The extremal plurisubharmonic function of the torus

The extremal plurisubharmonic function of the torus

Federico Piazzon^a

In honor of the 60th birthday of my PhD advisor and friend, Norm Levenberg.

Communicated by S. De Marchi

Abstract

We compute the extremal plurisubharmonic function of the real torus viewed as a compact subset of its natural algebraic complexification.

1 Introduction

Pluripotential Theory is the study of the complex Monge Ampere operator(dd^c)ⁿand plurisubharmonic functions. It is a non linear potential theory on multi-dimensional complex spaces that can be understood as the natural generalization of classical Potential Theory. Indeed, when the dimensionnof the ambient space is 1, the two theories coincide. We refer to the classical monograph[21]and to[24]for an extensive review on the subject, including the more recent topics and developments.

Pluripotential Theory is deeply related to complex analysis, approximation theory, algebraic and differential geometry, random polynomials and matrices, as discussed e.g.,[23,20,8,3,9]. Despite this rather wide range of applications, there are very few examples in which analytic formulas for the main quantities of Pluripotential Theory are available. The aim of the present work is to compute a formula for one of these objects, theextremal plurisubharmonic function V_T(·,T)(see Definition1.1below) of the set of real points of the two dimensional torus, i.e.,

T:=T ∩R³, considered as a compact subset of its algebraic complexification

T :={z∈C³:(z₁²+z₂²+z₃²−(R²+r²))²=4R²(r²−z₃²)}, 0<r<R<∞. (1) Geometrically, the setTis obtained as surface of revolution rotating the real circle{(z₁−R)²+z₂²=r²,z₃=0} ∩R³with respect to theℜz3axis. Note thatT can be equivalently described by

T ={z∈C³:(z₁²+z₂²+z₃²+R²−r²)²=4R²(z²₁+z²₂)}, 0<r<R<∞. (2) The motivation of this investigation is three-fold. First, building examples of explicit formulas for extremal plurisubharmonic functions has its own interest since the known cases are so few. Second, the torus is a very classical set in various branches of Mathematics, so it would be interesting to specialize to the case of the torus certain applications of Pluripotential Theory, e.g., approximation of functions, orthogonal expansions, polynomial sampling inequalities and optimization, random polynomials, random arrays, and determinantal point processes. Lastly we mention our original motivation. Let(M,g)be a compact real analytic Riemannian manifold. It has been proven that (an open bounded subset of) the Riemann foliation of the real tangent spaceT M admits a natural complex structure such that the leaves of the Riemannian foliation glue together to construct a complex (Stein) manifoldX in whichMembeds as a totally real submanifold[22]. The Stein manifoldX is termed theGrauert tubeofM. The most relevant two examples of unbounded Grauert tubes (see[26]and references therein), are relative to the real sphere and real projective space. Their construction is in some sense canonical, not only from the Riemannian perspective, but also from the pluripotential point of view. In this last situation two elements are crucial: the algebraicity of the starting real manifold and the fact that the Baran metric[12](a specific Finsler metric that can be defined starting from the extremal plurisubharmonic function) is Riemannian.

Analogous investigations for the torus, that will be the subject of our future studies, necessitate the computation of the extremal plurisubharmonic function for the torus. This is accomplished in the present paper as stated in the following theorem.

Theorem 1.1. The extremal plurisubharmonic function of the torus (of radii0<r<R<∞) is V_T(z,T):=logh

max

|p

1−(z3/r)²+1|+|p

1−(z3/r)²−1|

2 ,

Æz₁²+z₂²+z2

2(r+R) +

Æz₁²+z₂²−z2

2(r+R) +

Æz₁²+z₂² (r+R) −1

, (3)

where equality holds only for z∈T and h(z):=z+p

z²−1is the inverseJoukowski function, see(7)below.

Remark1. It is worth saying that, even though the maximization procedure of formula (3) may cause in principle a rather irregular behavior ofV_T^∗(·,T), the plot we made seem to reveal that indeedV_T^∗(·,T)is quite smooth away fromT. Since so far we have only partial results on this aspect, we display some sections of the graph ofV_T^∗(·,T)in Figure1. Note in particular that the apparent jump in the derivative in the last two pictures of Figure1is confined to the singular set ofT (two leaves comes together) and it is actually only due to the choice of the branch of the square root when defining the local coordinates (z1,z2)7→(z1,z2,z^(h)(z1,z2))forT that are used in the proof of formula (3). This is evident in Figure2below, where the two branches are plotted together.

Acknowledgements. Obviously the present work has been deeply influenced by the discussions that I had with Norm Levenberg, both during and after my PhD period. This is fairly evident also by the number of papers of Professor Levenberg appearing in the references. Norm was the first one teaching me Pluripotential Theory and he is responsible of my fascination for the subject, which is a direct consequence of his enthusiastic lectures.

The first version of the present work contained an error which has been pointed out by Mirosław Baran during themultivariate polynomial approximation and pluripotential theorysection of the DRWA18 workshop. Sione Ma‘u find out in few hours how to fix the issue: the last part of the proof of Theorem1.1is essentially due to him.

1.1 Pluripotential Theory in the Euclidean setting

Before proving equation (3), we recall for the reader’s convenience some notation, definitions, and basic facts from Pluripotential Theory in different settings.

Plurisubharmonic functionson a domainΩ⊂Cⁿare functions that are globally uppersemicontinuous and subharmonic on the intersection withΩof any complex line (or analytic disk inΩ). We denote this set of functions by PSH(Ω). Ifu∈C²(Ω)the conditionu∈PSH(Ω)reduces to the positivity of the(1, 1)differential form

dd^cu=2i∂∂¯u:=

n

X

j=1

∂²u

∂z_j∂z¯_j(z)dz_j∧d¯z_j. Here we used the classical notation for exterior and partial differentiation inCⁿ, i.e.,

d := (∂+∂¯), d^c:=i(∂¯−∂),

∂u:=

n

X

j=1

∂u

∂z_jdz_j, ∂¯u:=

n

X

j=1

∂u

∂¯z_jd¯z_j

If a functionuis uppersemicontinuous and locally integrable, we can still check if it is a plurisubharmonic functions by checking the positivity of thecurrent(i.e., differential form with distributional coefficients) dd^cu. We refer the reader to[21]for an extensive treatment of the subject.

The complexMonge Ampereoperator(dd^c)ⁿcan be defined forC²(Ω)functions by setting (dd^cu)ⁿ:=dd^cu∧dd^cu∧. . .· · · ∧dd^cu=c_ndet[∂h∂¯ku]h,kd Vol_Cn.

Note that, in contrast with dd^c, this is a fully non linear differential operator, therefore its extension to functions that are not C²(Ω)is highly non-trivial..

In the seminal paper[6], Bedford and Taylor extended such a definition to locally bounded plurisubharmonic functions by an inductive procedure on the dimensionn. In this more general setting(dd^cu)ⁿis ameasure. Further extensions have been carried out more recently[17].

Among all plurisubharmonic functions,maximal plurisubharmonic functionsplay a very special role: one can think of the relation of this subclass with PSH as an analogue to the relation of harmonic functions with subharmonic functions inC. Indeed a plurisubharmonic functionuis maximal inΩif for any subdomainΩ⁰⊂Ωand for anyv∈PSH(Ω)such thatu(z)≥v(z)for anyz∈∂Ω⁰, it follows thatu≥vinΩ⁰. Perhaps more importantly to our aims, a maximal plurisubharmonic functionuonΩis characterized by (being plurisubharmonic and)(dd^cu)ⁿ≡0 onΩin the sense of measures.

In Pluripotential Theory there is an analogue of the Green Function with pole at infinity. LetK⊆Cⁿbe a compact set. One can consider the upper envelope

V_K(z):=sup{u(z):u∈L(Cⁿ),u(w)≤0∀w∈K}, (4) whereL(Cⁿ)is the Lelong class of plurisubharmonic functions onCⁿhaving logarithmic growth, i.e.,u∈L(Cⁿ)ifuis plur- isubharmonic onCⁿand for anyMlarge enough there exists a constantCsuch thatu(z)≤log|z|+Cfor any|z|>M. There are

two possible scenarios. EitherV_Kis not locally bounded and in this case the setKis termed pluripolar (roughly speaking it is too small for pluripotential theory). OrV_Kis locally bounded and its uppersemicontiunuous regularization

V_K^∗(z):=lim sup

w→z

V_K(w), (5)

which is calledZaharjuta extremal plurisubharmonic function, is a locally bounded plurisubharmonic function onCⁿwhich is maximal onCⁿ\K, that is

(dd^cV_K^∗)ⁿ=0 onCⁿ\K.

One important fact aboutV_Kis that one can recover the same function by a different upper envelope. More in detail, following [30], we have

V_K(z) =log⁺ΦK(z), where theSiciak extremal functionΦKis defined by setting

ΦK(z):= lim

j→∞ sup{|p(z)|,p∈P^j(C),kpkK≤1}1/j

. (6)

HereP^j(Cⁿ)denotes the space of polynomial functions with complex coefficients onCⁿwhose total degree is at mostj∈Nand k · kKdenotes the uniform norm onK.

The functionV_K^∗(z) =logΦ^∗_K(z)is referred as thepluricomplex Green function with pole at infinityofCⁿ\Kor as theextremal plurisubharmonic functionofK.

Computing extremal plurisubharmonic functions is, in general, a very hard task. There are very few examples (see[11,7]) of settings in which it has been computed by various (hard to be generalized) techniques. Numerical methods for the approximation of extremal functions have been developed in[27]. An exceptional case is the one ofKbeing a centrally symmetricconvex real body. In such a case the Baran formula (see (8) below) gives an analytic expression forV_K^∗, moreover even deeper properties of this function have been studied (see[13,15,16]) as the structure of the Monge Ampere foliation ofCⁿ\Kand the density of (dd^cV_K^∗)ⁿwith respect to thendimensional Lebesgue measure onK,[2,1].

LetK⊂Rⁿbe a real convex body. We denote byK^∗its polar set

K^∗:={x∈Rⁿ:〈x;y〉 ≤1,∀y∈K}

and by ExtrK^∗the set of itsextremal points. That is,x∈ExtrK^∗ifx∈K^∗andxis not an interior point of any segment lying in K^∗(note thatK^∗is convex by definition but it may be not strictly convex).

We denote byh:C→Ctheinverse Joukowski map

h(z):z+p

z²−1. (7)

Baran[2]proved that, for any centrally symmetric convex real body, V_K^∗(z) = sup

y∈ExtrK^∗

log|h(〈z,y〉)|. (8)

1.2 Pluripotential Theory on algebraic varieties

LetAbe an algebraic variety having pure dimensionm, 1≤m<n. Take any set of local defining functionf1,f2, . . . ,f_kforAand consider the function

f(z):= max

j∈{1,2,...,k}log|fj(z)|.

This is a a plurisubharmonic function onCⁿsuch thatAis (locally) contained in the{−∞}set of it. This is (see for instance[21]) equivalent to the fact thatAis locally pluripolar, that istoo small for n dimensional pluripotential theory. On the other hand, the setA_regof the regular points ofAis anm−dimensional complex manifold som-dimensional pluripotential theory is well defined on it by naturally extending the definitions given in the Euclidean setting using local holomorphic coordinates. Then, using the nice properties of coordinate projections[18], it is possible to define pluripotential theory (and in particular the complex Monge Ampere operator and an associated capacity) on the setA,[32,19,5].

Up to this point, this procedure can be carried out on analytic sets. In contrast, if we want to deal with the restriction of polynomials ofCⁿtoA, the algebraicity ofAbecomes determining as shown by the following fundamental result.

Theorem 1.2(Sadullaev[29]). Let A⊂Cⁿbe a irreducible analytic subvariety ofCⁿof dimension1≤m<n.Then A is algebraic if and only if there exists a compact set K⊂A such that

V_K(z,A):=log⁺lim

j→∞ sup

|p|,p∈P^j(Cⁿ): kpkK≤1 ^1/j (9) is locally bounded on A.If this is the case, then V_L(·,A)is locally bounded for any compact set L⊂A such that L∩A_regis not pluripolar in A_regand the function

V_L^∗(z,A):=lim sup

A_reg3ζ→z

V_L(ζ,A) (10)

is maximal in A\L,i.e.,

(dd^cV_L^∗(z,A))^m=0, in A\L. (11)

The extremal plurisubharmonic functionV_K^∗(·,A)defined by Sadullaev can be understood as the natural counterpart of the (log of the) Siciak type extremal function defined in (6). The definition of a Zaharjuta type (i.e., built by an upper envelope of plurisubharmonic functions of a given growth) extremal function in this setting can be view as a particular case of thepluricomplex green function for Stein Spaces with parabolic potential, developed in[32]. LetA⊂Cⁿan algebraic irreducible variety of pure dimensionm<n, then we denote byL(A)the class of locally bounded functionsuonAthat are plurisubharmonic onA_regsuch that for some constant (depending onu) we have

u(z)< max

j=1,2,...,nlog|z_i|+c_u for anyz∈Aand|z|large enough. In thissimplyfiedsetting we have the following.

Theorem 1.3(Zeriahi[32]). Under the above assumption the Siciak and the Zaharjuta type extremal functions are the same, that is V_K^∗(z,A) =lim sup

ζ→z

sup{u(ζ)∈L(A):u≤0on K} (12)

for any compact set K⊂A.

Definition 1.1(Plurisubharmonic extremal function). Since in our setting of an irreducible algebraic variety of pure dimension m<nembedded inCⁿthe extremal functions (10) and (12) are the same, we refer to both of them asthe plurisubharmonic extremal functionofK.

Remark2. Wewarn the readerthat, for the sake of an easier presentation of the definitions and results, we restrict our attention to the case ofirreduciblealgebraic varieties. This avoids some ambiguity in the considered class of plurisubharmonic functions (namely any weakly plurisubharmonic function is actually plurisubharmonic[19]) and consequently in the definition of the extremal functions. Indeed, in our settingu∈PSH(A)equivalently means that

• uis the restriction toAof a plurisubharmonic function ˜u:Ω→[−∞,∞[, for some neighbourhoodΩofAinCⁿor

• the restriction ˆuof the uppersemicontinuous functionuto the regular pointsA_regofAis a plurisubharmonic function on the complex manifoldAreg.

1.3 Polynomial degrees and P-pluripotential Theory

When dealing with multivariate polynomials, the concept of degree of a polynomial needs to be specified. Although the standard choice is to use the so-calledtotal degreedeg (i.e., deg(z^α):=|α|1and deg(Pk

i=1c_iz^α) =max1≤i≤k{deg(z^αi):c_i6=0}), for various applications it may be convenient to use different definitions of "degree". Classical examples of this are the so calledtensor degree andEuclidean degree[31], corresponding to the use of`<sup>∞</sup>and`²norms instead of the`¹norm in the definition of the function deg, respectively.

It is clear that the definition of the Siciak type extremal function (6) depends on which definition of degree is used, but the Lelong classL(Cⁿ)used in defining the Zaharjuta type extremal function (4) a priori does not.

More in general one can define a degree deg_PonP(Cⁿ)depending on a subsetPofRⁿ₊:={x∈Rⁿ:x_i≥0∀i=1, 2, . . . ,n} satisfying certain geometric properties. The study of this variant of Pluripotential Theory has been started very recently in[10,4]

and it is termedP-pluripotential Theory. We recall a few facts about P-pluripotential Theory that we will need to use later on.

LetP⊂Rⁿ₊be a convex set containing a neighborhood of 0 (in the relative topology ofRⁿ₊). We denote consider the following polynomial complex vector spaces

Poly(kP):=span{z^α:α∈kP∩Nⁿ},∀k∈N. (13)

Equivalently one can set

deg_P(z^α):=inf

k∈N{k:α∈kP}, degP(X

α∈I

c_αz^α):= max

α∈I,c_α6=0deg_P(z^α), (14) andp∈Poly(kP)if and only ifpis a polynomial and deg_P(p)≤k. Let us recall that thesupport functionφPof the convex setPis defined as

φP(x):=sup

y∈P〈x;y〉,∀x∈Rⁿ₊.

We can use the support function ofPto introduce a dependence onPin (a modified version of) the definition of the Lelong class.

Namely, we define thelogarithmic support function H_Pand theP-Lelong class

H_P(z):=φP(log|z1|, log|z2|, . . . , log|zn|). (15) L_P(Cⁿ):={u∈PSH(Cⁿ): u−H_Pis bounded above for|z| → ∞}. (16) Note that the standard case of total degree polynomials corresponds to pickingP=Σ, the standard unit simplex. Indeed we have Σ={y∈Rⁿ₊:|y|1≤1}andφΣ(x) =|x|_∞, so thatH_P(z) =maxilog|z_i|andL_Σ(Cⁿ)reduces to the classical Lelong class.

We can introduce a Zaharjuta type extremal function for any suchPand any compact setK⊂Cⁿsetting

V_K,P(z):=sup{u(z):u∈L_P(Cⁿ),u|K≤0}, (17) V_K,P^∗ (z):=lim

ζ→zV_K,P(ζ), (18)

and a Siciak type extremal function by setting ΦK,P(z):=lim

k sup{|p|^1/k:p∈Poly(kP),kpkK}, (19)

Figure 1:Plots ofV_T^∗((z1,z2,z^h₃(z1,z2)),T). Here(z1,z2)7→z₃^h(z1,z2),h=1, 2, 3, 4 are the four leaves ofT, that is the four local inverses of the coordinate projectionT3z7→(z₁,z₂). By symmetry we need to look only at two of these leaves (i.e.,h=1, 2), as the graph ofV_T^∗is the same on the other two. From left to right and from above to below, we plotR²3(z₁,z₂)7→V_T^∗((z₁,z₂,z₃¹(z₁,z₂)),T),R²3(z₁,z₂)7→V_T^∗((z₁,z₂,z₃²(z₁,z₂)),T), R²3(ℜz₁,ℑz₁)7→V_T^∗((z₁, 0,z₃¹(z₁, 0)),T),R²3(ℜz₁,ℑz₁)7→V_T^∗((z₁, 0,z₃²(z₁, 0)),T).

where the existence of the limit is part of the statement (see[4]) and if the limit is continuous the convergence improves from point-wise to locally uniform.

For any non pluripolar setKwe have

V_K,P(z) =log⁺ΦK,P(z). (20)

Many results of Pluripotential Theory have been extended to the P-pluripotential setting. Among others, we recall for future use this version of the Global Domination Principle.

Proposition 1.4(P-Global Domination Principle[25]). Let u∈L_P(Cⁿ)and v∈L⁺_P(Cⁿ),i.e. v∈L_P(Cⁿ)and|v−H_P|(z)is bounded for|z| → ∞.Assume that u≤v(dd^cv)ⁿ-a.e., then

u(z)≤v(z),∀z∈Cⁿ.

2 Proof of Theorem 1.1

We divide the proof in some steps since they might have their own interest. LetΨ:C³→C³be defined by Ψ(z):=





z₁²+z²₂+z²₃+R²−r² 2R

z₁²+z₂² z₂²



=:

w1

w₂ w₃

! .

This can be used as a polynomial change of coordinates, indeed we haveΨ(T) =C, whereCis the parabolic cylinder C:={w∈C³:w²₁=w₂}.

Indeed, by elementary algebraic manipulation, we can derive by (1) the equivalent representation ofT as T =

¨ z∈C³:

z₁²+z₂²+z₃²+R²−r² 2R

2

=z₁²+z₂²

« .

Figure 2: The multifunctions R² 3 (ℜz1,ℑz1) 7→ {V_T^∗((z1, 0,z⁽₃¹⁾(z1, 0)),T),V_T^∗((z1, 0,z₃⁽²⁾(z1, 0)),T)} (above) andR² 3 (ℜz1,ℜz2) 7→

{V_T^∗((z1,z2,z₃⁽¹⁾(z1,z2)),T),V_T^∗((z1,z2,z₃⁽²⁾(z1,z2)),T)}(below) exhibit a nice smoothness away fromT.

Notice also thatΨhas 8 (possibly coinciding) inverses determined by

Ψ^←(w) =











α1

pw2−w3

α2pw₃ α3

p2Rw1−w2+r²−R²



:α∈ {−1, 1}³





 .

Lemma 2.1. In the above notation we have

V^∗

T(z,T) =1

2V_E^∗(Ψ(z),C), (21)

where

E:={w∈C∩R³: 0≤w₃≤w²₁,R−r≤w₁≤R+r}.

In particular, the function V^∗

T(·,T)is constant onΨ⁻¹(w)for any w∈C.

Proof. The mappingΨis a polynomial mapping having the same (total) degree in each component and whose homogeneous part

Ψ(ˆ z):=





z²₁+z²₂+z₃² 2R

z₁²+z²₂ z₂²





clearly satisfies theKlimek condition

Ψˆ^←(0) =0.

Moreover it is straightforward to check thatΨ(T) =EandΨ^←(E) =T.

Therefore we can apply[21, Th. 5.3.1]in its "equality case" to get (21).

Remark3. Note that[21, Th. 5.3.1]is not formulated in terms of varieties or complex manifolds, it holds for honest extremal functions in Euclidean space. Nevertheless we can apply it. Indeed the statement of the theorem does not require the compact set of which we are computing the extremal function to be non pluripolar and, due to the deep result of Sadullaev (see Prop.1.2and [29]), we can recover the extremal function of a compact subset of an algebraic variety by usingglobalpolynomials.

Proposition 2.2. Letπ:C→C²,π(w):= (w1,w3).Then we have

V_E^∗(w,C) =V_π(E),Σ2,1(π(w)), ∀w∈C, (22)

whereΣ2,1:={y∈R²₊:y₁/2+y₂≤1}.

Proof. By definition we haveV_E(w,C) =sup{u(w),u∈E}, where

E:={u∈L(C),u|E≤0}, whileV_π(E),Σ2,1(η) =sup{v(η),u∈F}, where

F:={v∈PSH(C²),v−max{2 log|w₁|, log|w₃|}bounded above as|w| → ∞,v|πE≤0}.

Here we used that the logarithmic support function (cf. (15)) forΣ2,1isH_Σ2,1(z):=max{2 log|η1|, log|η2|}.

We use the holomorphic coordinatesζ:= (w₂,w3)forC. These coordinates are a set of so called Rudin coordinates[28], i.e., C⊂ {w∈C³:|w1| ≤C(1+|ζ|)}for some constantC. This choice allows us to re-define the Lelong classL(C)as

L(C) ={u∈PSH(C):u−max{log|w2|, log|w3|}bounded above as|w| → ∞}

={u∈PSH(C):u−max{2 log|w₁|, log|w₃|}bounded above as|w| → ∞}.

Also note that the coordinate projectionπ:C→C²,π(w):= (w₁,w₃)is one to one. It follows that, ifu∈E, thenu◦π⁻¹∈F and, ifv∈F, thenv◦π∈E. Therefore the two upper envelopes coincide when composed with the coordinate projection map.

Proposition 2.3. Let

K:={x∈R²:−1≤x₁≤1,−(r x₁+R)≤x₂≤r x₁+R}, then we have

V_πE,Σ2,1(t) =2V_K(Φ(t)),∀t∈C², (23) where

ց t₁ t₂

‹

= _t

1−R

pr

t₂

and, by symmetry, we can chooseanybranch of the square root.

Figure 3:The setKin (25) is the intersection of a triangle and a strip.

Proof. Letv(t):=2V_K^∗(Φ(t)). This is a plurisubharmonic function onC²\ {t∈C²:t₂∈R,t₂<0}because it is the composition of a plurisubharmonic function with an holomorphic map. Note thatN:={t∈C²:t₂∈R,t₂≤0}is a pluripolar set inC².

Due to the symmetry ofV_K^∗(t1,t2) =V_K^∗(t1,−t2)(this is a byproduct of the proof of Theorem1.1below), the uppersemicontinu- ous extension toC²ofv|C²\Nis a continuous plurisubharmonic function, that indeed coincides withv. Hencevis plurisubharmonic onC².

It is a classical result (see for instance[21]) thatV_K^∗∈L⁺(C²). More precisely, there existC∈Rsuch that, for|w|large enough, we have

|V_K^∗(w)−max{log|w1|, log|w2|}|<C. (24) It follows that

|2VK(Φ(t))−2 max{log|Φ1(t)|, log|Φ2(t)|}| ∼ |2VK(Φ(t))−max{2 log|t1|, log|t2|}|<C, sincev∈PSH(C²)we getv∈L⁺_P(C²).

Now note thatv=0 onπE. This follows by the fact thatΦ(πE)⊆KandV_K^∗=0 onK. This last statement will clarified in the

"proof of Theorem1.1" below, where the functionV_K^∗is computed.

Now we notice that(dd^cv)²is zero on(C²\N)\πE. This follows by the fact thatΦis holomorphic and onC²\N and v=2V_K^∗◦Φ, whereV_K^∗is maximal onC²\K⊃C²\Φ(πE). SinceNis pluripolar andvis locally bounded we have(dd^cv)²=0 on C²\πE,[6]. Hence supp(dd^cv)²⊆πE.

We can apply Proposition1.4withu(t):=V_πE,^∗ _Σ

2,1(t)which is by definition a function inL_P(C²)q.e. vanishing onπE. Indeed, since the Monge Ampere of locally bounded PSH function does not charge pluripolar sets[6], we haveu≤valmost everywhere with respect to(dd^cv)². We conclude that

V_πE,Σ^∗

2,1(t)≤2V_K^∗(Φ(t)), ∀t∈C².

But, since the left end side is (q.e.) defined by an upper envelope containing the right hand side, equality must hold.

End of the proof of Theorem1.1. In order to conclude the proof, we are left to compute the extremal function of the trapezoidK.

Note that this is a convex real body but it is not centrally symmetric, hence the Baran formula (see eq. (8)) is not applicable for this case.

Instead we need to use another technique that has been suggested by Sione Ma‘u.

We want to show that

V_K^∗(ζ) =max{V_K^∗₁(ζ1),V_K^∗

2(ζ)} ∀ζ∈C², where (25)

K₁:={z∈R:|z| ≤1}, K₂:={ζ∈R²:−R/r≤z₁≤1,−rz₁−R≤z₂≤rz₁+R}. (26) Notice thatKis the intesection of the vertical stripK1×Rand the triangleK2, see Figure3.

Let us denote the function appearing in the right hand side of (25) byv. Notice thatvis a good candidate forV_K^∗, indeed it is fairly clear thatv∈PSH(C²)because it is the maximum of two plurisubharmonic functions onC², in the same wayv∈L⁺(C²). We prove that the functions indeed coincide using theextremal ellipses technique.

Let us pickz∈C²\K. SinceK is a real convex body, there exists at least one leafE_z3zof the Monge Ampere foliation relative toK. We recall[15,14,16]thatC²is foliated by a set of complex analytic curves{E_α}such thatV_K^∗|E_αis a subharmonic function that coincides withVE_α∩K(·,E_α). The curvesE_αareextremal ellipses, i.e., (possibly degenerate) complex ellipses whose area is maximal among all ellipses of given direction and eccentricity that are inscribed inK. We need the following lemma.

Lemma 2.4. IfEis an extremal ellipse for K, then at least one of the following holds true (i) Eis an extremal ellipse for K₂or

(ii) E3ζ7→V_K^∗

1(ζ1)is harmonic onE\C×K1and the orientation ofEis not in the directionζ2. We postpone the proof of this claim and we show first why this implies thatV_K^∗≡v.

Assume(i). Then we have

V_K^∗(ζ)|E=V_E^∗_∩K(ζ,E) =V_K^∗

2(ζ)|E ∀ζ∈E. (27)

Here the first equality is due to the fact thatEis extremal forKand the second to the fact that it is extremal forK₂. Assume now(ii)We define the functionu:E→[−∞,+∞[by settingu(ζ):=V_K^∗

1(ζ1). Notice thatu∈L⁺(E),u≤0 on E\(K1×C)and dd^cu=0 onE\(K1×C)(i.e., the support of dd^cuis inE∩(K1×C). Therefore, by the Global Domination Principle on the smooth algebraic varietyE, we haveu(ζ) =V_E^∗_∩(K

1×C)(·,E). Then it follows that V_K^∗(ζ)|E=V_E∩(K^∗

1×C)(ζ,E) =u(ζ) =V_K^∗

1(ζ1), ∀ζ∈E. (28)

For any (possibly degenerate) extremal ellipseEforKwe denote byEthe (possibly degenerate)filled-inreal ellipse relative toE.

Note thatV_K^∗(·)|E=V_E^∗|E=V_E∩K^∗ (·,E)sinceEis trivially an extremal ellipse forE.

If we assume(i), then the monotonicity of extremal functions with respect to the set inclusion implies V_E^∗(ζ)≥V_K^∗

1(ζ1), ∀ζ∈C², (29)

while if we assume(ii)the monotonicity property implies V_E^∗(ζ)≥V_K^∗

2(ζ), ∀ζ∈C². (30)

Finally, assumption(i)implies (27) and (29), thus we have V_K^∗(ζ) =V_K^∗

2(ζ)≥V_K1(ζ1), ∀ζ∈E, and assumption(ii)implies (28) and (30), thus we have

V_K^∗(ζ) =V_K^∗

1×C(ζ)≥V_K2, ∀ζ∈E.

Therefore, on any extremal ellipseV_K^∗coincides withv, but, since extremal ellipses forKare a foliation ofC², it follows that V_K^∗≡veverywhere.

Now we need to computeV_K^∗

2.

Let us mention a possible way to compute based on[2, Example 3.9]. Note thatV_K^∗

2(z) =V_K^∗

2+(R/r,0)(z+ (R/r, 0))and set K˜₂:=K₂+ (R/r, 0). Then we can easily see that

K˜₂:={ζ∈R²: 2ζ·y_k−1∈[−1, 1],∀k∈ {1, 2, 3}}, y1:= 1

2(R+r)

 r 1

‹

, y2:= 1 2(R+r)

 r

−1

‹

,y3:=y1+y2.

V_K˜^∗

2(ζ) =logh

–

1max≤k≤3

‚ ₂ X

l=1

A_k,l|yl·ζ|+|yk·ζ−1|

Ϊ

, whereA:=





1 0

0 1

1 1



. (31) Therefore we have

V_K^∗

2(ζ) =logh

–

1max≤k≤3

‚ ₂ X

l=1

A_k,l|yl·(ζ+ζ⁽⁰⁾)|+|yk·(ζ+ζ⁽⁰⁾)−1|

Ϊ

, whereζ⁽⁰⁾:= (R/r, 0). (32) On the other hand, it is probably more easy to consider the linear mapR²→R²represented by the invertible matrix

L:=

_r

2(R+r) 1 2(R+r) r

2(R+r) −1 2(R+r)

and to notice thatL(K₂+ζ⁽⁰⁾) =LK₂=Σ:={x∈R²:x_i≥0,x₁+x₂≤1}, the standard simplex. Hence we have V_K^∗

2(ζ) =V_Σ^∗(L(ζ+ζ⁽⁰⁾)) =logh(|s1|+|s2|+|s1+s₂−1|)

_{s=L(ζ+ζ(0))}. (33)

Here the last equality is a classical result, see for instance[16]. Note that in particular this shows thatV_K^∗is continuous and V_K^∗((t₁,t₂)) =V_K^∗((t₁,−t2)), as assumed in the proof of Proposition2.3.

By Proposition2.3, and usingV_K^∗

1(ζ1) =logh€_|ζ

1+1| 2 +^|ζ12^−1|

Š, we get

V_πE,Σ2,1(t) =2V_K(Φ(t)) =2 max

– logh

‚|^t1−_r^R+r|

2 +|^t1−_r^R−r| 2

Œ ,V_Σ^∗

t₁+p t₂ 2(R+r),t₁−p

t₂ 2(R+r)

‹‹™

,∀t∈C². By equation (22) we get

V_E^∗(w,C) =2 max

– logh

‚|^w1−R+r_r |

2 +|^w1−R−r_r | 2

Œ ,V_Σ^∗

w₁+p w₃ 2(R+r) ,w₁−p

w₃ 2(R+r)

‹‹™

,∀w∈C.

By equation (21) we obtain,∀z∈T, V^∗

T(z,T) =max

– logh

‚|^Ψ1(z)−R+r_r |

2 +|^{Ψ1(z)−R−r}_r | 2

Œ ,V_Σ^∗

Ψ1(z) +p Ψ3(z)

2(R+r) ,Ψ1(z)−p Ψ3(z) 2(R+r)

™

=logh

max |p

1−(z₃/r)²+1|+|p

1−(z₃/r)²−1|

2 ,

Æz₁²+z₂²+z₂ 2(r+R)

+

Æz₁²+z₂²−z₂ 2(r+R)

+

Æz₁²+z₂² (r+R) −1

. Here we used the monotonicity ofhon the positive real semi-axis, equation (33), the definition ofΨ, and the equation of the torus in the form (2).

Proof of Lemma2.4. Let us first consider non-degenerate ellipses. It is easy to see that, ifEis an extremal ellipse forK, then card(E∩∂K)≥2. Note that, ifEintersects∂Konly on the two diagonal edges or in the oblique edges and on the side lying on ζ1=−1 it can not be extremal forK. For, we can simply translate the ellipse along thez₁axis by a small but positive displacement (e.g.,E⁰:=E+εe₁) and then slightly dilate the ellipse to get a larger ellipseE⁰⁰:= (1+δ)E⁰that is still inK; this essentially follows by the strict convexity of the ellipse. It is even more clear that ifEintersects∂Konly on two adjacent edges can not have maximal area. As a consequence, for a non-degenerate maximal ellipse forKthere are only two possible configurations

1. card(E∩∂K) =4, one point on each edge ofK

2. card(E∩∂K) =3, two points on oblique edges and one onζ1=1

3. card(E∩∂K) =3, two points on vertical edges and one on an oblique edge 4. card(E∩∂K) =2, two points on vertical edges and major axis onζ2=0.

It is fairly clear that(3)and(4)implies (ii). Indeed, in each of this cases,ζ7→V_K^∗

1(ζ1)is pluriharmonic onC²\(K₁×C). On the other hand, in the cases(1)and(2)the ellipseEis tangent on at least three sides of∂K2, so it needs to be maximal for it.

Let us consider the case of degenerate maximal ellipses, that are line segments. IfE is a maximal line forK, thenE∂K can not be the set of one point on a oblique edge and one point on the vertical edgez₁=−1. For, consider any small enough ε >0 and letE⁰=E+εn, wherenis the unit normal toEpointing the half plane containing the barycenter ofK. Then we have l eng th(E∩K)<l eng th(E⁰∩K). Also, ifEis a vertical line , then it must be the side of∂Kwithz₁=1. Therefore, for a maximal line forKthere are only the following cases

1. E∩∂K={z∈K:z₁=1}

2. E∩∂K={z∈∂K,z₂=±rz₁±R}

3. card(E∩∂K) =2, two points on the oblique sides 4. card(E∩∂K) =2, two points on the vertical sides

In the cases (1) and (3)Eis evidently extremal forK₂, so (i) holds true. In the cases (2) and (4), since the functionζ7→V_K^∗

1(ζ)is pluriharmonic onC²\(K1×C)and(E∩K)⊂(K1×C), (ii) holds.

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