to highlight the nature of the singularity in 0 of (2), thus (3) becomes

Hν =−div₍Sⁿ⁻1,˜gr)(ã) +r² <∇ãrã,ã >_˜gr −br²II_θ^r(ã,ã) +bH_θ^r+b∇rãb (4) 1.3 The mean curvature of a normal graph

In this paragraph we interpret formulae (3) and (4) in the particular case of hypersurfacesNu. Let

Wu :=

q

1 +k−→

∇i^∗_u(g)uk²_i^∗_u_(g). Then

b= ₁

Wu < ν, θ >≥0 ν outward

−W¹u < ν, θ >≤0 ν inward a=−buIθ,u(θ)θ

−→

∇^guu

(5) whereIθ,u(θ)θ is the identification ofRⁿ∼=TpM ofTθTpMwithT_u(θ)θTpM that is induced by the chosen system of normal coordinates centered in p (notation inspired by [Cha95]). In the sequel we always make the choice of b=−_W¹_u, hence

a= u

WuIθ,u(θ)θ

−→

∇^guu

. (6)

Forr sufficiently small, formula (3) becomes H_ν^N_inw(u, θ) = −div₍Sⁿ⁻¹,gu)(

−

→∇guu Wu

)− 1

W_u² <∇⁻^→_∇

guu(u−→

∇guu Wu

),−→

∇guu >_g(7)_u + u²

W_u³II_θ^u(−→

∇guu,−→

∇guu)− 1 Wu

H_θ^u(u, θ)

+ 1

W_u <−→

∇^gu( 1 W_u), u

−

→∇guu W_u )>gu .

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Formula (4) becomes

H_ν^N_inw(u, θ) = −div₍Sⁿ⁻1,˜gu)(

−

→∇˜guu u²Wu

)− 1

u²W_u² <∇⁻^→_∇_˜

guu(

−

→∇˜guu uWu

),−→

∇˜guu >(8)_˜gu

+ 1

u²W_u³II_θ^u(−→

∇˜guu,−→

∇˜guu)

− 1 Wu

H_θ^u(u, θ) + 1 Wu

<−→

∇g˜u( 1 Wu

),

−

→∇˜guu uWu

)>_˜gu. 1.4 Linearization of the modified mean curvature

In this paragraph we denote byT_p¹M=Sⁿ−1TpMthe unit sphere ofTpM. Definition 1.1. Let F^k,α be the fiber bundle on M where the fiber over p is the space of C^k,α functions on the unit tangent sphere T_p¹Mand Γ(F^k,α) the topological space of C^∞ sections of F^k,α.

In other words, if y ∈ Γ(F^k,α), p ∈ M, then x = y(p) is a function on T_p¹M. Lety₀ denote the zero section.

In the rest of this section we are interested in the local behaviour of certain functions on this fiber bundle in view of the application of the implicit function theorem to the solution of equations in a neighborhood of the zero section y₀. In order to do this, it is convenient to denote y = (p, x) and to identify a neighborhood in the fiber bundle F^k,α with the trivialization U ×C^2,α(Sⁿ⁻¹,R) (U open set of M) with the aid of an atlas of the differ- entiable structure ofM.

We define, now, the domains and codomains of the functionals of modified curvature.

Definition 1.2. We let Ψ :

R×Γ(F^2,α) → Γ(F^0,α)

(r, y) 7→ r H(p, r(1 +x))−ⁿ⁻¹r

.

In other words, if p ∈ M, r ∈ R and x is a C^2,α function on T_p¹M, then Ψ(p, r, x) is the C^0,α function on T_p¹M defined by

Ψ(p, r, x) :=r

H(p, r(1 +x))−n−1 r

.

Proposition 1.1. Ψ is C^∞ on the open subset where ||x||∞ < 1. (here

|| · ||∞ is the L^∞-norm)

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Proof: We putu=r(1 +x) in the formula (8) and we get 1. W_r(1+x)=q

1 +_r2(1+x)¹ ²||−→

∇g˜_r(1+x)r(1 +x)||²g_˜_r(1+x) =h₀(p, r, x), h₀(p,0, x) =

q

1 +_(1+x)¹ 2||−→

∇g˜0x||²_g_˜₀

2. −div₍Sⁿ⁻¹,˜g_r(1+x))(

−

→∇˜gr(1+x)r(1+x)

r²(1+x)²W_r(1+x)) = ¹_rh₁(p, r, x), h₁(p,0, x) =−div₍Sⁿ⁻¹,can=g0)(

−

→∇˜g0x (1+x)²h0(p,0,x)) 3. _r₂_(1+x)₂¹_W2

r(1+x)

<∇⁻^→_∇_˜

gr(1+x)rx(

−

→∇˜gr(1+x)rx r(1+x)W_r(1+x)),−→

∇˜g_r(1+x)rx >˜g_r(1+x)=h2(p, r, x), h₂(p,0, x) = _(1+x)₂_h¹

0(0,x)² <∇⁻^→_∇_˜_g

0x(

−

→∇˜g0x

(1+x)h0(p,0,x)),−→

∇˜g0x >_˜_g₀ 4. _r₂_(1+x)₂¹_W3

r(1+x)

II_θ^r(1+x)(−→

∇˜g_r(1+x)rx,−→

∇g˜_r(1+x)rx) = ¹_rh3(p, r, x), h₃(p,0, x) = _(1+x)3h¹0(p,0,x)³(−→

∇g˜₀x,−→

∇g˜₀x)Rⁿ⁻1

5. −_W_r(1+x)¹ H_θ^r(1+x)(r(1 +x), θ) = ¹_rh₄(p, r, x) h₄(p,0, x) = _(1+x)hⁿ⁻¹

0(p,0,x)

6. _W ¹

r(1+x) <−→

∇g˜_r(1+x)(_W ¹

r(1+x)),

−

→∇˜gr(1+x)rx

r(1+x)W_r(1+x))>g_˜_r(1+x)=h₅(p, r, x), h₅(p,0, x) = _h ¹

0(p,0,x)² <−→

∇g˜0(_h ¹

0(p,0,x)),

−

→∇g˜0x (1+x))>_˜g0.

The functionsh₀,h₁,h₂,h₃,h₄,h₅are inC^∞(R×Γ(F^2,α),Γ(F^0,α)) provided

||x||∞<1. From the following formulae

rH(p, r(1 +x)) =h1(r, x) +rh2(r, x) +h3(r, x) +h4(r, x) +rh5(r, x), (9)

Ψ(p, r, x) =rH(p, r(1 +x))−(n−1), (10) we get that Ψ isC^∞. 2

We can compare proposition 1.1 with the calculation of the papers [Ye91, page 383] and [MPM04].

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Lemma 1.1. Let P be the orthogonal projector of L²(T_p¹M) on the first eigenspace of the Laplacian on the unit sphere Sⁿ⁻¹ and let Q be Id−P. Denote by

L:

C^k+2,α(Sⁿ⁻¹) → C^k,α(Sⁿ⁻¹)

v 7→ −△Sn−1v−(n−1)v Then _∂

∂xQ(Ψ(p, r, x))

r=0,x=0 =L.

Furthermore, let C₁^l,α(Sⁿ⁻¹) := Ker(L)^⊥ ∩C^l,α(Sⁿ⁻¹) where Ker(L)^⊥ is taken in the L² sense. Then L:C₁^k+2,α(Sⁿ−1) 7→ C₁^k,α(Sⁿ−1) is an isomorphism.

Proof: The following straightforward calculation shows that L(v) =−△Sn−1v−(n−1)v.

With the notations of proposition 1.1,

Ψ(p,0, tv) =h₁(0, tv) +h₃(0, tv) +h₄(0, tv)−(n−1).

From h₀(0, tv) = 1 +O(t²) we argue that 1. h₃(0, tv) =O(t²),

2. h1(0, tv) = −△Sn−1v

t+O(t²), 3. h₄(0, tv) = (n−1)−(n−1)tv+O(t²), and hence

Ψ(p,0, tv) = −△Sn−1v−(n−1)v

t+O(t²). (11) It follows that

∂

∂xΨ(p, r, x))

r=0,x=0

(v) =−△Sn−1v−(n−1)v. (12) Concerning the bijectivity, the proof is an immediate application of the following lemma withT =L, and this completes the proof. 2

Lemma 1.2. [Bes87, Cor. 32 Appendix, page 464]

Let E,F be two complex or real vector bundles on M.

Let T :C^∞(E)→C^∞(F) be a linear differential operator of orderk.

We consider an extension ofT :L²(E)→L²(F).

If T is elliptic or underdetermined elliptic then

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• Ker(T^∗)⊂C^∞(F)

• dimC(Ker(T^∗))<∞

• W^k,p(F) =T(W^k+l,p(E))L

Ker(T^∗) (1< p <+∞)

• C^l,α(F) =T(C^k+l,α(E))L

Ker(T^∗)

• C^∞(F) =T(C^∞(E))L

Ker(T^∗)

1.5 Differentiability of center of mass Lemma 1.3. There exists a smooth map

c:

R×Γ(F^2,α) → M (r, p, x) 7→ c(r, p, x) defined implicitly by the equation

Z

Np,r,x

exp⁻_c¹zdV ol(z) = 0 (13) in TcMwhere

Np,r,x ={expp(r(1 +x(θ))θ)|θ∈Sⁿ⁻¹}.

Remark: c(r, p, x) is the center of mass of Nr,p,x andc(0, p, x) =p.

Proof: We rewrite formula (13) in the following more suitable form for our purposes:

Z

Sⁿ⁻¹TpM

exp⁻_c¹(expp(r(1 +y(p)(θ))θ))f^∗(dV ol_N_r,p,y(p))(θ) = 0 (14) inTcMwhere

f^∗(dV ol_N_r,p,y(p)) =σdV ol₍Sⁿ⁻¹,can) (15) f :

Sⁿ⁻¹ → Np,r,x

θ 7→ expp(r(1 +x(θ))θ).

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Leto∈ Mbe such thatpandcare in a normal neighborhood ofo, we choose a local field of orthonormal frame, this choice gives an isometry ˜θ7→θ(p,θ)˜ ofSⁿ−1ToMonSⁿ−1TpM. Then (14) becomes

Z

Sⁿ⁻¹ToM

exp⁻¹_c (expp(r(1 +y(p)(θ))θ))f^∗(dV ol_N_r,y)JdV ol₍Sⁿ⁻1,can)(˜θ) = 0, (16) because|det(dθ(˜θ, p))|=J = 1.

We set

f˜(c, r, p, x) :=

Z

Sⁿ⁻¹TpM

exp⁻¹_c (expp(r(1 +y(p)(θ))θ))σdV olSⁿ⁻¹(θ) (17)

F :

M ×R×Γ(F^2,α) → TM (c, r, p, x) 7→ f(c, r, p, x)˜ We verify thatF isC^∞ with respect to all variables,

• by exchanging the operation of derivation and integration, and the differentiability of the exponential map on TMone gets the differentiability with respect to cof F,

• x7→dx is smooth from C^2,α to C^1,α by continuity and linearity,

• θ7→r(1 +x(θ))θ isC^∞(Sⁿ⁻¹, TpM),

• (r, p, x) 7→ f^∗(dV ol_Np,r,x) is C^∞(R× F^2,α, C^1,α(Sⁿ−1,R)) because it is the norm of a multivector of ∧n−1TpM whose components are the determinants in smooth functions ofr, p, x, dx,

• exp_p(r(1 +x(θ))) isC^∞,

• we take composition withexp⁻¹_c ,

• we multiply byσ,

• at last we integrate onSⁿ⁻¹, that is a linear continuous operation.

Now, we can see thatF is divisible byrⁿ⁻¹without changing the smoothness of the resulting functionGand finally, that we can apply the implicit function theorem toG= _rn−1^F . From the preceding arguments we see that G isC^∞ with respect to all variables.

∂G

∂c(c(0, p,0),0, p,0) = ∂

∂c

"

Z

Sⁿ⁻¹TpM

exp⁻¹_c (p)dV ol₍Sⁿ⁻¹,can)(θ)

#

c=p

=αn−1 △

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whereαn−1 :=V ol₍Sⁿ⁻1,can)(Sⁿ⁻¹) is the (n−1)-dimensional volume of the unit sphere with respect to the canonical metric induced by the Euclidean one ofRⁿ and

△:

T_pM → T_pM ×T_pM y 7→ (y,−y).

Let us chose a trivialisation ofT_pMand compose Gwith the projectionπ₁ on vertical fibers. So

∂π₁◦G

∂c (c(0, p,0),0, p,0) =π₁(α_n₋₁△) =−α_n₋₁Id.

Hence the implicit function theorem applies, there exists a uniquec(r, y) of class C^∞ such that π1(G(c(r, y), r, y)) = 0, i.e. G(c(r, y), r, y) is the zero section. 2

Lemma 1.4. There exists a smooth map A such that

exp⁻_p¹(c(r, y)) =rA(r, y) (18) and

A(0, y) = R

Sⁿ⁻¹(1 +x(θ))ⁿ⁻¹θp

||dx||²+ (1 +x)² R

Sⁿ⁻¹(1 +x(θ))ⁿ⁻²p

||dx||²+ (1 +x)² . (19) Proof: In order to show (18) it suffices to remark that

c(0, y) =p.

We now show (19). We choose a trivialisation of the tangent bundle.

Lemma 1.5.

exp⁻_c¹(exp_p(v)) =exp⁻_c¹(p) +v+o(||p−c||+||v||). (20) This reflects the fact that Riemannian manifolds are Euclidean at small scale.

Proof: In an arbitrary system of coordinates of a neighborhood of p, (20) becomes

exp⁻¹_c (expp(v)) =p−c+v+o(||p−c||+||v||). (21) In order to show this, we setq:=p−c andH(p, q, v) =exp⁻¹_p₋_q(exp_p(v)).

Expanding theC^∞functionH at first order in a neighborhood of (0,0), for

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allp, in function of q and v.

H(p,0, v) =v implies that _∂v^∂H(p,0,0) =id.

H(p, q,0) =exp⁻_p₋¹_q(p) =q+o(|q|) because ifw:=exp⁻_p₋¹_q(p) thenexpp−q(w) = p−q+w+o(|w|) =p due to the fact that d₀expz =id for all z and hence q=w+o(|w|) which implies that w=q+o(q). 2

We choose a trivialisation of the tangent bundle and we identify y = (p, x(p)), then from the equation G(c(r, y), r, y) = 0 we deduce

A(r, y) = R

Sⁿ⁻¹(1 +x(θ))ⁿ⁻¹θp

||dx||²+ (1 +x)²+o(r) R

Sⁿ⁻¹(1 +x(θ))ⁿ⁻²p

||dx||²+ (1 +x)²+o(r) +o(1).

If, we put in the preceding equationr = 0 we obtain (19).

This follows from the fact that

σ= (1 +x(θ))ⁿ⁻²rⁿ⁻¹p

||dx||²+ (1 +x)²+o(r) (22) and

exp⁻_c¹(exp_p(r(1 +y(p)(θ))θ))∼exp⁻_c¹(p) +r(1 +x)θ+o(r) (23) from which it follows

1 rⁿ

Z

Sⁿ⁻1TpM

(exp⁻_c¹(p) +r(1 +x)θ+o(r))σdV ol₍Sⁿ⁻¹,can)= 0 (24) and finally

Z

Sⁿ⁻¹TpM−A(r, y) σ rⁿ⁻¹ +

Z

Sⁿ⁻¹TpM

(1 +x) σ

rⁿ⁻¹ =o(1) (25) which proves the lemma. 2

1.6 Existence and uniqueness of pseudo-balls Lemma 1.6. Let

Φ :

R×Γ(F^2,α) → TM × C^0,α(Sⁿ⁻¹)∩(ker(L))^⊥ (r, p, x) 7→ (A(r, y), Q◦Ψ(r, p, x))

Then, forr sufficiently small,

there exists a unique x(p, r) in C^2,α(Sⁿ⁻¹) of small C^2,α norm, solution of the implicit equation

Φ(r, p, x(p, r)) = Φ(r, yr) = (y0,0).

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Here y₀ is the zero section of TM.

Furthermore, x depends smoothly on p andr.

Proof: Like in the preceding lemma, we remark that Φ isC^∞ in all its arguments. TrivialisingTMin the usual manner we have:

∂Φ

∂x(p,0,0) = (d

dtA(0, tv)_|_t=0, L)

∂Φ

∂x(p,0,0) :

C^2,α(Sⁿ⁻¹) → T_pM ×C^0,α(Sⁿ⁻¹)∩(ker(L))^⊥

v 7→ (_αⁿ

n−1

R

Sⁿ⁻1v(θ)θdV ol₍Sⁿ⁻¹,can), L(v)).

Remember that hereL=−△^Sⁿ⁻¹ −(n−1).

In order to obtain this result we combine the lemma with the following calculations:

A(0, tv) = ntR

Sⁿ⁻1v(θ)θ+O(t²) α_n₋₁+R

Sⁿ⁻¹(n−1)tvdV ol_can+O(t²) (26) that follows by puttingx=tv in (19).

Hence

d

dtA(0, tv)_|_t=0 = n αn−1

Z

Sⁿ⁻1

v(θ)θdV ol₍Sⁿ⁻¹,can). (27) If we identify TpM and ker(L) via the isomorphism of vector spaces that maps vectors of T_pM to the restriction of linear functions to the sphere Sⁿ−1TpM, it can be easily seen that ^∂Φ_∂x(p,0,0) is an isomorphism. From the previous discussion we conclude the existence of a smooth functionx(p, r) in the two variables such thatc(r, p, x(p, r)) =pandH(p, r(1+x(p, r)))−ⁿ⁻_r¹ ∈ Ker(L). Furthermore the implicit function theorem also asserts thatx(p, r) is the unique small solution of these equations. 2

Lemma 1.7. There exists ρ : M×]r₀, r₀[→ (−ρ₀, ρ₀) of class C^∞ defined implicitly by

V oln(x(p, r))V oln(N_p,r,x(p,r)⁺ ) =ωnρ(p, r)ⁿ, and there exists r:M×]ρ₀, ρ₀[defined by

V oln(x(p, r(p, ρ)) =ωnρⁿ. Here N_p,r,x(p,r)⁺ ={expp(tθ)|0≤t≤r(1 +x(p, r))}.

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Proof: Because V oln(x(p, r)) = ωnrⁿh(p, r) where h is aC^∞ function with h(p,0) = 1, then we can write ρ(p, r) := rh(p, r)ⁿ¹. Furthermore it is easy to see that ^∂ρ_∂r(p,0) = 1, hence we can solve

rh(p, r)ⁿ¹ =ρ inr and obtain a function r(p, ρ(p, r)) =r. 2

Theorem 5. There existρ₀ and a smooth mapβ :M ×R→ F^2,α such that for all p ∈ M, and for all ρ₀ > ρ > 0, the hypersurface expp(β(p, ρ)(θ)θ) is the unique pseudo-ball which has its center of mass at p and enclosing a volumeωn|ρ|ⁿ.

Proof: β(p, ρ) := r(p, ρ)(1 +x(p, r(p, ρ))) where x(p, r) is produced by the preceding lemma. 2

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2 Approximate solutions of the isoperimetric prob- lem for small volumes are nearly round spheres

2.1 Introduction

In this section it is assumed that

1. Mhas bounded geometry (|K| ≤Λ andinj_M≥ε >0) whereinj_M is the injectivity radius ofM,

2. the domainsDj ∈τ_M are approximate solutions i.e. ^{V ol}_{I(V ol}ⁿ⁻¹^(∂D^j⁾

n(Dj)) → 1 forj→+∞.

We prove in this section the following theorem.

Theorem 2.1. Let (M, g) be a Riemannian manifold with bounded geometry, Dj a sequence of approximate solutions of the isoperimetric problem such thatV olg(Dj)→0. Then there exist pj ∈ M, and radii Rj such that

j→lim+∞

V ol(Dj∆B(pj, Rj))

V ol(Dj) →0. (28)

The proof of theorem 2.1 occupies the rest of the section.

2.2 Taylor’s theorem revisited

Jean Taylor has shown that polyhedral chains inRⁿ which are approximate solutions of the isoperimetric problem are close to balls in the mass norm, as stated in the following theorem.

Definition 2.1. We denote by cn := ^{V ol}ⁿ⁻¹⁽^Sⁿ⁻¹⁾

[V oln(Bⁿ)]ⁿ⁻¹ⁿ the constant in the Eu- clidean isoperimetric profile.

Theorem 2.2. Let W be the n-ball of Rⁿ centered to the origin, of volume 1.

Let {S_j} ⊂ Pn(Rⁿ) be a sequence of polyhedral chains (i.e. of density 1) contained in a big ball of Rⁿ, of barycenter at the origin, M(Sj) = 1 and satisfying

j→lim+∞M(∂S_j) =M(∂W).

Then

M(Sj −W)→0.

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Proof: Apply Taylor’s theorem as stated in pages 420-421 of [Tay75] to the constant functionF equal to 1. 2

It turns out that the same theorem it’s true if the minimizing sequence is composed of more general currents than polyhedral chains, for example of integral currents (by the strong approximation theorem of Federer 4.2.20) and also if the minimizing sequence is not of bounded diameter. This follows from arguments which are somehow hidden in [Alm76]. A good reference for the following theorem is [LR03].

Theorem 2.3. Let {Tj} ⊂I_n(Rⁿ) be a sequence of integral currents, satisfying

j→+∞lim

M(∂T_j)

M(Tj)ⁿ⁻¹ⁿ =cn. Then there exist ballsW_j such that

M(Tj−Wj) M(Wj) →0.

2.3 Lebesgue numbers

Let (M, g) be a Riemannian manifold with bounded geometry. We can construct a good covering ofMby balls having the same radius.

Lemma 2.1. Let(M, g) be a Riemannian manifold with bounded geometry.

There exist an integerN, some constants C, ǫ >0 and a covering U of M by balls having the same radius 3ǫ and having also the following properties.

1. ǫ is a Lebesgue number for U, i.e. every ball of radius ǫ is entirely contained in at least one element ofU and meets at most N elements of U.

2. For every ball B of this covering, there exist a C bi-Lipschitz diffeomorphism on an Euclidean ball of the same radius.

Proof: Let ǫ = ^inj₂^M. Let B = {B(p, ǫ)} be a maximal family of balls of Mof radius ǫ that have the property that any pair of distinct members of B have empty intersection. Then the family 2B :={B(p,2ǫ)} is a covering of M. Furthermore, for all y ∈ M, there exist B(p, ε) ⊂ B such that y∈B(y,2ǫ) and thus B(y, ε)⊆B(p,3ǫ). Hence ǫis a Lebesgue number for the covering 3B. Let B(p,3ǫ) and B(p^′,3ǫ) be two balls of 3B having non

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empty intersection. Thend(p, p^′)<6ǫ, henceB(p^′, ǫ)⊆B(p,7ǫ). The ratios V ol(B(p,7ǫ))/V ol(B(p, ǫ)) are uniformly bounded because the Ricci curvature ofMis bounded from below, and hence the Bishop-Gromov inequality applies. The number of disjoints balls of radius ǫ, contained in B(p,7ǫ), is bounded and does not depend on p. Thus the number of balls of 3B that intersect one of these balls is uniformly bounded by an integer N. We conclude the proof by takingU := 3B. In fact by Rauch’s comparison theorem, for every ballB(p, ǫ), the exponential map isC bi-Lipschitz with a constant C that depends only on ǫ and on upper bounds for the sectional curvature K. 2

2.4 Cutting domains in small diameter subdomains

This section is inspired by the article of B´erard and Meyer [BM82] lemme II.15 and the theorem of appendix C page 531.

Proposition 2.1. Let I be the isoperimetric profile ofM. Then lim sup

a→0

I(a) aⁿ⁻ⁿ¹ ≤cn. Proof: Fix a point p∈ M.

lim sup

a→0

I(a)

aⁿ⁻¹ⁿ ≤lim sup

a→0

V ol(∂B(p, r(a))) V ol(B(p, r(a)))ⁿ⁻¹ⁿ withr(a) such thatV ol(B(p, r(a))) =a.

Changing variables in the limits, we find lim sup

a→0

V ol(∂B(p, r(a)))

V ol(B(p, r(a)))ⁿ⁻¹ⁿ = lim sup

r→0

V ol(∂B(p, r)) V ol(B(p, r))ⁿ⁻¹ⁿ lim sup

r→0

rⁿ⁻¹V ol(Sⁿ⁻¹) +· · ·

[rⁿV ol(Bⁿ) +· · ·]ⁿ⁻¹ⁿ = cn. 2

Definition 2.2. Let r >0. We define the unit grid of Rⁿ and we denote byG1 the set of points which have at least one integer coordinate (∈Z). We call grid of mesh r in Rⁿ a set G of the form v+rG₁ where v ∈ Rⁿ. We denote by Gr:= ([0, r]ⁿ,Lⁿ) the set of all grids of mesh r, endowed with its natural Lebesgue mesure.

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Proposition 2.2. Let D be an open set of Rⁿ. 1

rⁿ Z

G^r

V oln−1(D∩G)Lⁿ(dG) = n

rV oln(D).

Proof: We observe that every grid G decomposes as a union of n sets G⁽ⁱ⁾ of the type v+tG⁽ⁱ⁾₁ where G⁽ⁱ⁾₁ is the set of points with integer i−th coordinate.

Moreover G⁽ⁱ⁾ ∩G^(j) has (n−1)-dimensional Hausdorff measure equal to zero.

1 rⁿ

Z

Gr

V ol_n₋₁(D∩G)Lⁿ(dG) = 1 rⁿ

n

X

i=1

Z

[0,r]ⁿ

V ol_n₋₁(D∩G⁽ⁱ⁾)Lⁿ(dG)

= 1

rⁿ

n

X

i=1

Z r 0

rⁿ⁻¹V oln−1(D∩G⁽ⁱ⁾)Lⁿ(dG)

= n

rV oln(D).

2

Corollary 2.1. Let r >0. Let D be an open set ofRⁿ. There exists a grid Gof mesh r such that

V oln−1(D∩G)≤ n

rV oln(D). (29)

Proposition 2.3. We denote DG,k the connected components of D\ G.

Then P

kV ol(∂DG,k)−V ol(∂D) V ol(D)ⁿ⁻ⁿ¹ →0 for V ol(D)ⁿ¹

r →0.

Proof: For every grid G, X

k

V ol(∂D_G,k)−V ol(∂D) = 2V oln−1(D∩G).

By corollary 2.2, there exists a gridGsuch thatV ol_n₋₁(D∩G)≤ ⁿ_rV ol_n(D).

We deduce that 0≤

P

kV ol(∂DG,k)−V ol(∂D) V ol(D)ⁿ⁻ⁿ¹ ≤

2n

r V oln(D)

V ol(D)ⁿ⁻ⁿ¹ = 2nV oln(D)¹ⁿ

r .

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Thus ifr is very large with respect to V ol(D)ⁿ¹ then P

kV ol(∂DG,k)−V ol(∂D) V ol(D)ⁿ⁻ⁿ¹

is close to 0. 2

Proposition 2.4. LetMbe a Riemannian manifold with bounded geometry.

Let Dj be a sequence of domains ofM so that 1. V oln(Dj)→0.

2. lim sup_j_→₊_∞^{V ol}ⁿ⁻¹^(∂D^j⁾

V ol(Dj)ⁿ⁻ⁿ¹ ≤cn.

For any sequence (r_j) of positive real numbers that tends to zero (r_j → 0 ) and ^{V ol(D}^j⁾

n1

rj → 0, there exists a cutting Dj = S

kDj,k of Dj in domains Dj,k with Diam(Dj,k)≤const_M·rj such that

lim sup

j→+∞

P

kV oln−1(∂Dj,k) (P

kV ol(Dj,k))ⁿ⁻ⁿ¹ ≤c_n.

Proof: We apply lemma 2.1 and we take a covering{U} of Mby balls of radius 3ǫ, of multiplicity N and Lebesgue number ǫ > 0. For every ball B(p,3ǫ) of this family, we fix a diffeomorphismφp:B(p,3ǫ)→BRⁿ(0,3ǫ) of Lipschitz constant C.

For every j we fix also a radius r_j >> V ol_n(D_j)ⁿ¹ and we map the grids of mesh rj of Rⁿin B(p,3ǫ) via φp, i.e. forG∈ Grj, we have

Gp =φ⁻¹_p (G).

Let us denote byDj,kthe connected components ofDj\(∪pGp). We are look- ing for an estimate of the supplementary volume introduced by the cutting in thisD_j,k,

X

k

V ol_n₋₁(∂D_j,k)−V ol_n₋₁(∂D_j) = 2V ol_n₋₁(D_j ∩(∪lG_l)).

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First estimate the average m = _r¹n j

R

GrjV oln−1(Dj∩(∪^lGl))Lⁿ(dG) of this volume over all possible choices of the gridsG∈ Grj.

m ≤ 1

rⁿ_j X

p

Z

Grj

V oln−1(Dj∩Gp)Lⁿ(dG)

≤ 1 rⁿ_j

X

p

Z

G_rj

V oln−1(Rⁿ,φ⁻¹p

∗(g))(φp(Dj)∩G)Lⁿ(dG)

≤ C rⁿ_j

X

p

Z

Grj

V ol_n₋₁₍_Rn,can)(φ_p(D_j∩ Up)∩G)Lⁿ(dG)

≤ Cn r_j

X

p

V oln(φp(Dj∩B(p,3ǫ)))

≤ C²n rj

X

p

V oln(Dj∩B(p,3ǫ))

≤ C²n rj

N V oln(Dj).

This is true because every point of M is contained in at most N balls B(p,3ǫ). Then there existsGinGrj such that

V oln−1(Dj∩(∪^pGp))≤C²n

r_jN V oln(Dj), and so

0≤ P

kV ol_n₋₁(∂D_j,k)−V ol_n₋₁(∂D_j)

V oln(Dj)ⁿ⁻¹ⁿ ≤2C²n rj

N V oln(Dj)ⁿ¹. From the last inequality we obtain

lim sup

j→+∞

P

kV ol_n^M₋₁(∂Dj,k) (P

kV ol^M_n (D_j,k))ⁿ⁻¹ⁿ = lim sup

j→0

V ol^M_n₋₁(∂Dj) V ol_n^M(D_j)ⁿ⁻¹ⁿ ≤cn.

Now, fix x ∈ Dj. By construction, ǫ is a Lebesgue number of the covering {U}, and there exists a ball B(p,3ǫ) that contains B_M(x, ǫ). Let D_j,k denote the connected components ofD\(∪pGp) that containsx, andD_j,k^′ the connected components of φp(B(p, ǫ))\G that contains φp(x). We observe thatD_j,k^′ is a cube of edger_j, ifj is large enough so thatr_j ≤ǫ/C√

n, then D_j,k^′ is contained in φp(B(p, ǫ)), hence Dj,k is contained in φ⁻_p¹D_j,k^′ , which have diameter at mostC rj. 2

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2.5 Selecting a large subdomain

We first show that an almost Euclidean isoperimetric inequality can be applied to small domains.

Lemma 2.2. Let M be a Riemannian manifold with bounded geometry.

Then

V ol(∂D)

V ol(D)ⁿ⁻ⁿ¹ ≥c_n(1−η(diam(D))) (30) withη→0 asdiam(D)→0.

Proof: In a ball of radius r < inj(M), we reduce to the Euclidian isoperimetric inequality via the exponential map, that is a C bi-Lipschitz diffeomorphism withC= 1+O(r²). This implies for all domains of diameter

< r,

V ol(∂D)

V ol(D)ⁿ⁻ⁿ¹ ≥c_nC⁻²ⁿ⁺² =c_n(1− O(r²)).

2

Second, we have a combinatorial lemma that tells us how in a cutting the largest domain contains almost all the volume.

Lemma 2.3. Let fj,k ∈ [0,1] be numbers such that for all j, P

kfj,k = 1.

Then

lim sup

j→+∞

X

k

f

n−1 n

j,k ≤1 implies that

j→+∞lim max

k fj,k = 1.

Proof: We argue by contradiction. Suppose there existsε >0 for which there exists jε∈Nso that for all j≥jε, we have maxk{fj,k} ≤1−ε. Then for all j≥jε, we have f_j,k ≤1−ε. From this inequality,

X

k

f

n−1 n

j,k =X

k

fj,kf

−1 n

j,k ≥ P

kf_j,k

(1−ε)ⁿ¹ ≥ 1 (1−ε)ⁿ¹ , hence

lim sup

j→+∞

X

k

f

n−1 n

j,k ≥ 1

(1−ε)¹ⁿ >1,

The isoperimetric profile of a compact Riemannian Manifold for small volumes

HAL Id: hal-00177259

https://hal.archives-ouvertes.fr/hal-00177259

Manifold for small volumes

Stefano Nardulli

To cite this version:

hal-00177259, version 1 - 6 Oct 2007

Manifold for Small Volumes

Contents

1 Construction of pseudo-balls

2 Approximate solutions of the isoperimetric prob- lem for small volumes are nearly round spheres