ANNALS OF
M ATHEMATICS
anmaah
SECOND SERIES, VOL. 172, NO. 2
September, 2010
Elliptic functions, Green functions and the mean field equations on tori
ByChang-Shou Lin and Chin-Lung Wang
Elliptic functions, Green functions and the mean field equations on tori
By CHANG-SHOULINand CHIN-LUNGWANG
Abstract
We show that the Green functions on flat tori can have either three or five critical points only. There does not seem to be any direct method to attack this problem.
Instead, we have to employ sophisticated nonlinear partial differential equations to study it. We also study the distribution of the number of critical points over the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori.
1. Introduction and statement of results
The study of geometric or analytic problems on two dimensional tori is the same as the study of problems onR2with doubly periodic data. Such situations occur naturally in sciences and mathematics since early days. The mathematical foundation of elliptic functions was subsequently developed in the 19th century. It turns out that these special functions are rather deep objects by themselves. Tori of different shapes may result in very different behavior of the elliptic functions and their associated objects. Arithmetic on elliptic curves is perhaps the eldest and the most vivid example.
In this paper, we show that this is also the case for certain nonlinear partial differential equations. Indeed, researches on doubly periodic problems in mathe- matical physics or differential equations often restrict the study to rectangular tori for simplicity. This leaves the impression that the theory for general tori may be much the same as for the rectangular case. However this turns out to be false.
We will show that the solvability of the mean field equation depends on the shape of the Green function, which in turn depends on the geometry of the tori in an essential way.
Recall that the Green function G.z; w/ on a flat torus T DC=Z!1CZ!2is the unique function on T T which satisfies
4zG.z; w/D ıw.z/ 1 jT j 911
andR
TG.z; w/ dAD 0, where ıw is the Dirac measure with singularity at zD w.
Because of the translation invariance of4z, we have G.z; w/D G.z w; 0/ and it is enough to consider the Green function G.z/WD G.z; 0/.
Not surprisingly, G can be explicitly solved in terms of elliptic functions. For example, using theta functions we have (cf. Lemmas 2.1 and 7.1)
G.z/D 1
2 logj#1.z/j C 1
2by2C C. /
where z D x C iy and WD !2=!1D a C ib. The structure of G, especially its critical points and critical values, will be the fundamental objects that interest us.
The critical point equationrG.z/ D 0 is given by
@G
@z 1 4
.log #1/zC 2iy b
D 0:
In terms of Weierstrass’ elliptic functions }.z/, .z/WD Rz
} and by the relation .log #1/z D .z/ 1z and with i D .z C !i/ .z/ the quasi-periods, the equation takes the simpler form: zD t!1C s!2is a critical point of G if and only if the following linear relation (Lemma 2.3) holds:
(1.1) .t !1C s!2/D t1C s2:
Since G is even, it is elementary to see that half-periods 12!1, 12!2 and
1
2!3D .!1C !2/=2 are the three obvious critical points and other critical points must appear in pairs. The question is: Are there other critical points? or How many critical points mightG have? It turns out that this is a delicate question and cannot be attacked easily from the simple looking equation (1.1). One of our chief purposes in this paper is to understand the geometry of the critical point set over the moduli space of flat toriM1DH=SL.2;Z/ and to study its interaction with the nonlinear mean field equation.
The mean field equation on a flat torus T takes the form (2RC)
(1.2) 4u C euD ı0:
This equation has its origin in the prescribed curvature problem in geometry like the Nirenberg problem, cone metrics etc. It also comes from statistical physics as the mean field limits of the Euler flow, hence the name. Recently it was shown to be related to the self dual condensation of the Chern-Simons-Higgs model. We refer to [3], [4], [2], [5], [6], [8], [9] and [10] for the recent development of this subject.
When ¤ 8m for any m 2Z, it has been recently proved in [4], [2], [5]
that the Leray-Schauder degree is nonzero; so the equation always has solutions, regardless of the actual shape of T .
The first interesting case remaining is when D 8 where the degree theory fails completely. Instead of the topological degree, precise knowledge on the Green function plays a fundamental role in the investigation of (1.2). The first main result
of this paper is the following existence criterion whose proof is given in Section 3 by a detailed manipulation on elliptic functions:
THEOREM1.1 (Existence). For D 8, the mean field equation on a flat torus has solutions if and only if the Green function has critical points other than the three half-period points. Moreover, each extra pair of critical points corresponds to a one-parameter scaling family of solutions.
It is known that for rectangular tori G.z/ has precisely the three obvious crit- ical points; hence for D 8 equation (1.2) has no solutions. However we will show in Section 2 that for the case !1D 1 and D !2D e i=3 there are at least five critical points and the solutions of (1.2) exist.
Our second main result is the uniqueness theorem.
THEOREM1.2 (Uniqueness). For D 8, the mean field equation on a flat torus has at most one solution up to scaling.
In view of the correspondence in Theorem 1.1, an equivalent statement of Theorem 1.2 is the following result:
THEOREM1.3. The Green function has at most five critical points.
Unfortunately we were unable to find a direct proof of Theorem 1.3 from the critical point equation (1.1). Instead, we will prove uniqueness theorem first, and then Theorem 1.3 is an immediate corollary. Our proof of Theorem 1.2 is based on the method of symmetrization applied to the linearized equation at a particularly chosen, even solution in the scaling family. In fact we study in Section 4 the one parameter family
4u C euD ı0; 2 Œ4; 8
on T within even solutions. This extra assumption allows us to construct a double cover T ! S2via the Weierstrass } function and to transform equation (1.2) into a similar one on S2but with three more delta singularities with negative coefficients.
The condition 4 is to guarantee that the original singularity at 0 still has a nonnegative coefficient of delta singularity.
The uniqueness is proved for this family via the method of continuity. For the starting point D 4, by a construction similar to the proof of Theorem 1.1 we sharpen the result on the nontrivial degree to the existence and uniqueness of solution (Theorem 3.2). For 2 Œ4; 8, the symmetrization reduces the problem on the nondegeneracy of the linearized equation to the isoperimetric inequality on domains inR2 with respect to certain singular measure:
THEOREM1.4 (Symmetrization Lemma). Let R2be a simply connected domain and letv be a solution of
4v C evDXN
j D12 ˛jıpj
in. Suppose that the first eigenvalue of4 C ev is zero on with ' the first eigenfunction. If the isoperimetric inequality with respect tods2D evjdxj2:
(1.3) 2`2.@!/ m.!/.4 m.!//
holds for all level domains!D f' > tg with t > 0, then Z
evdx 2:
Moreover, (1.3) holds if there is only one negative ˛j and˛j D 1.
The proof of the number of critical points appears to be one of the very few instances that one needs to study a simple analytic equation, here the critical point equation (1.1), by way of sophisticated nonlinear analysis.
To get a deeper understanding of the underlying structure of solutions, we first notice that for D 8, (1.2) is the Euler-Lagrange equation of the nonlinear functional
(1.4) J8.v/D1 2
Z
T jrvj2dA 8 log Z
T
ev 8G.z/dA on H1.T /\ fv jR
TvD 0g, the Sobolev space of functions with L2-integrable first derivative. From this viewpoint, the nonexistence of minimizers for rectangular tori was shown in [5]. Here we sharpen the result to the nonexistence of solutions.
Also for 2 .4; 8/ we sharpen the result on the nontrivial degree of equation (1.2) in [4] to the uniqueness of solutions within even functions. We expect the uniqueness holds true without the even assumption, but our method only achieves this at D 4. Obviously, uniqueness without the even assumption fails at D 8
due to the existence of scaling.
Naturally, the next question after Theorem 1.2 is to determine those tori whose Green functions have five critical points. It is the case if the three half-periods are all saddle points. A strong converse is proved in [7]:
THEOREMA. If the Green function has five critical points then the extra pair of critical points are minimum points and the three half-periods are all saddle points.
Together with Theorem 1.1, this implies that a minimizer of J8 exists if and only if the Green function has more than three critical points. In fact we show in [7]
that any solution of equation (1.2) must be a minimizer of the nonlinear functional J8. Thus we completely solve the existence problem on minimizers, a question raised by Nolasco and Tarantello in [9].
By Theorem A, we have reduced the question on detecting a given torus to have five critical points to the technically much simpler criterion on (non)-local minimality of the three half-period points. In this paper, however, no reference to Theorem A is needed. Instead, it motivates the following comparison result, which also simplifies the criterion further:
M1
0 12 1
i
1 2(1 + i)
1 2 + b1i
Figure 1. 5contains a neighborhood of e i=3. THEOREM1.5. Let z0andz1be two half-period points. Then
G.z0/ G.z1/ if and only if j}.z0/j j}.z1/j:
For general flat tori, a computer simulation suggests the following picture:
Let 3 (resp. 5) be the subset of the moduli space M1[ f1g Š S2 which corresponds to tori with three (resp. five) critical points. Then 3[ f1g is a closed subset containing i , 5 is an open subset containing e i=3, both of them are simply connected and their common boundary C WD @3D @5is a curve homeomorphic to S1 containing1. Moreover, the extra critical points are split out from some half-period point when the tori move from 3to 5across C .
We propose to prove the experimental observation by the method of deforma- tions inM1. The degeneracy analysis of critical points, especially the half-period points, is a crucial step. In this direction we have the following partial result on tori corresponding to the line Re D 1=2. These are equivalent to the rhombus tori and D 12.1C i/ is equivalent to the square torus where there are only three critical points.
THEOREM1.6 (Moduli Dependence). Let !1D 1 and !2D D 12C ib with b > 0. Then
(1) There exists b0< 12< b1<p
3=2 such that 12!1is a degenerate critical point ofG.zI / if and only if b D b0orbD b1. Moreover, 12!1is a local minimum point ofG.zI / if b0< b < b1and it is a saddle point ifb < b0orb > b1. (2) Both 12!2and 12!3are nondegenerate saddle points ofG.
(3) G.zI / has two more critical points ˙z0. / when b < b0orb > b1. They are nondegenerate global minimum points ofG and in the former case,
Re z0. /D1
2I 0 < Im z0. / <b 2:
Part (1) gives a strong support of the conjectural shape of the decomposition M1D 3[5. Part (3) implies that minimizers of J8exist for tori with D12Cib where b < b0or b > b1.
0.2 0.4 0.6 0.8 1 1.2 1.
-20 -10 10
Figure 2. Graphs of 1(the left one) and e1in b where e1is increasing.
0.2 0.4 0.6 0.8 1 1.2 1.4
-40 -30 -20 -10 10
Figure 3. Graphs of e1C 1(the upper one) and 12e1 1. Both functions are increasing in b and 12e1 1% 0.
The proofs are given in Section 6, notably in Lemmas 6.1, 6.2, 6.6 and The- orem 6.7. They rely on two fundamental inequalities of special values of elliptic functions and we would like to single out the statements (recall that ei D }.12!i/ and iD 2.12!i/):
THEOREM1.7 (Fundamental Inequalities). Let !1D 1 and !2D D12C ib withb > 0. Then
(1) dbd .e1C 1/D 4dbd22 logj#2.0/j > 0.
(2) 12e1 1D 4dbd logj#3.0/j < 0 and dbd22logj#3.0/j > 0. The same holds for#4.0/D #3.0/. In particular, e1increases inb.
These modular functions come into play due to the explicit computation of Hessian at half-periods along Re D12 (cf. (6.3) and (6.21)):
42det D2G!1
2
D .e1C 1/ e1C 1
2
b
; 42det D2G!2
2
D
1 1
2e12
b C1
2e1 1
jIm e2j2:
Although ei’s and i’s are classical objects, we were unable to find an ap- propriate reference where these inequalities were studied. Part of (2), namely
1
2e1 1< 0, can be proved within the Weierstrass theory (cf. (6.22)). The whole theorem, however, requires theta functions in an essential way. Theta functions are recalled in Section 7 and the theorem is proved in Theorems 8.1 and 9.1. The proofs make use of the modularity of special values of theta functions (Jacobi’s imaginary transformation formula) as well as the Jacobi triple product formula. Notice that the geometric meaning of these two inequalities has not yet been fully explored.
For example, the variation on signs from #2 to #3 is still mysterious to us.
2. Green functions and periods integrals
We start with some basic properties of the Green functions that will be used in the proof of Theorem 1.1. Detailed behavior of the Green functions and their critical points will be studied in later sections.
Let T DC=Z!1CZ!2 be a flat torus. As usual we let !3D !1C !2. The Green function G.z; w/ is the unique function on T which satisfies
(2.1) 4zG.z; w/D ıw.z/ 1
jT j andR
T G.z; w/ dAD 0. It has the property that G.z; w/ D G.w; z/ and it is smooth in .z; w/ except along the diagonal zD w, where
(2.2) G.z; w/D 1
2 logjz wj C O.jz wj2/C C
for a constant C which is independent of z and w. Moreover, due to the translation invariance of T we have G.z; w/D G.z w; 0/. Hence it is also customary to call G.z/WD G.z; 0/ the Green function. It is an even function with the only singularity at 0.
There are explicit formulae for G.z; w/ in terms of elliptic functions, either in terms of the Weierstrass } function or the Jacobi-Riemann theta functions #j. Both are developed in this paper since they have different advantages. We adopt the first approach in this section.
LEMMA2.1. There exists a constant C. /, D !2=!1, such that
(2.3) 8G.z/D 2
jT j Z
T
logj}./ }.z/j dA C C. /:
It is straightforward to verify that the function of z, defined in the right-hand side of (2.3), satisfies the equation for the Green function. By comparing with the behavior near 0 we obtain Lemma 2.1. Since the proof is elementary, also an equivalent form in theta functions will be proved in Lemma 7.1, we skip the details here.
In view of Lemma 2.1, in order to analyze critical points of G.z/, it is natural to consider the following periods integral
(2.4) F .z/D
Z
L
}0.z/
}./ }.z/d ;
where L is a line segment in T which is parallel to the !1-axis.
Fix a fundamental domain T0D f s!1Ct!2j 12 s; t 12g and set LD L.
Then F .z/ is an analytic function, except at 0, in each region of T0 divided by L[ L. Clearly, }0.z/=.}./ }.z// has residue˙1 at D ˙z. Thus for any fixed z, F .z/ may change its value by˙2i if the integration lines cross z. Let T0n.L [ L/D T1[ T2[ T3, where T1 is the region above L[ L, T2 is the region bounded by L and L and T3 is the region below L[ L. Recall that
0.z/D }.z/ and i D .z C !i/ .z/ for i2 f1; 2; 3g. Then we have LEMMA2.2. Let C1D 2i, C2D 0 and C3D 2i. Then for z 2 Tk, (2.5) F .z/D 2!1.z/ 21zC Ck:
Proof.For z2 T1[ T2[ T3, we have F0.z/D
Z
L
d dz
}0.z/
}./ }.z/
d :
Clearly, z and z are the only (double) poles of dzd }0.z/
}./ }.z/
as a meromorphic function of and dzd
}0.z/
}./ }.z/
has zero residues at D z and z. Thus the value of F0.z/ is independent of L and it is easy to see F0.z/ is a meromorphic function with the only singularity at 0.
By fixing L such that 062 L [ L, a straightforward computation shows that F .z/D2!1
z 21zC O.z2/ in a neighborhood of 0. Therefore
F0.z/D 2!1}.z/ 21: By integrating F0, we get
F .z/D 2!1.z/ 21zC Ck:
Since F .!2=2/D 0; F .!1=2/D 0 and F . !2=2/D 0, Ckis as claimed. Here we have used the Legendre relation 1!2 2!1D 2i.
From (2.3), we have
(2.6) 8GzD 1
jT j Z
T
}0.z/
}./ }.z/dA:
LEMMA2.3. Let G be the Green function. Then for zD t!1C s!2,
(2.7) GzD 1
4..z/ 1t 2s/:
In particular, z is a critical point of G if and only if (2.8) .t !1C s!2/D t1C s2:
Proof. We shall prove (2.7) by applying Lemma 2.2. Since critical points appear in pair, without loss of generality we may assume that zD t!1C s!2with s 0. We first integrate (2.6) along the !1direction and obtain
f .s0/W D Z
L1.s0/
}0.z/
}./ }.z/d D
8
<
:
2!1.z/C 21z if s0> s;
2!1.z/C 21z 2 i if s < s0< s;
2!1.z/C 21z if s0< s;
where L1.s0/D f t0!1C s0!2j jt0j 12g. Thus, 8GzD
Z 12
1 2
Z
L1.s0/
}0.z/
}./ }.z/dt0ds0D !11
Z 12
1 2
f .s0/ ds0
D !11.. 2!1.z/C 21z/.1 2s/C . 2!1.z/C 21z 2 i /2s/
D !11. 2!1.z/C 21z 4si /
D !11. 2!1.z/C 21t !1C 2s.1!2 2 i //
D 2.z/ C 21tC 2s2;
where the Legendre relation is used again.
COROLLARY2.4. Let G.z/ be the Green function. Then 12!k, k2 f1; 2; 3g are critical points of G.z/. Furthermore, if z is a critical point of G then both periods integrals
F1.z/W D 2.!1.z/ 1z/ and (2.9)
F2.z/W D 2.!2.z/ 2z/
are purely imaginary numbers.
Proof. The half-periods 12!1, 21!2 and 12!3 are obvious solutions of (2.8).
Alternatively, the half-periods are critical points of any even functions. Indeed for G.z/D G. z/, we get rG.z/ D rG. z/. Let p D 12!i for some i2 f1; 2; 3g, then pD p in T and so rG.p/ D rG.p/ D 0.
If zD t!1C s!2is a critical point, then by Lemma 2.3,
!1.z/ 1zD !1.t 1C s2/ 1.t !1C s!2/D s.!12 !21/D 2si:
The proof for F2is similar.
Example2.5. When D !2=!12 iR, by symmetry considerations it is known (cf. [5, Lemma 2.1]) that the half-periods are all the critical points.
Example2.6. There are tori such that equation (2.8) has more than three so- lutions. One such example is the torus with !1D 1 and !2D12.1Cp
3i /. In this case, the multiplication map z7! !2z is simply the counterclockwise rotation by angle =3, which preserves the latticeZ!1CZ!2; hence } satisfies
(2.10) }.!2z/D }.z/=!22:
Similarly in }02D 4}3 g2} g3, g2D 60X0 1
!4 D 60X0 1
.!2!/4 D !22g2; which implies that g2D 0 and
(2.11) }00D 6}2:
Let z0 be a zero of }.z/. Then }00.z0/D 0 too. By (2.10), }.!2z0/D 0;
hence either !2z0D z0or !2z0D z0 on T since }.z/D 0 has zeros at z0 and z0 only. From here, it is easy to check that either z0is one of the half-periods or z0D ˙13!3. But z0cannot be a half-period because }00.z0/¤ 0 at any half-period.
Therefore, we conclude that z0D ˙13!3and }00.˙13!3/D }.˙13!3/D 0.
We claim that 13!3is a critical point. Indeed from the addition formula
(2.12) .2z/D 2.z/ C1
2 }00.z/
}0.z/
we have
(2.13) 2!3
3
D 2!3
3
: On the other hand,
2!3
3
D !3
3
C .1C 2/D !3
3
C 1C 2: Together with (2.13) we get
!3
3
D1
3.1C 2/:
That is, 13!3satisfies the critical point equation.
Thus G.z/ has at least five critical points at 12!k, kD 1; 2; 3 and ˙13!3when
D !2=!1D 12.1Cp 3i /.
By way of Theorem 1.3, these are precisely the five critical points, though we do not know how to prove this directly.
To conclude this section, let u be a solution of (1.2) with D 8 and set
(2.14) v.z/D u.z/ C 8G.z/:
Then v.z/ satisfies
(2.15) 4v.z/ C 8
e 8G.z/ev.z/ 1 jT j
D 0
in T . By (2.2), it is obvious that v.z/ is a smooth solution of (2.15). An important fact which we need is the following: Assume that there is a blow-up sequence of solutions vj.z/ of (2.15). That is,
vj.pj/D max
T vj ! C1 as j ! 1:
Then the limit pD limj !1pj is the only blow-up point offvjg and p is in fact a critical point of G.z/:
(2.16) rG.p/ D 0:
We refer the reader to [3, p. 739, Estimate B] for a proof of (2.16).
3. The criterion for existence via monodromies Consider the mean field equation
(3.1) 4u C euD ı0; 2RC
in a flat torus T , where ı0is the Dirac measure with singularity at 0 and the volume of T is normalized to be 1. A well known theorem due to Liouville says that any so- lution u of4uCeuD 0 in a simply connected domain Cmust be of the form
(3.2) uD c1C log jf0j2
.1C jf j2/2;
where f is holomorphic in . Conventionally f is called a developing map of u. Given a torus T DC=Z!1CZ!2, by gluing the f ’s among simply connected domains it was shown in [5] that for D 4l, l 2N, (3.2) holds on the wholeC with f a meromorphic function. (The statement there is for rectangular tori with lD 2, but the proof works for the general case.)
It is straightforward to show that u and f satisfy
(3.3) uzz
1
2u2zDf000 f0
3 2
f00 f0
2
:
The right-hand side of (3.3) is the Schwartz derivative of f . Thus for any two developing maps f and Qf of u, there exists S D pq pNNq 2 PSU.1/ (i.e. p, q 2C andjpj2C jqj2D 1) such that
(3.4) fQD Sf WD pf Nq
qf C Np:
Now we look for the constraints. The first type of constraint is imposed by the double periodicity of the equation. By applying (3.4) to f .zC !1/ and f .zC !2/, we find S1and S2in PSU.1/ with
f .zC !1/D S1f;
(3.5)
f .zC !2/D S2f:
These relations also force that S1S2D S2S1(up to a sign, as A A in PSU.1/).
The second type of constraint is imposed by the Dirac singularity of (3.1) at 0.
A straightforward local computation with (3.2) shows that
LEMMA3.1. (1) If f .z/ has a pole at z0 0 .mod !1; !2/, then the order kD l C 1.
(2) If f .z/D a0C akzkC is regular at z0 0 .mod !1; !2/ with ak ¤ 0 thenkD l C 1.
(3) If f .z/ has a pole at z06 0 .mod !1; !2/, then the order is 1.
(4) If f .z/ D a0C ak.z z0/kC is regular at z0 6 0 .mod !1; !2/ with ak¤ 0 then k D 1.
Now we are in a position to prove Theorem 1.1, namely the case lD 2.
Proof. We first prove the “only if” part. Let u be a solution and f be a developing map of u. By the above discussion, we may assume, after conjugating a matrix in PSU.1/, that S1D e0 ei 0i for some 2R. Let S2D pq pNNq and then (3.5) becomes
f .zC !1/D e2if .z/;
(3.6)
f .zC !2/D S2f .z/:
Since S1S2D S2S1, a direct computation shows that there are three possibil- ities:
(1) pD 0 and ei D ˙i;
(2) qD 0;
(3) ei D ˙1.
Case1. By assumption we have
(3.7) f .zC !1/D f .z/; f .zC !2/D .Nq/2 f .z/: For any l2N, the logarithmic derivative
gD .log f /0Df0 f
is a nonconstant elliptic function on zT DC=Z!1CZ2!2which has a simple pole at each zero or pole of f . By Lemma 3.1, if f or 1=f is singular at zD 0 then g
has no zero, which is not possible. So f must be regular at zD 0 with f .0/ ¤ 0.
Moreover g has two zeros of order l at 0 and !2. Let }.z/, .z/ and .z/D expRz
.w/ dw D z C be the Weierstrass elliptic functions on zT . Recall that is odd with a simple zero at each lattice point.
Moreover, for!z1D !1,!z2D 2!2and!z3D !1C 2!2, (3.8) .z˙ z!i/D ei.z˙12!zi/ .z/:
Now lD 2. By the standard representation of elliptic functions, (3.9) g.z/D A 2.z/2.z !2/
.z a/ .z b/ .z c/ .z d /
with four distinct simple poles such that aC b C c C d D 2!2. We will show that such a function g.z/ does not exist.
By (3.7), we have g.zC!2/D g.z/. Hence we may assume that c a C!2 and d b C !2 modulo !1; 2!2. Thus aC b 0 modulo 12!1; !2 and we arrive at two inequivalent cases. (i) .a; b; c; d /D .a; a; a C !2; aC !2/. (ii) .a; b; c; d /D .a; a C12!1; aC !2; a 12!1C !2/. Using (3.8), it is easily checked that (i) leads to g.zC !2/D g.z/ and (ii) leads to g.z C !2/D g.z/.
Hence we are left with (ii).
The residues of g at a; b; c and d are equal to Ar, Ar0, Ar and Ar0re- spectively, where
rD 2.aC !2/2.a/
.!2/ .2a 12!1C !2/ .2aC12!1/ and
r0D 2.a 12!1/2.a 12!1C !2/
.2a 12!1/ .2a 12!1C !2/ .!1 !2/: We claim that ArD ˙1 and Ar0D ˙1: Since
f .z/D exp Z z
g.w/ dw
is well-defined, by the residue theorem, we must have ArD m for some m 2Z. Moreover one of a, b is a pole of order jmj of f and then by Lemma 3.1 we conclude that mD ˙1. Similarly Ar0D ˙1.
In particular we must have r=r0D ˙1. Using (3.8), this is equivalent to (3.10) 2.aC !2/2.a/
2.aC12!1/2.a 12!1C !2/ D e1. 12!1C!2/: To solve a from this equation, we first recall that
}.z/ }.y/D .zC y/.z y/
2.z/2.y/ :
By substituting yD12!zi into it and using (3.8), we get (3.11) }.z/ ei D 2.zC12!zi/
2.z/2.12!zi/e iz: With (3.11), the “C” case in equation (3.10) simplifies to
} a !1
2 C !2
e1D }.a/ e1:
That is, 2a12!1 !2. But this implies that b c, a contradiction.
Similarly, the “ ” case in (3.10) simplifies to (using the Legendre relation) }
aC!1
2
e3D }.a/ e3:
That is, 2aC12!1 0. This leads to c d , which is again a contradiction.
Case2. In this case we have
f .zC !1/D e2i1f .z/;
(3.12)
f .zC !2/D e2i2f .z/:
The logarithmic derivative gD .log f /0D f0=f is now elliptic on T which has a simple pole at each zero or pole of f . As in Case 1, it suffices to investigate the situation when f is regular at 0 and f .0/¤ 0. Since g has 0 as its only zero (of order 2), we have
(3.13) g.z/D A 2.z/
.z z0/ .zC z0/
where .z/ is the Weierstrass sigma function on T . Without loss of generality we may assume that f has a zero at z0 and a pole at z0. In particular z0¤ z0in T and we conclude that z0¤ !k=2 for any k2 f1; 2; 3g.
Notice that if a meromorphic function f satisfies (3.12), then ef also satis- fies (3.12) for any 2R. Thus
(3.14) u.z/D c1C log e2jf0.z/j2 .1C e2jf .z/j2/2 is a scaling family of solutions of (3.1).
Clearly u.z/! 1 as ! C1 for any z such that f .z/ ¤ 0 and u.z0/! C1 as ! C1. Hence z0is the blow-up point and we have by (2.16) that
rG.z0/D 0:
Namely, it is a critical point other than the half-periods.
Case3. In this case we get that S1is the identity. So by another conjugation in PSU.1/ we may assume that S2is in diagonal form. But this case is then reduced to Case 2. The proof of the “only if” part of Theorem 1.1 is completed.
Now we prove the “if” part. Suppose that there is a critical point z0of G.z/
with z0¤12!k for any k2 f1; 2; 3g.
For any closed curve C such that z0and z062 C , the residue theorem implies that
(3.15)
Z
C
}0.z0/
}./ }.z0/d D 2 mi for some m2Z. Hence
(3.16) f .z/WD exp
Z z 0
}0.z0/ }./ }.z0/d
is well-defined as a meromorphic function. Notice that f is nonconstant since }0.z0/¤ 0.
Let L1 and L2 be lines in T which are parallel to the !1-axis and !2-axis respectively and with˙z062 L1; L2. Then for j D 1; 2,
(3.17) f .zC !j/D f .z/ exp Z
Lj
}0.z0/ }./ }.z0/d
! : By Lemma 2.2,
Z
Lj
}0.z0/
}./ }.z0/d D Fj.z0/C Ck
for some Ck2 f2i; 0; 2ig. Also, by Corollary 2.4, Fj.z0/D 2.!j.z0/ jz0/D 2ij 2 iR: Hence for j D 1; 2,
(3.18) f .zC !j/D f .z/e2ij holds. Set
u.x/D c1C log e2jf0.z/j2 .1C e2jf .z/j2/2:
Then u.x/ satisfies (3.1) for any 2Rand uis doubly periodic by (3.18). There- fore, solutions have been constructed and the proof of Theorem 1.1 is completed.
A similar argument leads to
THEOREM3.2. For D 4, there exists a unique solution of (3.1).
Proof.By the same procedure of the previous proof, there are three cases to be discussed. For Case 2, the subcases that f or 1=f is singular at zD 0 leads to contradiction as before. For the subcase that f is regular at zD 0 and f .0/ ¤ 0 we see that f .z/=f0.z/ is an elliptic function with 0 as its only simple pole (since now k 1D l D 1). Hence Case 2 does not occur. Similarly Case 3 is not possible.
Now we consider Case 1. By (3.7), the function gD .log f /0D f0=f is elliptic on zT DC=Z!1CZ2!2and g has a simple pole at each zero or pole of f .
By Lemma 3.1, if f or 1=f is singular at zD 0 then g has no zero and we get a contradiction. So f is regular at zD 0, f .0/ ¤ 0 and g has two simple zeros at 0 and !2.
Let be the Weierstrass sigma function on zT . Then
(3.19) g.z/D A .z/ .z !2/
.z a/ .z b/
for some a, b with aC b D !2.
From (3.7), we have g.zC !2/D g.z/. So a C !2D b .mod !1; 2!2/.
Since the representation of g in terms of sigma functions is unique up to the lattice f!1; 2!2g, there is a unique solution of .a; b/:
(3.20) aD !1
2 ; bD!1
2 C !2:
Notice that the residues of g at a and b are equal to Ar and Ar respectively, where
rD .12!1/ .12!1C !2/
.!1C !2/ : We claim that ArD ˙1. Since
f .z/D f .0/ exp Z z
0
g.w/ dw
is well-defined, by the residue theorem, we must have ArD m for some m 2Z. Moreover one of a, b is a pole of orderjmj and then by Lemma 3.1 we conclude that mD ˙1.
Conversely, by picking up a, b and AD 1=r as above, f .z/ is a uniquely defined meromorphic function up to a factor f .0/. There is an unique choice of f .0/ up to a norm one factor such that c WD f .!2/f .0/ has jcj D 1. Thus by integrating g.zC !2/D g.z/ we get f .z C !2/D c=f .z/.
By integrating g.zC !1/D g.z/ we get f .z C !1/D c0f .z/ where c0D f .!1/
f .0/ D exp Z !1
0
g.z/ dz:
To evaluate the period integral, notice that g.12!1C u/ D g.12!1 u/. By using the Cauchy principal value integral and the fact that the residue of g at 12! is˙1, we get
(3.21)
Z !1
0
g.z/ dzD ˙1
2 2i D ˙i and so c0D 1.
Thus f gives rise to a solution of (3.1) for D 4. The developing map for the other choice AD 1=r is 1=f .z/ which leads to the same solution. The proof
is completed.
Since (3.1) is invariant under z7! z, the unique solution is necessarily even.
4. A uniqueness theorem for 2 Œ4; 8 via symmetrizations
From the previous section, for D 8, solutions to the mean field equation exist in a one-parameter scaling family in with developing map f and centered at a critical point other than the half-periods. By choosing D logjf .0/j we may assume that f .0/D 1. Since g D .log f /0is even by (3.13), we have log f . z/D
log f .z/ and then
f . z/D 1 f .z/: Consider the particular solution
u.z/D c1C log jf0.z/j2 .1C jf .z/j2/2:
It is easy to verify that u. z/D u.z/ and u is the unique even function in this family of solutions. In order to prove uniqueness up to scaling, it is equivalent to prove uniqueness within the class of even functions.
The idea is to consider the following equations
(4.1) 4u C euD ı0 and
u. z/D u.z/ on T
where 2 Œ4; 8. We will use the method of symmetrization to prove
THEOREM4.1. For 2 Œ4; 8, the linearized equation of (4.1) is nondegen- erate. That is, the linearized equation has only trivial solutions.
Together with the uniqueness of solution in the case D 4 (Theorem 3.2), we conclude the proof of Theorem 1.2 by the inverse function theorem.
We first prove Theorem 1.4, the Symmetrization Lemma. The proof will con- sist of several lemmas. The first step is an extension of the classical isoperimetric inequality of Bol for domains inR2 with metric ewjdxj2 to the case when the metric becomes singular.
Let R2be a domain and w2 C2./ satisfy
(4.2)
8
<
:
4w C ew 0 in and Z
ewdx 8:
This is equivalent to saying that the Gaussian curvature of ewjdxj2is
K 1
2e w4w 1 2: For any domain ! b , we set
(4.3) m.!/D
Z
!
ewdx and `.@!/D Z
@!
ew=2ds:
Bol’s isoperimetric inequality says that if is simply connected then
(4.4) 2`2.@!/ m.!/.8 m.!//:
We first extend it to the case when w acquires singularities:
8
<
:
4w C ewDPN
j D12 ˛jıpj in and Z
ewdx 8;
(4.5)
with ˛j > 0, j D 1; 2; : : : ; N .
LEMMA 4.2. Let be a simply connected domain and w be a solution of (4.5), or more generally a sub-solution with prescribed singularities:
w.x/ X
j
˛jlogjx pjj 2 C2./:
Then for any domain! b ,
(4.6) 2`2.@!/ m.!/.8 m.!//:
Proof.Define v and w"by w.x/DX
j
˛jlogjx pjj C v.x/;
w".x/DX
j
˛j
2 log.jx pjj2C "2/C v.x/:
By straightforward computations, we have 4w".x/C ew".x/
DX
j
2˛j"2
.jx pjj2C "2/2C ev
Y
j
.jx pjj2C "2/˛j=2 Y
j
jx pjj˛j
0:
Let `"and m"be defined as in (4.3) with respect to the metric ew".x/jdxj2. Then we have
2`2".@!/ m".!/.8 m".!//:
By letting "! 0 we obtain (4.6).
Next we consider the case that some of the ˛j’s are negative. For our purpose, it suffices to consider the case with only one singularity p1 with negative ˛1(and we only need the case that ˛1D 1). In view of (the proof of) Lemma 4.2, the problem is reduced to the case with only one singularity p1. In other words, let w satisfy
(4.7) 4w C ewD 2ıp1 in :
LEMMA4.3. Let w satisfy (4.7) with simply connected. Suppose that (4.8)
Z
ewdx 4I then
(4.9) 2`2.@!/ m.!/.4 m.!//:
Proof.We may assume that p1D 0. If 0 62 ! then
2`2.@!/ m.!/.8 m.!// > m.!/.4 m.!//
by Bol’s inequality, trivially. If 0 2 !, we consider the double cover z of branched at 0. Namely we set zD f 1./ where
xD f .z/ D z2 for z2C: The induced metric evjdzj2on z satisfies
ev.z/jdzj2D ew.x/jdxj2D ew.z2/4jzj2jdzj2: That is, the metric potential v is the regular part
v.z/WD w.x/ C log jxj C log 4 D w.z2/C 2 log jzj C log 4:
By construction, v satisfies
4v C evD 0 in znf0g:
Since v is bounded in a neighborhood of 0, by the regularity of elliptic equations, v.z/ is smooth at 0. Hence v satisfies
4v C evD 0 in z:
Let!zD f 1.!/. Clearly @!z f 1.@!/. Also
l.@!/z 2l.@!/ and m. Qw/D 2m.!/;
where
l.@!/z D Z
@ z!
ev=2ds and m.!/z D Z
z
!
evdx:
By Bol’s inequality, we have 4`2.@!/ `2.@!/z 1
2m.!/.8z m.!//z D m.!/.8 2m.!//:
Thus 2`2.@!/ m.!/.4 m.!//.
LEMMA4.4 (Symmetrization I). Let R2be a simply connected domain with02 and let v be a solution of
4v C evD 2ı0
in. If the first eigenvalue of4 C ev is zero on then (4.10)
Z
evdx 2:
Proof.Let be the first eigenfunction of4 C ev:
(4.11) 4 C ev D 0 in and
D 0 on @:
InR2, let U and ' be the radially symmetric functions
(4.12)
8 ˆˆ
<
ˆˆ :
U.x/D log 2
.1C jxj/2jxj and '.x/D 1 jxj
1C jxj: From4 D@r@22 C1r@r@ Cr12
@2
@2, it is easy to verify that U and ' satisfy
(4.13) 4U C eU D 2ı0 and
4' C eU'D 0 inR2:
For any t > 0, set t D fx 2 j .x/ > tg and r.t/ > 0 such that (4.14)
Z
Br.t /
eU.x/dxD Z
f >tg
ev.x/dx;
where Br.t /is the ball with center 0 and radius r .t /. Clearly r .t / is strictly decreas- ing in t for t 2 .0; max /. In fact, r.t/ is Lipschitz in t. Denote by .r/ the symmetrization of with respect to the measure eU.x/dx and ev.x/dx. That is,
.r/D supft j r < r.t/g:
Obviously .r/ is decreasing in r and for t2 .0; max /, .r/D t if and only if r .t /D r. Thus by (4.14), we have a decreasing function
(4.15) f .t /WD Z
f >t g
eU.x/dxD Z
f >tg
ev.x/dx:
By Lemma 4.3, for any t > 0,
(4.16) 2`2.f D tg/ f .t/.4 f .t //:
We will use inequality (4.16) in the following computation: For any t > 0, by the Co-Area formula,
d dt
Z
t
jr j2dxD Z
@t
jr j ds and f0.t /D d
dt Z
t
evdxD Z
@t
ev jr jds
hold almost everywhere in t . Thus d
dt Z
tjr j2dxD Z
f Dtgjr j ds (4.17)
Z
f Dtg
ev=2ds
2Z
f Dtg
ev jr jds
1
D `2.f D tg/f0.t / 1
1
2f .t /.4 f .t //f0.t / 1:
It is known that 2 H01.Br.0// and the same procedure for leads to (4.18) d
dt Z
f >t gjr j2dxD Z
f Dtgjr j ds D
Z
f Dtg
eU=2ds
2Z
f Dtg
eU jr jds
1
D `2.f D tg/f0.t / 1
D 1
2f .t /.4 f .t //f0.t / 1; with all inequalities being equalities. Hence
d dt
Z
f >tgjr j2dx d dt
Z
f >t gjr j2ds holds almost everywhere in t . By integrating along t , we get
Z
jr j2dx Z
Br.0/jr j2dx:
Since and have the same distribution (or by looking at R f0.t /t2dt di- rectly), we have
Z
Br.0/
eU.x/ 2dxD Z
ev.x/ 2dx:
Therefore 0D
Z
jr j2dx Z
ev 2dx Z
Br.0/
jr j2dx Z
Br.0/
eU 2dx:
This implies that the linearized equation4 C eU.x/has nonpositive first eigenvalue.
By (4.12), this happens if and only if r .0/ 1. Thus Z
ev.x/dxD Z
Br.0/
eU.x/dx 2:
Remark4.5 (see [1]). By applying symmetrization to4v C evD 0 in with
1.4 C ev/D 0, the corresponding radially symmetric functions are U.x/D 2 log
1C1 8jxj2
and '.x/D 8 jxj2 8C jxj2: The same computations leads toR
evdx 4.
A closer look at the proof of Lemma 4.4 shows that it works for more general situations as long as the isoperimetric inequality holds:
LEMMA4.6 (Symmetrization II). Let R2be a simply connected domain and letv be a solution of
4v C evDXN
j D12 ˛jıpj
in. Suppose that the first eigenvalue of4 C ev is zero on with ' the first eigenfunction. If the isoperimetric inequality with respect tods2D evjdxj2,
2`2.@!/ m.!/.4 m.!//
holds for all level domains!D f' tg with t 0, then Z
evdx 2:
Notice that we do not need any further constraint on the sign of ˛j.
For the last statement of Theorem 1.4, the limiting procedure of Lemma 4.2 implies that the isoperimetric inequality (Lemma 4.3) and symmetrization I (Lemma 4.4) both hold regardless on the presence of singularities with nonnega- tive ˛.
Indeed the proof of Lemma 4.3 can be adapted to the case 4w C ewD 2ıp1CXN
j D22 ˛jıpj
with ˛j > 0 for j D 2; : : : ; N . On the double cover N! branched over p1D 0, the metric potential v.z/ again extends smoothly over zD 0 and satisfies
4v C evDXN
j D22 ˛j.ıqjC ıqj0/;
where qj; qj02 N are points lying over pj. The remaining argument works by using Lemma 4.2 and we still conclude 2`2.@!/ m.!/.4 m.!//.
Thus the proof of Theorem 1.4 is complete.
Proof of Theorem4.1. Let u be a solution of equation (4.1). It is clear that we must haveR
TeuD 1. Suppose that '.x/ is a nontrivial solution of the linearized equation at u:
(4.19) 4' C eu'D 0 and
'.z/D '. z/ in T : We will derive from this a contradiction.
Since both u and ' are even functions, by using x D }.z/ as a two-fold covering map of T onto S2DC[ f1g, we may require that since } is an isometry:
eu.z/jdzj2D ev.x/jdxj2D ev.x/j}0.z/j2jdzj2: Namely, we set
(4.20) v.x/WD u.z/ log j}0.z/j2 and .x/WD '.z/:
There are four branch points onC[ f1g, namely p0D }.0/ D 1 and pj D ej WD }.!j=2/ for j D 1; 2; 3. Since }0.z/2 D 4Q3
j D1.x ej/, by construction v.x/
and .x/ then satisfy (4.21)
(
4v C evDX3
j D1. 2/ıpj and 4 C ev D 0 inR2:
To take care of the point at infinity, we use coordinate yD 1=x or equivalently we consider T ! S2via yD 1=}.z/ z2. The isometry condition reads as
eu.z/jdzj2D ew.y/jdyj2D ew.y/j}0.z/j2 j}.z/j4jdzj2: Near yD 0 we get
w.y/D u.z/ logj}0.z/j2 j}.z/j4
4 1 logjyj:
Thus 4 implies that p0 is a singularity with nonnegative ˛0: 4w C ewD ˛0ı0CX3
j D1. 2/ı1=pj:
In dealing with equation (4.1) and the above resulting equations, by replacing u by uC log etc., we may (and will) replace the on the left-hand side by 1 for simplicity. The total measures on T andR2are then given by
Z
T
eudzD 8 and Z
R2
evdxD 2 4:
The nodal line of decomposes S2into at least two connected components and at least two of them are simply connected. If there is a simply connected com- ponent which contains no pj’s, then the symmetrization (Remark 4.5) leads to
Z
evdx 4;
which is a contradiction becauseR2n ¤ ∅. If every simply connected component
i, iD 1; : : : ; m, contains only one pj, then Lemma 4.4 implies that Z
i
evdx 2 for iD 1; : : : ; m:
The sum is at least 2m, which is again impossible unless mD 2 andR2D S1[S2. So without lost of generality we are left with one of the following two situations:
R2[ f1gnf D 0g D C[ where
C fx j .x/ > 0g and fx j .x/ < 0g:
Both Cand are simply connected.
(1) Either contains p1; p2and p32 C or (2) p12 , p32 Cand p22 C D @CD @ .
Assume that we are in case (1). By Lemma 4.3, we have on C (4.22) 2`2.f D tg/ m.f tg/.4 m.f tg//
for t 0.
We will show that the similar inequality
(4.23) 2`2.f D tg/ m.f tg/.4 m.f tg//
holds on for all t 0.
Let t 0 and ! be a component of f tg.
If ! contains at most one point of p1 and p2, then Lemma 4.3 implies that 2`2.@!/ .4 m.!//m.!/:
If ! contains both p1and p2, thenR2[ f1gn! is simply connected which contains p3only. Thus by Lemma 4.3
2`2.@!/ .4 m.R2n!//m.R2n!/
(4.24)
D .4 =2C m.!//.=2 m.!//
D .4 m.!//m.!/C .4 =2/.=2 2m.!//:
Since 8 andR
Cevdx 2, we get
2 D
Z
R2
ev Z
C
evC Z
!
ev 2 C m.!/ 2m.!/:
Then again
2`2.@!/ .4 m.!//m.!/
with equality hold only when D 8 and m.!/ D 2.
Now it is a simple observation that domains which satisfy the isoperimetric inequality (4.9) have the addition property. Indeed, if 2a2 .4 m/m and 2b2 .4 n/n, then
2.aC b/2D 2a2C 4ab C 2b2> .4 m/mC .4 n/n
D 4.m C n/ .m C n/2C 2mn > .4 .mC n//.m C n/:
Hence (4.23) holds for all t 0.
