published in Adv. Math. 354 (2019), Article ID 106747, 36 pp.

Minimal Lagrangian Connectionson Compact Surfaces

THOMAS METTLER

ABSTRACT. We introduce the notion of a minimal Lagrangian connection on thetangent bundle of a manifold and classify all such connections in the case wherethe manifold is a compact oriented surface of non-vanishing Euler characteristic.Combining our classification with results of Labourie and Loftin, we conclude thatevery properly convex projective surface arises from a unique minimal Lagrangianconnection.

1. Introduction

1.1. Background

A projective manifold is a pair .M; p/ consisting of a smooth manifold M anda projective structure p, that is, an equivalence class of torsion-free connectionson the tangent bundle TM , where two such connections are called projectivelyequivalent if they share the same geodesics up to parametrisation. A projectivemanifold .M; p/ is called properly convex if it arises as a quotient of a properlyconvex open set QM � RPn by a group � � PSL.nC 1;R/ of projective transform-ations which acts discretely and properly discontinuously. The geodesics of p arethe projections to M D �n QM of the projective line segments contained in QM . Inparticular, locally the geodesics of a properly convex projective structure p can bemapped diffeomorphically to segments of straight lines, that is, p is flat.

It follows from the work of Cheng–Yau [9, 10] that the universal cover QM of aproperly convex projective manifold .M; p/ determines a unique properly embeddedhyperbolic affine sphere f W QM ! RnC1, which is asymptotic to the cone overQM in RnC1. The Blaschke metric and Blaschke connection induced by f descend

to the quotient � n QM and equip M with a complete Riemannian metric g andprojectively flat connection r 2 p, see the work of Loftin [30]. The differencebetween r and the Levi-Civita connection of the Blaschke metric is encoded interms of a cubic form, the so-called Fubini–Pick form of f . For an introduction toaffine differential geometry the reader may consult [33] as well as [29] for a nicesurvey on affine spheres.

Properly convex projective surfaces are of particular interest, as they may be seen– through the work of Hitchin [25], Goldman [20] and Choi–Goldman [12] – as thenatural generalisation of the notion of a hyperbolic Riemann surface. In the case ofa properly convex oriented surface .†; p/, the Fubini–Pick form is the real part of acubic differential that is holomorphic with respect to the Riemann surface structureon† defined by the orientation and the conformal equivalence class of the Blaschke

Date: March 4, 2019.1

2 T. METTLER

metric. Conversely, Wang [36] observed that a holomorphic cubic differential Con a closed hyperbolic Riemann surface .†; Œg�/ determines a unique conformalRiemannian metric g whose Gauss curvature Kg satisfies

(1.1) Kg D �1C 2jC j2g ;

where jC jg denotes the point-wise tensor norm of C with respect to the Hermitianmetric induced by g on the third power of the canonical bundle of †. Furthermore,the pair .g;Re.C // can be realized as the Blaschke metric and Fubini–Pick form ofa complete hyperbolic affine sphere f W QM ! R3 defined on the universal coverQM of M . In particular, combining Wang’s work with the work of Loftin estab-

lishes – on a compact oriented surface of negative Euler characteristic – a bijectivecorrespondence between properly convex projective structures and pairs .Œg�; C /consisting of a conformal structure Œg� and a cubic holomorphic differential C ,see [30]. This correspondence was also discovered independently by Labourie [26].Since then, Benoist–Hulin [1] have extended the correspondence to noncompactprojective surfaces with finite Finsler volume and Dumas–Wolf [14] study the caseof polynomial cubic differentials on the complex plane.

In [28], Libermann constructs a para-Kähler structure .h0; �0/ on the opensubmanifold A0 � RPn � RPn� consisting of non-incident point-line pairs. Apara-Kähler structure may be thought of as a split-complex analogue of the notionof a Kähler structure. In particular, h0 is a pseudo-Riemannian metric of split-signature .n; n/ and �0 a symplectic form, so that there exists an endomorphismof the tangent bundle relating h0 and �0 which squares to become the identitymap. In [22, 23], Hildebrand – see also the related work [17, 19, 35] – observedthat proper affine spheres f W M ! RnC1 correspond to minimal Lagrangianimmersions Of W M ! A0. Thus, the result of Hildebrand, combined with thework of Cheng–Yau, associates a minimal Lagrangian immersion to every properlyconvex projective manifold.

1.2. Minimal Lagrangian connections

Here we propose a generalization of the notion of a properly convex projectivesurface which arises naturally from the concept of a minimal Lagrangian connection.In joint work with Dunajski the author has shown that the construction of Libermannis a special case of a more general result: In [15], it is shown that a projectivestructure p on an n-manifoldM canonically defines an almost para–Kähler structure.hp; �p/ on the total space of a certain affine bundle A ! M , whose underlyingvector bundle is the cotangent bundle of M . The bundle A! M has the crucialproperty that its sections are in one-to-one correspondence with the representativeconnections of p. Therefore, fixing a representative connection r 2 p gives asection sr W M ! A and hence an isomorphism r W T

�M ! A, by declaringthe origin of the affine fibre Ap to be sr.p/ for all p 2 M . Correspondingly, weobtain a pair .hr ; �r/ D �

r.hp; �p/ on the total space of the cotangent bundle.

Besides being a geometric structure of interest in itself (see [15] for details), thepair .hr ; �r/ has the natural property

o�hr D .sr/�hp D �

�1

n � 1

�RicC.r/

MINIMAL LAGRANGIAN CONNECTIONS 3

and

o��r D .sr/��p D

�1

nC 1

�Ric�.r/;

where o W M ! T �M denotes the zero-section and Ric˙.r/ the symmetric(respectively, the anti-symmetric) part of the Ricci curvature Ric.r/ of r. Con-sequently, we call r Lagrangian if the Ricci tensor of r is symmetric, or equival-ently, if the zero-section o is a Lagrangian submanifold of .T �M;�r). Likewise,we call r timelike/spacelike if˙RicC.r/ is positive definite, or equivalently, if thezero-section o is a timelike/spacelike submanifold of .T �M;hr/. The upper signcorresponds to the timelike case and lower sign to the spacelike case. Moreover, wecall r minimal if the zero-section is a minimal submanifold of .T �M;hr/.

We henceforth restrict our considerations to the case of oriented surfaces. Weshow (see Theorem 4.4 below) that a timelike/spacelike Lagrangian connection ris minimal if and only if

Rij�2riRjk � rkRij

�D 0;

where Rij denotes the Ricci tensor of r and Rij its inverse. We then show that aminimal Lagrangian connectionr on an oriented surface† defines a triple .g; ˇ; C /on †, consisting of a Riemannian metric g, a 1-form ˇ and a cubic differential C ,so that the following equations hold

(1.2) Kg D ˙1C 2 jC j2g C ıgˇ; @C D

�ˇ � i ?g ˇ

�˝ C; dˇ D 0:

As usual, i Dp�1, @ denotes the “del-bar” operator with respect to the integrable

almost complex structure J induced on † by Œg� and the orientation, ?g , ıg andKg denote the Hodge-star, co-differential and Gauss curvature with respect to g,respectively. Recall that jC jg denotes the point-wise tensor norm of C with respectto the Hermitian metric induced by g on the third power of the canonical bundleof †. As a consequence, we use a result of Labourie [26] to prove that if r is aspacelike minimal Lagrangian connection on a compact oriented surface † defininga flat projective structure p.r/, then .†; p/ is a properly convex projective surface.Moreover, the zero-section is a totally geodesic submanifold of .T �†; hr/ if andonly if r is the Levi-Civita connection of a hyperbolic metric.

We also show that a minimal Lagrangian connection defines a flat projectivestructure if and only if ˇ vanishes identically. In particular, we recover Wang’sequation (1.1) in the projectively flat case. In the projectively flat case Labourie [26]interpreted the first two equations as an instance of Hitchin’s Higgs bundle equa-tions [24]. In the case with ˇ ¤ 0 the above triple of equations falls into thegeneral framework of symplectic vortex equations [13] (see also [3]). Furthermore,it appears likely that the above equations also admit an interpretation in terms ofaffine differential geometry, but this will be addressed elsewhere.

The last two of the equations (1.2) say that locally there exists a (real-valued)function r so that e�2rC is holomorphic. As a consequence of this we show that theonly examples of minimal Lagrangian connections on the 2-sphere are Levi-Civitaconnections of metrics of positive Gauss curvature.

Furthermore, if .†; Œg�/ is a compact Riemann surface of negative Euler char-acteristic �.†/, then the metric g of the triple .g; ˇ; C / is uniquely determined in

4 T. METTLER

terms of .Œg�; ˇ; C /. This leads to a quasi-linear elliptic PDE of vortex type, whichbelongs to a class of equations solved in [17] using the technique of sub – andsupersolutions (see also [14] for the case when ˇ vanishes identically). Here instead,we use the calculus of variations and prove existence and uniqueness of a smoothminimum of the following functional defined on the Sobolev space W 1;2.†/

E�;� W W1;2.†/! R; u 7!

1

2

Z†

jduj2g0� 2u � �e2u C �e�4udAg0

;

where �; � 2 C1.†/ satisfy � < 0, � > 0 and g0 denotes the hyperbolic metric inthe conformal equivalence class Œg�.

An immediate consequence of (1.2) is that the area of a spacelike minimalLagrangian connection r – by which we mean the area of o.†/ � .T �†; hr/ –satisfies the inequality

Area.r/ > �2��.†/:Therefore, we call a spacelike minimal Lagrangian connection with area �2��.†/area minimising. We obtain:

Theorem A. Let † be a compact oriented surface with �.†/ < 0. Then we have:

(i) there exists a one-to-one correspondence between area minimising Lag-rangian connections on T† and pairs .Œg�; ˇ/ consisting of a conformalstructure Œg� and a closed 1-form ˇ on †;

(ii) there exists a one-to-one correspondence between non-area minimisingminimal Lagrangian connections on T† and pairs .Œg�; C / consisting ofa conformal structure Œg� and a non-trivial cubic differential C on † thatsatisfies @C D

�ˇ � i ?g ˇ

�˝ C for some closed 1-form ˇ.

Using the classification of properly convex projective structures by Loftin [30]and Labourie [26], it follows that every properly convex projective structure on† arises from a unique spacelike minimal Lagrangian connection, paralleling theresult of Hildebrand.

1.3. Related work

After the first version of this article appeared on the arXiv, Daniel Fox informedthe author about his interesting paper [17], which contains some closely relatedresults. Here we briefly compare our results which were arrived at independently.In a previous eprint [16] (see also [18]), Fox introduced the notion of an AH (af-fine hypersurface) structure which is a pair comprising a projective structure anda conformal structure satisfying a compatibility condition which is automatic intwo dimensions. He then proceeds to postulate Einstein equations for AH struc-tures, which are motivated by Calderbank’s work on Einstein–Weyl structures onsurfaces [5]. Subsequently in [17], Fox classifies the Einstein AH structures onclosed oriented surfaces and in particular, observes that in the case of negative Euler-characteristic they precisely correspond to the properly convex projective structures.On the 2-sphere S2 he recovers the Einstein–Weyl structures of Calderbank. OnS2, the Einstein AH structures and the minimal Lagrangian connections “overlap”in the space of metrics of constant positive Gauss–curvature. In the case of negativeEuler characteristic, the minimal Lagrangian connections are however strictly more

MINIMAL LAGRANGIAN CONNECTIONS 5

general than the Einstein AH structures. Indeed, in this case, the projective struc-tures arising from minimal Lagrangian connections provide a new and previouslyunstudied class of projective structures.

This new class of (possibly) curved projective structures may be thought ofas a generalization of the notion of a (flat) properly convex projective structure.One would expect that this class exhibits interesting properties, similar to those ofproperly convex projective structures. As a first result in this direction, it is shownin [32], that the geodesics of a minimal Lagrangian connection naturally give riseto a flow admitting a dominated splitting (a certain weakening of the notion of anAnosov flow). In particular, this flow provides a generalization of the geodesic flowinduced by the Hilbert metric on the quotient surface of a divisible convex set.

Furthermore, in joint work in progress by the author and A. Cap [8], the notion ofa minimal Lagrangian connection is extended to all so-called j1j-graded parabolicgeometries. This is a class of geometric structures which, besides projective geo-metry, includes (but is not restricted to) conformal geometry, (almost) Grassmanniangeometry and (almost) quaternionic geometry.

Acknowledgements

The author is grateful to Andreas Cap, Daniel Fox, Gabriel Paternain, Nigel Hitchin,Norbert Hungerbühler, Tobias Weth and Luca Galimberti for helpful conversationsor correspondence. The author also would like to thank the anonymous referees forseveral helpful suggestions.

2. Preliminaries

Throughout the article † will denote an oriented smooth 2-manifold without bound-ary. All manifolds and maps are assumed to be smooth and we adhere to theconvention of summing over repeated indices.

2.1. The coframe bundle

We denote by � W F ! † the bundle of orientation preserving coframes whose fibreat p 2 † consists of the linear isomorphisms f W Tp†! R2 that are orientationpreserving with respect to the fixed orientation on † and the standard orientationon R2. Recall that � W F ! † is a principal right GLC.2;R/-bundle with rightaction defined by the rule Ra.f / D f � a D a�1 ı f for all a 2 GLC.2;R/. Thebundle F is equipped with a tautological R2-valued 1-form ! D .!i / defined by!f D f ı �

0f

, and this 1-form satisfies the equivariance property R�a! D a�1!.

A torsion-free connection r on T† corresponds to a gl.2;R)-valued connection1-form � D .� ij / on F satisfying the structure equations

d! D �� ^ !;(2.1)

d� D �� ^ � C‚;(2.2)

where ‚ denotes the curvature 2-form of � . The Ricci curvature of r is the (notnecessarily symmetric) covariant 2-tensor field Ric.r/ on † satisfying

Ric.r/.X; Y / D tr�Z 7! rZrXY � rXrZY � rŒZ;X�Y

�; Z 2 �.TM/;

6 T. METTLER

for all vector fields X; Y on †. Denoting by Ric˙.r/ the symmetric (respectively,the anti-symmetric) part of the Ricci curvature of r, so that Ric.r/ D RicC.r/CRic�.r/, the (projective) Schouten tensor of r is defined as

Schout.r/ D RicC.r/ �1

3Ric�.r/:

Since the components of ! are a basis for the �-semibasic forms on F ,1 it followsthat there exist real-valued functions Sij on F such that

��Schout.r/ D !tS! D Sij!i ˝ !j ;

where S D .Sij /. Note thatR�aS D a

tSa

for all a 2 GLC.2;R/, since !tS! is invariant under Ra. In terms of the functionsSij the curvature 2-form ‚ D .‚ij / can be written as2

(2.3) ‚ij D�ıiŒkSl�j � ı

ijSŒkl�

�!k ^ !l ;

or explicitly

(2.4) ‚ D

�2S21 � S12 S22�S11 S21 � 2S12

�!1 ^ !2:

2.2. The orthonormal coframe bundle

Recall that if g is a Riemannian metric on the oriented surface †, the Levi-Civitaconnection .'ij / of g is the unique connection on the coframe bundle � W F ! †

satisfying

d!i D �'ij ^ !j ;

dgij D gik'kj C gkj'

ki ;

where we write ��g D gij!i ˝ !j for real-valued functions gij D gj i on F .

Differentiating these equations implies that there exists a unique function Kg , theGauss curvature of g, so that

d'ij C 'ik ^ '

kj D gjkKg!

i^ !k :

We may reduce F to the SO.2/-subbundle Fg consisting of orientation preservingcoframes that are also orthonormal with respect to g , that is, the bundle definedby the equations gij D ıij . On Fg the identity dgij D 0 implies the identities'11 D '

22 D 0 as well as '12 C '

21 D 0. Therefore, writing ' WD '21 , we obtain the

structure equations

d!1 D �!2 ^ ';

d!2 D �' ^ !1;

d' D �Kg!1 ^ !2;(2.5)

1Recall that a 1-form ˛ 2 �1.M/ is semibasic for the projection � W M ! N if ˛ vanishes onvector fields that are tangent to the �-fibres.

2For a matrix S D .Sij / we denote by S.ij / its symmetric part and by SŒij � its anti-symmetricpart, so that Sij D S.ij / C SŒij �.

MINIMAL LAGRANGIAN CONNECTIONS 7

where !i D ıij!j . Continuing to denote the basepoint projection Fg ! † by � ,the area form dAg of g satisfies ��dAg D !1 ^ !2. Also, note that a complex-valued 1-form ˛ on † is a .1;0/-form for the complex structure J induced on † byg and the orientation if and only if ��˛ is a complex multiple of the complex-valuedform ! D !1 C i!2. In particular, denoting by K† the canonical bundle of † withrespect to J , a section A of the `-th tensorial power of K† satisfies ��A D a!`

for some unique complex-valued function a on Fg . Denote by S30 .T�†/ the

trace-free part of S3.T �†/ with respect to Œg�, where S3.T �†/ denotes the thirdsymmetrical power of the cotangent bundle of †. The proof of the following lemmais an elementary computation and thus omitted.

Lemma 2.1. Suppose W 2 ��S30 .T

�†/�. Then there exists a unique cubic

differential C 2 �.K3†/ so that Re.C / D W . Moreover, writing ��W D

wijk!i ˝ !j ˝ !k for unique real-valued functions wijk on Fg , totally symmetricin all indices, the cubic differential satisfies ��C D .w111 C iw222/!3.

In complex notation, the structure equations of a cubic differential C 2 �.K3†/can be written as follows. Writing ��C D c!3 for a complex-valued function c onFg , it follows from the SO.2/-equivariance of c!3 that there exist complex-valuedfunctions c0 and c00 on Fg such that

dc D c0! C c00! C 3ic';

where we write ! D !1 � i!2. Note that the Hermitian metric induced by g on K3†has Chern connection D given by

c 7! dc � 3ic':

In particular, the .0;1/-derivative of C with respect to D is represented by c00, thatis, ��.D0;1C/ D c00!3 ˝ !. Since @ D D0;1, we obtain

(2.6) ���@C�D c00!3 ˝ !:

Also, we record the identity��jC j2g D jcj

2:

Moreover, recall that for u 2 C1.†/ we have the following standard identityfor the change of the Gauss curvature of a metric g under conformal rescaling

Ke2ug D e�2u�Kg ��gu

�;

where �g D ��ıgdC dıg

�is the negative of the Laplace–Beltrami operator with

respect to g. Also,dAe2ug D e2udAg

for the change of the area form dAg ,

�e2ug D e�2u�g

for �g acting on functions and

ıe2ug D e�2uıg

for the co-differential acting on 1-forms. Finally, the norm of C changes as

jC j2e2ugD e�6ujC j2g :

8 T. METTLER

2.3. The cotangent bundle and induced structures

Recall that we have a GLC.2;R/ representation % on R2 – the real vector spaceof row vectors of length two with real entries – defined by the rule %.a/� D �a�1

for all � 2 R2 and a 2 GLC.2;R/. The cotangent bundle of † is the vectorbundle associated to the coframe bundle F via the representation %, that is, thebundle obtained by taking the quotient of F �R2 by the GLC.2;R/-right actioninduced by %. Consequently, a 1-form on† is represented by an R2-valued function� on F which is GLC.2;R/-equivariant, that is, � satisfies R�a� D �a for alla 2 GLC.2;R/.

Using � we may define a Riemannian metric hr as well as a symplectic form�r on T �† as follows. Let

� W F �R2 ! .F �R2/ =GLC.2;R/ ' T �†

denote the quotient projection. Writing

D d� � �� � �!� � !tS t

or in components D . i / with

i D d�i � �j �ji � �j!

j �i � Sj i!j ;

we consider the covariant 2-tensor field Tr D ! WD i ˝ !i . Note that the

�-semibasic 1-forms on F � R2 are given by the components of ! and d� � �� ,or, equivalently, by the components of ! and . Indeed, the components of ! aresemibasic by the definition of !. Moreover, if Xv for v 2 gl.2;R/ is a fundamentalvector field for the coframe bundle F ! †, that is, the vector field associated tothe flow Rexp.tv/, then

.d� � ��/.Xv/ D ���.Xv/C limt!0

1

t

�.Rexp.tv//

�� � ��

D ��v C limt!0

1

t.� exp.tv/ � �/ D ��v C �v D 0;

where we have used that the connection � maps a fundamental vector field Xv to itsgenerator v 2 gl.2;R/. Since the fundamental vector fields span the vector fieldstangent to fibres of � , it follows that the components of d� � �� are �-semibasic.Moreover,

R�a D d� a � �aa�1�a � �aa�1!�a � !t .a�1/tatS ta D a;(2.7)

R�a! D a�1!;(2.8)

for all a 2 GLC.2;R/, it follows that the �-semibasic tensor field Tr is invariantunder the GLC.2;R/-right action and hence there exists a unique symmetric covari-ant 2-tensor field hr and a unique anti-symmetric covariant 2-tensor field �r onT �† such that

�� .hr C�r/ D Tr :

MINIMAL LAGRANGIAN CONNECTIONS 9

Using the structure equation (2.1), we compute

���r D d�i ^ !i � �j �ji ^ !

i� �i�j!

j^ !i � Sj i!

j^ !i

D d�i ^ !i C �id!i � Sij!i ^ !j

D d.�i!i / � SŒij �!i^ !j :

The 1-form �! D �i!i on F � R2 is �-semibasic and Ra invariant, hence the

�-pullback of a unique 1-form � on T �† which is the tautological 1-form of T �†.Recall that the canonical symplectic form on T �† is �0 D d� , hence �r defines asymplectic structure on T �† which is the canonical symplectic structure twistedwith the (closed) 2-form 1

3Ric�.r/

�r D �0 C ‡�

�1

3Ric�.r/

�;

where ‡ W T �†! † denotes the basepoint projection. In particular, denoting byo W †! T �† the zero ‡ -section, the definition of the Schouten tensor gives

o��r D1

3Ric�.r/:

This shows:

Proposition 2.2. The zero section of T �† is a �r-Lagrangian submanifold if andonly if r has symmetric Ricci tensor.

Which motivates:

Definition 2.3. A torsion-free connectionr on T† is called Lagrangian if Ric�.r/vanishes identically.

Also, we obtain for the symmetric part

��hr D ı ! WD 1 ı !1C 2 ı !

2

where ı denotes the symmetric tensor product. Since the four 1-forms 1; 2; !1; !2

are linearly independent, it follows that hr is non-degenerate and hence defines apseudo-Riemannian metric of split signature .1; 1;�1;�1/ on T �†.

Remark 2.4. The motivation for introducing the pair .hr ; �r/ is its projectiveinvariance, i.e., suitably interpreted, the pair .hr ; �r/ does only depend on theprojective equivalence class of the connection r. Moreover, the metric hr isanti-self-dual and Einstein. We refer the reader to [15] as well as [6] for furtherdetails.

From the definition of the Schouten tensor and hr we immediately obtain

(2.9) o�hr D �RicC.r/:

Following standard pseudo-Riemannian submanifold theory, we call a tangent vectorv timelike if hr.v; v/ < 0 and spacelike if hr.v; v/ > 0. Thus (2.9) motivates:

Definition 2.5. A torsion-free connection r on T† is called timelike if RicC.r/ ispositive definite and spacelike if RicC.r/ is negative definite.

10 T. METTLER

3. Twisted Weyl connections

We will see that timelike/spacelike minimal Lagrangian connections are twisted Weylconnections. In this section we study some properties of this class of connectionsthat we will need later during the classification of spacelike minimal Lagrangianconnections.

Let Œg� be a conformal structure on the smooth oriented surface †. By a Œg�-Weyl connection on † we mean a torsion-free connection on T† preserving theconformal structure Œg�. It follows from Koszul’s identity that a Œg�-Weyl connectioncan be written in the following form

.g;ˇ/r D

gr C g ˝ ˇ] � ˇ ˝ Id � Id˝ ˇ;

where g 2 Œg�, ˇ 2 �1.†/ is a 1-form and ˇ] denotes the g-dual vector field to ˇ.We will use the notation Œg�r to denote a general Œg�-Weyl connection.

Definition 3.1. A twisted Weyl connection r on .†; Œg�/ is a connection on thetangent bundle of † which can be written as r D Œg�r C ˛ for some Œg�-Weyl con-nection Œg�r and some 1-form ˛ with values in End.T†/ satisfying the followingproperties:

(i) ˛.X/ is trace-free and Œg�-symmetric for all X 2 �.T†/;(ii) ˛.X/Y D ˛.Y /X for all X; Y 2 �.T†/.

Note that if ˛ satisfies the above properties, then Œg�r C ˛ is torsion-free.Moreover, the covariant 3-tensor obtained by lowering the upper index of ˛ with ametric g 2 Œg� gives a section of �.S30 .T

�†//. Conversely, every End.T†/-valued1-form on † satisfying the above properties arises in this way. In other words,fixing a Riemannian metric g 2 Œg� allows to identify the twist term ˛ with a cubicdifferential.

Fixing a metric g 2 Œg�, the connection form � D .� ij / of a twisted Weylconnection is given by

� ij D 'ij C

�bkg

kigjl � ıij bl � ı

ilbj C a

ijl

�!l ;

where the map .gij / W F ! S2.R2/ represents the metric g, the map .bi / W F !R2 represents the 1-form ˇ and the map .ai

jk/ W F ! R2 ˝ S2.R2/ represents the

1-form ˛. Moreover, .'ij / denote the Levi-Civita connection forms of g. Reducingto the bundle Fg of g-orthonormal orientation preserving coframes, the connectionform becomes

� D

��ˇ ?gˇ � '

' � ?gˇ �ˇ

�C

�a111!1 C a

112!2 a112!

1 C a122!2a211!1 C a

212!2 a212!1 C a

222!2

�;

where we use the identity ���?gˇ

�D �b2!1 C b1!2. By definition, on Fg the

functions aijk

satisfy the identities

aijk D aikj ; akkj D 0; ıkia

kjl D ıkja

kil :

Thus, writing c1 D a111 and c2 D a222, we obtain

� D

��ˇ ?gˇ � '

' � ?gˇ �ˇ

�C

�c1!1 � c2!2 �c2!1 � c1!2�c2!1 � c1!2 �c1!1 C c2!2

�:

MINIMAL LAGRANGIAN CONNECTIONS 11

In order to compute the curvature form of � we first recall that we write ��ˇ D bi!iand since bi!i is SO.2/-invariant, it follows that there exist unique real-valuedfunctions bij on Fg such that

db1 D b11!1 C b12!2 C b2';

db2 D b21!1 C b22!2 � b1':

Recall also that the area form of g satisfies ��dAg D !1 ^ !2 and since ?g1 DdAg , we get

��ıgˇ D �.b11 C b22/;

as well as���d?gˇ

�D .b11 C b22/!1 ^ !2:

Since c1 C ic2 represents a cubic differential on †, there exist unique real-valuedfunctions cij on Fg such that

dc1 D c11!1 C c12!2 � 3c2';

dc2 D c21!1 C c22!2 C 3c1':

Consequently, a straightforward calculation shows that the curvature form ‚ D

d� C � ^ � satisfies

(3.1)

‚ D

��dˇ KgdAg C d ?g ˇ � 1

2j˛j2g!1 ^ !2

�KgdAg � d ?g ˇ C 12j˛j2g!1 ^ !2 �dˇ

�C

�2.b1c2 C b2c1/ � .c12 C c21/ 2.b1c1 � b2c2/C .c22 � c11/

2.b1c1 � b2c2/C .c22 � c11/ 2.�b1c2 � b2c1/C .c12 C c21/

�!1^!2;

where we use the identity ��j˛j2g D 4�.c1/

2 C .c2/2�.

3.1. A characterisation of twisted Weyl connections

We obtain a natural differential operator DŒg� acting on the space A.†/ of torsion-free connections on T†

DŒg� W A.†/! �2.†/; r 7! trg Ric.r/dAg :

Note that this operator does indeed only depend on the conformal equivalence classof g. A twisted Weyl connection r on .†; Œg�/ can be characterised by minimisingthe integral of DŒg� among its projective equivalence class p.r/.

Proposition 3.2. Suppose r 0 D Œg�r C ˛ is a twisted Weyl connection on thecompact Riemann surface .†; Œg�/. Then

infr2p.r0/

Z†

DŒg�.r/ D 4��.†/ � k˛k2g

and 4��.†/ � k˛k2g is attained precisely on r 0.

Remark 3.3. Note that

k˛k2g D

Z†

j˛j2g dAg

does only depend on the conformal equivalence class of g.

12 T. METTLER

Proof of Proposition 3.2. Write r 0 D .g;ˇ/r C ˛ for some Riemannian metricg 2 Œg�, some 1-form ˇ and some End.T†/-valued 1-form ˛ on † satisfying theproperties of Definition 3.1. From (2.4) and the definition of the Schouten tensor itfollows that

���trg Ric.r 0/dAg

�D ‚12 �‚

21;

where ‚ D .‚ij / denotes the curvature form of r 0 pulled-back to Fg . Thus,equation (2.4) gives

(3.2) trg Ric.r 0/dAg D 2Kg C 2d ?g ˇ � j˛j2gdAg

and hence

(3.3)Z†

trg Ric.r 0/dAg D 4��.†/ � k˛k2g

by the Stokes and the Gauss–Bonnet theorem.It is a classical result due to Weyl [37] that two torsion-free connections r1;r2

on T† are projectively equivalent if and only if there exists a 1-form on † suchthat r1 � r2 D ˝ IdC Id˝ . It follows that the connections in the projectiveequivalence class of r 0 can be written as

r D r0C ˝ IdC Id˝

with 2 �1.†/. A simple computation gives

(3.4) Ric.r/ D Ric.r 0/C 2 � Symr 0 C 3 d ;

where Sym W �.T �†˝ T �†/! �.S2.T �†// denotes the natural projection. Wecompute

trg Symr 0 dAg D trg Sym�gr C g ˝ ˇ] � ˇ ˝ Id � Id˝ ˇ C ˛

� dAg

D d ?g C�2 .ˇ]/ � .ˇ]/ � .ˇ]/

�dAg

D d ?g ;

where we used that ˛.X/ is trace-free and Œg�-symmetric for all X 2 �.T†/. Sincethe last summand of the right hand side of (3.4) is anti-symmetric, we obtainZ

†

trg Ric.r/dAg DZ†

trg Ric.r/CZ†

trg 2dAg �Z†

trg Symr 0 dAg

D 4��.†/ � k˛k2g C k k2g �

Z†

d ?g ;

thus the claim follows from the Stokes theorem. �

In [31] the following result is shown, albeit phrased in different language:

Proposition 3.4. Let .†; Œg�/ be a Riemann surface. Then every torsion-free con-nection on T† is projectively equivalent to a unique twisted Œg�-Weyl connection.

Let P.†/ denote the space of projective structures on †. Using Proposition 3.2and Proposition 3.4 we immediately obtain:

Theorem 3.5. Let .†; Œg�/ be a compact Riemann surface. Then

supp2P.†/

infr2p

Z†

trg Ric.r/dAg D 4��.†/:

MINIMAL LAGRANGIAN CONNECTIONS 13

Remark 3.6. A twisted Weyl connection r on .†; Œg�/ defines an AH structure.p.r/; Œg�/ in the sense of [16, 17, 18], where p.r/ denotes the projective equi-valence class arising from r. Moreover, the twisted Weyl connection agrees withthe aligned representative of the associated AH structure .p.r/; Œg�/. In particular,the equations (3.2) and (3.3) have counterparts in the equations (5.8) and (7.12)of [17]. Also, the Proposition 3.4 corresponds to the existence of a unique alignedrepresentative for an AH structure from [17].

4. Submanifold theory of Lagrangian connections

We now restrict attention to torsion-free connections on T† having symmetric Riccitensor, so that the zero-section o W † ! T �† is a Lagrangian submanifold. Iffurthermore r is timelike/spacelike, we obtain an induced metric g D �o�hr D˙Ric.r/ and the immersion o W †! T �† has a well-defined normal bundle andsecond fundamental form. In particular, we want to compute when o W †! T �†

is minimal, that is, the trace with respect to g of its second fundamental formvanishes identically.

4.1. Algebraic preliminaries

Before we delve into the computations, we briefly review the relevant algebraicstructure of the theory of oriented surfaces in an oriented Riemannian – and orientedsplit-signature Riemannian four manifold .M; g/. We refer the reader to [4] and [11]for additional details.

First, let X W †! .M; g/ be an immersion of an oriented surface † into an ori-ented Riemannian 4-manifold. The bundle of orientation compatible g-orthonormalcoframes of .M; g/ is an SO.4/-bundle � W FCg !M . The GrassmannianGC2 .R

4/

of oriented 2-planes in R4 is a homogeneous space for the natural action of SO.4/and the stabiliser subgroup is SO.2/ � SO.2/. Consequently, the pullback bundleX�FCg ! † admits a reduction FX � X�FCg with structure group SO.2/�SO.2/,where the fibre of FX at p 2 † consists of those coframes mapping the orientedtangent plane to † at X.p/ to some fixed oriented 2-plane in R4, while preservingthe orientation.

The second fundamental form of X is a quadratic form on T† with values inthe rank two normal bundle of X . Therefore, it is represented by a map FX !S2.R2/˝R2 which is equivariant with respect to some suitable representation ofSO.2/ � SO.2/ on S2.R2/ �R2. The relevant representation is defined by the rule

%.r˛; rˇ /.A/.x; y/ D r�ˇA.r˛x; r˛y/; x; y 2 R2;

where A 2 S2.R2/˝ R2 is a symmetric bilinear form on R2 with values in R2

and r˛; rˇ denote counter-clockwise rotations in R2 by the angle ˛; ˇ, respectively.As usual, we decompose the SO.2/� SO.2/-module S2.R2/˝R2 into irreduciblepieces. This yields

% D &0;�1 ˚ &2;1 ˚ &2;�1;

where for .n;m/ 2 Z2 the complex one-dimensional SO.2/�SO.2/-representation&n;m is defined by the rule

&n;m.r˛; rˇ / D ei.n˛Cmˇ/:

14 T. METTLER

Explicitly, the relevant projections S2.R2/˝R2 ! C are

p0;�1.A/ D1

2

�A111 C A

122

�C

i2

�A211 C A

222

�;(4.1)

p2;�1.A/ D1

4

�A111 � A

122 C 2A

212

�C

i4

�A211 � A

222 � 2A

112

�;(4.2)

p2;1.A/ D1

4

�A122 � A

111 C 2A

212

�C

i4

�A211 � A

222 C 2A

112

�(4.3)

and where we write A.ei ; ej / D Akij ek with respect to the standard basis .e1; e2/of R2.

The canonical bundle K† of † with respect to the complex structure inducedby the metric X�g and orientation is the bundle associated to the representation&1;0. Moreover, the conormal bundle, thought of as a complex line bundle, isthe bundle N �X associated to the representation &0;1. Consequently, the secondfundamental form of X defines a section H of the normal bundle which is themean curvature vector of X , as well as a quadratic differential QCwith values inthe conormal bundle and a quadratic differential Q� with values in the normalbundle. Consequently, we obtain a quartic differentialQCQ� on† which turns outto be holomorphic, provided X is minimal and g has constant sectional curvature,see [11].

If we instead consider a split-signature oriented Riemannian 4-manifold .M; g/,the bundle of orientation compatible g-orthonormal coframes of .M; g/ is anSO.2; 2/-bundle � W FCg ! M . Here, as usual, we take SO.2; 2/ to be thesubgroup of SL.4;R/ stabilising the quadratic form

q.x/ D .x1/2C .x2/

2� .y1/

2� .y2/

2;

where .x; y/ 2 R2;2. Now the action of SO.2; 2/ on the Grassmannian GC2 .R2;2/

of oriented 2-planes in R2;2 is not transitive, it is however transitive on the opensubmanifolds of oriented timelike/spacelike 2-planes. In both cases, the stabil-iser subgroup is SO.2/ � SO.2/ as well. Therefore, the submanifold theory ofa timelike/spacelike oriented surface in an oriented split-signature Riemannianfour manifold is entirely analogous to the Riemannian case. In particular, we alsoencounter the mean curvature vector H and the quadratic differentials Q˙.

4.2. The mean curvature form

Knowing what to expect, we now carry out the submanifold theory of time-like/spacelike Lagrangian connections. Note however, that in addition to the split-signature metric hr , we also have a symplectic form �r . The symplectic formallows to identify the conormal bundle to a Lagrangian spacelike/timelike immersionwith the cotangent bundle of †. In particular we may think of the mean curvaturevector H as a 1-form, the conormal-bundle valued quadratic differential QC asa cubic differential and the normal-bundle valued quadratic differential Q� as a.1;0/-form.

The product P WD F � R2 is a principal right GLC.2;R/-bundle over T �†,where the GLC.2;R/-right action is given by .f; �/ � a D .a�1 ı f; �a/ for all

MINIMAL LAGRANGIAN CONNECTIONS 15

a 2 GLC.2;R/ and .f; �/ 2 P . We define two R2-valued 1-forms on P

� WD1

2

� t C !

�and � WD

1

2

� t � !

�;

so that the metric hr satisfies

��hr D �t� � �t� D .�1/2 C .�2/2 � .�1/2 � .�2/2:

From the equivariance properties (2.7) and (2.8) of and !, we compute

(4.4) R�a

��

�

�D1

2

�at C a�1 at � a�1

at � a�1 at C a�1

���

�

�:

Remark 4.1. The reader may easily verify that the representation GLC.2;R/ !GL.4;R/ defined by (4.4) embeds GLC.2;R/ as a subgroup of SO.2; 2/.

Recall that the Lie algebra of the split-orthogonal group O.2; 2/ consists ofmatrices of the form �

� �

�t #

�where � and # are skew-symmetric. Consequently, there exist unique o.2/-valued1-forms �; # on F � R2 and a unique gl.2;R/-valued 1-form � on F � R2 suchthat

(4.5) d��

�

�D �

�� �

�t #

�^

��

�

�:

In order to compute these connection forms we first remark that since the functionS D .Sij / represents the (symmetric) Ricci tensor of r, there must exist uniquereal-valued functions Sijk D Sj ik on F so that

dSij D Sijk!kC Sik�

kj C Skj �

ki :

Clearly, the function .Sijk/ W F ! S2.R2/˝R2 represents r Ric.r/ with k beingthe derivative index.

Lemma 4.2. We have

�ij D ��Œi!j � C �Œij � � SkŒij �!k;

�ij D ��k!kıij � �.i!j / � �.ij / C SkŒij �!k;

#ij D ��Œi!j � C �Œij � C SkŒij �!k!k;

where we write !i D ıij!j and �ij D ıik�kj .

Proof. Since the connection forms are unique, the proof amounts to plugging theabove formulae into the structure equations (4.5) and verify that they are satisfied.This is tedious, but an elementary computation and hence is omitted. �

Recall that the components of and ! – and hence equivalently the componentsof � and � – span the 1-forms on P that are semi-basic for the projection � WP ! T �†. In particular, if � is a 1-form on T �†, then there exists a unique map.e1; e2/ W P ! R2;2 so that ��� D e1� C e2�. Since ��� is invariant under theGLC.2;R/ right action, the function .e1; e2/ satisfies the equivariance propertydetermined by (4.4). Phrased differently, the cotangent bundle of T �† is the bundle

16 T. METTLER

associated to � W P ! T �† via the representation % W GLC.2;R/! R2;2 definedby the rule

(4.6) %.a/��1 �2

�D��1 �2

� 12

�at C a�1 at � a�1

at � a�1 at C a�1

�for all a 2 GLC.2;R/ and .�1; �2/ in R2;2.

We will next use this fact to exhibit the conormal bundle of the immersiono W †! T �† as an associated bundle to a natural reduction of the pullback bundleo�P . Note that by construction, the pullback bundle o�P ! † is just the framebundle � W F ! † and that on o�P ' F we have t D �S!, thus

� D1

2.I2 � S/! and � D �

1

2.I2 C S/!:

If we assume that r is timelike/spacelike, then the Ricci tensor of r is posit-ive/negative definite and hence the equations S D ˙I2 define a reduction Fr ! †

with structure group SO.2/ whose basepoint projection we continue to denote by � .Note that by construction, Fr ! † is the bundle of orientation preserving orthonor-mal coframes of the induced metric g D ˙Ric.r/. In particular, from (4.6) we seethat the pullback bundle o�.T �.T �†// is the bundle associated to � W Fr ! † viathe SO.2/-representation on R2;2 defined by the rule

(4.7) %.a/��1 �2

�D��1a

t �2at�

for all a 2 SO.2/ and .�1; �2/ in R2;2. Furthermore, on Fr we obtain D �!t

and hence

(4.8)��

�

�D

�0

�!

�in the timelike case and

(4.9)��

�

�D

�!

0

�in the spacelike case. Recall that if ˛ is a 1-form on† then there exists a unique R2-valued function a on F – and hence on Fr as well – so that ��˛ D a!. It followsas before that T �† is the bundle associated to Fr via the SO.2/-representationdefined by the rule

(4.10) %.a/.�/ D �at

for all a 2 SO.2/ and � 2 R2. Using (4.7), (4.8) and (4.9), we see that the conormalbundle

N �o WD o�.T �.T �†//=T �†

of o is (isomorphic to) the bundle associated to Fr via the representation (4.10) aswell. We thus have an isomorphism N �o ' T

�† between the conormal bundle ofthe immersion o W †! T �† and the cotangent bundle T �†. Of course, the metricg on † provides an isomorphism T �† ' T† and hence No ' T �†, where Nodenotes the normal bundle of o. The second fundamental form of o is a quadraticform on T† with values in the normal bundle, thus here naturally a section ofS2.T �†/˝ T �†.

MINIMAL LAGRANGIAN CONNECTIONS 17

Lemma 4.3. Let o W †! T �† be timelike/spacelike. Then the second fundamentalform A of o is represented by the functions Aijk D Aikj , where

(4.11) Aijk D �1

2

�Skji � Sijk � Sikj

�:

Proof. We will only treat the spacelike case, the timelike case is entirely analogousup to some sign changes. In our frame adaption on Fr we have � D 0 and � D !.Consequently,

0 D d� D ��t ^ � � # ^ � D ��t ^ !

or in components0 D �j i ^ !j :

Cartan’s lemma implies that there exist unique real-valued functions Aijk D Aikjon Fr so that

�j i D Aijk!k

and by standard submanifold theory the functions Aijk represent the second funda-mental form of o. In order to compute the functions Aijk , we use that in our frameadaption Sij D �ıij and hence

0 D dSij D Sijk!k � ıik�kj � ıkj �

ki D Sijk!k � 2�.ij /:

Since � D 0 we thus get from Lemma 4.2

�j i D ��.j i/ C SkŒj i�!k D

��1

2Sijk C

1

2Skji �

1

2Sikj

�!k;

and the claim follows. �

Denoting by S ij the functions onF representing the inverse of the Ricci curvatureof r, so that S ijSjk D ıik , we thus have:

Theorem 4.4. A timelike/spacelike Lagrangian connection r is minimal if and onlyif

(4.12) S ij�2Skij � Sijk

�D 0:

Proof. By standard submanifold theory, the immersion o W †! T �† is minimal ifand only if the trace of the second fundamental form with respect to the inducedmetric g D ˙Ric.r/ vanishes identically. �

Remark 4.5. Note that in index notation the minimality condition (4.12) is equivalentto

�k WD1

2Rij

�2riRjk � rkRij

�D 0;

where Rij denotes the Ricci tensor of r and Rij its inverse. We call the 1-form �

the mean curvature form of r.

Example 4.6. Let .†; g/ be a two-dimensional Riemannian manifold. The Levi-Civita connection r of g has Ricci tensor Ric.g/ D Kg, where K denotes theGauss curvature of g. Thus, if K is positive/negative, then r is a timelike/spacelikeLagrangian connection and Theorem 4.4 immediately implies that r is minimal. Infact, we will show later (see Proposition 6.1) that on the 2-sphere metrics of positiveGauss curvature are the only examples of minimal Lagrangian connections.

18 T. METTLER

Recall that a (non-degenerate) submanifold of a (pseudo-)Riemannian manifoldis called totally geodesic if its second fundamental form vanishes identically. Wecall a timelike/spacelike connection r totally geodesic if o W †! .T �†; hr/ is atotally geodesic submanifold. We also get:

Corollary 4.7. Let r be a timelike/spacelike Lagrangian connection. Then r istotally geodesic if and only if its Ricci tensor is parallel with respect to r.

Proof. The second fundamental form vanishes identically if and only if

Sjki D Sijk � Sikj :

Since the Ricci tensor is symmetric, the left hand side is symmetric in j; k, but theright hand side is anti-symmetric in j; k, thus Skji vanishes identically. �

4.3. Minimality and the Liouville curvature

The minimality condition for a timelike/spacelike Lagrangian connection can alsobe expressed in terms of the Liouville curvature of r. To this end we decomposethe structure equation3

(4.13) dSij D Sijk!kC Sik�

kj C Skj �

ki

into (we compute modulo � ij )

dSij D�1

3.Sijk C Sikj C Sjki /C

2

3Sijk �

1

3.Sikj C Sjki /

�!k

D

�Rijk C

2

3L.i"j /k

�!k;

where we define

Rijk D1

3

�Sijk C Sikj C Sjki

�and

Li D2

3"jk

�Sijk �

1

2.Sikj C Sjki /

�:

Remark 4.8. The equivariance properties of the function S D .Sij / yield R�aL DLa det a, where we write L D .Li /. Since

R�a�!1 ^ !2

�D .det a�1/!1 ^ !2;

it follows that the there exists a unique 1-form �.r/ on† taking values inƒ2.T �†/,such that

���.r/ D�L1!

1C L2!

2�˝ !1 ^ !2:

The ƒ2.T �†/-valued 1-form was discovered by R. Liouville and hence we call itthe Liouville curvature of r. Liouville showed that the vanishing of �.r/ is thecomplete obstruction to r being projectively flat.

3We define " D ."ij / by "ij D �"j i with "12 D 1 and "ij denote the components of the transposeinverse of ".

MINIMAL LAGRANGIAN CONNECTIONS 19

Writing g D ˙Ric.r/ for the induced metric, we define ˇ 2 �1.†/ by

ˇ D3

8trg Symrg;

where Sym W ��T �†˝ S2.T �†/

�! �

�S3.T �†/

�denotes the natural projec-

tion. We have:

Proposition 4.9. A timelike/spacelike Lagrangian connection r on T† is minimalif and only if

(4.14) �.r/ D � 2 ?gˇ ˝ dAg :

Proof. In order to prove the claim we work on the orthonormal coframe bundleof g which is cut out of the coframe bundle F by the equations Sij D ˙ıij . Bydefinition, the functions Sijk represent r Ric.r/ and hence the functions Rijkrepresent˙Symrg. Therefore, on Fg , writing ��ˇ D bi!i , the components bi ofˇ are

(4.15) bk D ˙3

8ıijRijk :

Now on Fg the equation (4.14) becomes

.L1!1 C L2!2/˝ !1 ^ !2 D �2 .�b2!1 C b1!2/˝ !1 ^ !2

which is equivalent to

L1 D3

4.R112 CR222/ and L2 D �

3

4.R111 CR221/ ;

where we have used that ��.?gˇ/ D �b2!1C b1!2 as well as ��dAg D !1^!2and (4.15). On the other hand Theorem 4.4 implies that the minimality is equivalentto

ıij�2Skij � Sijk

�D ıijRijk C ı

ij

�4

3L.k"i/j �

2

3L.i"j /k

�D ıijRijk C ı

ij

�2

3.Lk"ij C Li"kj / �

1

3.Li"jk C Lj "ik/

�D ıijRijk �

4

3ıijLi"jk D 0:

Written out, this gives the two conditions

(4.16) R111 CR221 C4

3L2 D R112 CR222 �

4

3L1 D 0;

which proves the claim. �

Remark 4.10. Proposition 4.9 shows that a timelike/spacelike minimal Lagrangianconnection is projectively flat if and only if the 1-form ˇ vanishes identically.

5. Minimal Lagrangian connections

We will next compute the structure equations of a timelike/spacelike minimalLagrangian connection r. As before, we work on the orthonormal coframe bundleFg of the induced metric g D ˙Ric.r/. The submanifold theory discussed in §3tells us that the second fundamental form of o W † ! .T �†; hr/ is described in

20 T. METTLER

terms of a cubic differential and a .1;0/-form. Using the definition (4.2) of the .1;0/-form, the expression (4.11) for the second fundamental form and the minimalityconditions (4.16), one easily computes that on Fg the .1;0/-form is represented bythe complex-valued function

b D ˙3

8..R111 CR221/ � i.R112 CR222// :

Hence comparing with (4.15), we conclude that b D b1� ib2. Note that if we definethe .1;0/-form

ˇ1;0 WD ˇ C i ?g ˇ;

then we have ��ˇ1;0 D .b1 � ib2/.!1 C i!2/, thus the .1;0/-form obtained fromthe normal bundle valued quadratic differential Q� by using the symplectic form�r is ˇ1;0. Likewise, QC gives the cubic differential C on † which is representedon Fg by the complex-valued function

c D �1

8..R111 � 3R122/C i .�3R112 CR222// :

From Lemma 2.1 and

�� .Sym0Ric.r// D�Rijk �

3

2ı.ijRk/lmı

lm

�!i ˝ !j ˝ !k

we easily compute that the cubic differential C satisfies Re.C / D �12

Sym0rg,where the subscript 0 denotes the trace-free part with respect to g.

The structure equations can now be summarised as follows:

Proposition 5.1. Let † be an oriented surface and r a timelike/spacelike minimalLagrangian connection on T†. Then we obtain a triple .g; ˇ; C / on † consistingof a Riemannian metric g D ˙Ric.r/, a 1-form ˇ D 3

8trg Symrg and a cubic

differential C so that Re.C / D �12

Sym0rg. Furthermore, the triple .g; ˇ; C /satisfies the following equations

Kg D ˙1C 2 jC j2g C ıgˇ;(5.1)

@C D�ˇ � i ?g ˇ

�˝ C;(5.2)

dˇ D 0:(5.3)

Proof. In our frame adaption where Sij D ˙ıij on Fg , we obtain from (4.13)

0 D dSij D�Rijk C

2

3L.i"j /k

�!k ˙ ıik�

kj ˙ ıkj �

ki :

Therefore, writing �ij D ıik�kj , we have

(5.4) �.ij / D �1

2

�Rijk C

2

3L.i"j /k

�!k :

MINIMAL LAGRANGIAN CONNECTIONS 21

For later usage we introduce the notation c1 D �.18R111 �38R122/ and c2 D

�.�38R112 C

18R222/, so that c D c1 C ic2. Equation (5.4) written out gives

�11 D �1

2R111!1 �

�3

4R112 C

1

4R222

�!2;

1

2.�12 C �21/ D �

�3

8R112 �

1

8R222

�!1 �

�3

8R122 �

1

8R111

�!2;

�22 D �

�3

4R122 C

1

4R111

�!1 �

1

2R222!2:

Defining

' D �21 �1

2R222!1 ˙

�1

4R111 C

3

4R122

�!2;

we compute

(5.5) � D

��ˇ ?gˇ � '

' � ?gˇ �ˇ

�C

�c1!

1 � c2!2 �c2!

1 � c1!2

�c2!1 � c1!

2 �c1!1 C c2!

2

�:

The motivation for the definition of ' is that we have

d!1 D �!2 ^ ' and d!2 D �' ^ !1;

hence ' is the Levi-Civita connection form of g. In particular, we see that time-like/spacelike minimal Lagrangian connections are twisted Weyl connections. SinceRic.r/ D ˙g, it follows that the curvature 2-form of � must satisfy

(5.6) ‚ D d� C � ^ � D�

0 ˙!1 ^ !2�!1 ^ !2 0

�:

In order to evaluate this condition we first recall that we write ��ˇ D bi!i and

db1 D b11!1 C b12!2 C b2';

db2 D b21!1 C b22!2 � b1';

for unique real-valued functions bij on Fg . From (5.5) and (5.6) we obtain

dˇ D �1

2.d�11 C d�22/ D

1

2.�12 ^ �21 C �21 ^ �12/ D 0

showing that ˇ is closed, hence (5.3) is verified. Likewise, we also obtain

d' D1

2.d�21 � d�12/C d?gˇ

D .b11 C b22/!1 ^ !2 C1

2..�11 � �22/ ^ .�21 C �12//� !1 ^ !2

D ��2�.c1/

2C .c2/

2�� .b11 C b22/˙ 1

�!1 ^ !2:

Writing Kg for the Gauss curvature of g, this last equation is equivalent to

Kg D ˙1C 2 jC j2g C ıgˇ;

which verifies (5.1).In order to prove (5.2), we use

���ˇ � i ?g ˇ

�D .b1 C ib2/.!1 � i!2/:

22 T. METTLER

In light of (2.6) the condition (5.2) is equivalent to the condition

(5.7) dc ^ ! D bc! ^ ! C 3ic' ^ !;

where we use the complex notation b D b1 � ib2, c D c1C ic2 and ! D !1C i!2.Again, from (5.5) we compute

c! D1

2Œ.�11 � �22/ � i .�12 C �21/� ;

hence

dc ^ ! D d.c!/ � cd! D ��12 ^ �21

Ci2.�11 ^ .�12 � �21/C �22 ^ .�21 � �12// � .c1 C ic2/.d!1 C id!2/:

Using (5.5) and the structure equations (2.5) this gives

dc ^ ! D 3c2!1 ^ ' C 3c1!2 ^ ' � 2.b1c2 C b2c1/!1 ^ !2C i .�3c1!1 ^ ' C 3c2!2 ^ ' C 2.b1c1 � b2c2/!1 ^ !2/ ;

which is equivalent to

dc ^ ! D .b1 C ib2/.c1 C ic2/.!1 � i!2/ ^ .!1 C i!2/

C 3i.c1 C ic2/' ^ .!1 C i!2/;

that is, equation (5.7). This completes the proof. �

Conversely, unravelling our computations backwards, we also get:

Proposition 5.2. Suppose a triple .g; ˇ; C / on an oriented surface † satisfiesthe equations (5.1),(5.2),(5.3). Then the connection form (5.5) on Fg defines atimelike/spacelike minimal Lagrangian connection r on T† with Ric.r/ D ˙g.

We immediately obtain:

Corollary 5.3. Let † be an oriented surface. Then there exists a one-to-onecorrespondence between timelike/spacelike minimal Lagrangian connections onT† and triples .g; ˇ; C / satisfying (5.1),(5.2),(5.3).

Proof. Clearly, the map sending a torsion-free minimal Lagrangian connection rinto the set of triples .g; ˇ; C / satisfying the above structure equations, is surjective.Now suppose the two triples .g1; ˇ1; C1/ and .g2; ˇ2; C2/ on † satisfy the abovestructure equations and define the same torsion-free spacelike minimal Lagrangianconnection r on T†. Then g1 D ˙Ric.r/ D g2 and consequently we obtainˇ1 D ˇ2 as well as C1 D C2, since these quantities are defined in terms ofrRic.r/by using the metric g1 D g2. �

Remark 5.4. Remark 4.10 immediately implies that a minimal Lagrangian connec-tion is projectively flat if and only if the cubic differential C is holomorphic.

Another consequence of the structure equation is:

Proposition 5.5. Letr be a timelike/spacelike Lagrangian connection that is totallygeodesic. Then r is the Levi-Civita connection of a metric g of Gauss curvatureKg D ˙1.

MINIMAL LAGRANGIAN CONNECTIONS 23

Proof. The Lagrangian connection r is totally geodesic if and only if the secondfundamental form vanishes identically or equivalently, if ˇ and C vanish identically.In this case (5.5) implies that � is the Levi-Civita connection of g and the structureequation (5.1) gives that g has Gauss curvature˙1. �

Remark 5.6. As we have mentioned previously in Remark 3.6, a twisted Weylconnection on .†; Œg�/ defines an AH structure .p.r/; Œg�/. Moreover, a twistedWeyl connection arising from a triple .g; ˇ; C / satisfying @C D .ˇ � i ?g ˇ/˝C defines an associated AH structure .p.r/; Œg�/ which is naive Einstein in theterminology of [16, 17, 18]. Therefore, every minimal Lagrangian connectiondefines a naive Einstein AH structure.

6. The spherical case

The system of equations governing minimal Lagrangian connections are easy toanalyse on the 2-sphere S2:

Proposition 6.1. A connection on the tangent bundle of S2 is minimal Lagrangianif and only if it is the Levi-Civita connection of a metric of positive Gauss curvature.

Proof. Let r be a minimal Lagrangian connection on TS2 with associated triple.g; ˇ; C /. Since ˇ is closed and H 1.S2/ D 0, the 1-form ˇ is exact and hencethere exists a smooth real-valued function r on S2 such that ˇ D dr . Hence wehave

@C D�dr � i ?g dr

�˝ C:

Observe that dr � i ?g dr D 2@r , therefore, the cubic differential e�2rC is holo-morphic. Since, by Riemann-Roch, there are no non-trivial cubic holomorphicdifferentials on the 2-sphere, C must vanish identically. The connection form (5.5)of r thus becomes

� D

��dr ?gdr � '

' � ?gdr �dr

�;

where ' denotes the Levi-Civita connection form of g. We conclude that r is aWeyl connection given by

r Dgr C g ˝ g

rr � dr ˝ Id � Id˝ dr;

where grr denotes the gradient of r with respect to g. Since the Levi-Civitaconnection of a Riemannian metric g transforms under conformal change as [2,Theorem 1.159]

exp.2f /gr D

gr � g ˝ g

rf C df ˝ IdC Id˝ df;

we obtain r D exp.�2r/gr, thus showing that r is the Levi-Civita connection of aRiemannian metric. Moreover, since Ric.r/ must be positive or negative definite,the Gauss curvature of the metric e�2rg cannot vanish and hence is positive by theGauss–Bonnet theorem. Finally, Example 4.6 shows that conversely the Levi-Civitaconnection of a Riemannian metric of positive Gauss curvature defines a minimalLagrangian connection, thus completing the proof. �

24 T. METTLER

7. The case of negative Euler-characteristic

Before we address the classification of minimal Lagrangian connections on compactsurfaces of negative Euler characteristic, we observe that every projectively flatspacelike minimal Lagrangian connection defines a properly convex projectivestructure. Indeed, Labourie gave the following characterisation of properly convexprojective manifolds:

Theorem 7.1 (Labourie [26], Theorem 3.2.1). Let .M; p/ be an oriented flat pro-jective manifold. Then the following statements are equivalent:

(i) p is properly convex;(ii) there exists a connection r 2 p preserving a volume form and whose Ricci

curvature is negative definite.

We immediately obtain:

Corollary 7.2. Let r be a projectively flat spacelike minimal Lagrangian con-nection on the oriented surface †. Then r defines a properly convex projectivestructure.

Proof. In Remark 4.10 we have seen that a minimal Lagrangian connection r isprojectively flat if and only if ˇ vanishes identically. In the projectively flat case theconnection 1-form � of r thus is (see (5.5))

� D

�0 �'

' 0

�C

�c1!

1 � c2!2 �c2!

1 � c1!2

�c2!1 � c1!

2 �c1!1 C c2!

2

�:

In particular, the trace of � vanishes identically and hence r preserves the volumeform of g. Since Ric.r/ D �g, the claim follows by applying Labourie’s result.

�

7.1. Classification

In Section 5 we have seen that a triple .g; ˇ; C / on an oriented surface † satisfying(5.1),(5.2),(5.3) uniquely determines a minimal Lagrangian connection on T†. Inthis section we will show that in the case where† is compact and has negative Eulercharacteristic �.†/, the conformal equivalence Œg� of g and the cubic differential Calso uniquely determine .g; ˇ; C / and hence the connection, provided C does notvanish identically. In the case where C does vanish identically the connection isdetermined uniquely in terms of Œg� and ˇ.

We start by showing that there are no timelike minimal Lagrangian connectionson a compact oriented surface of negative Euler-characteristic (the reader may alsocompare this with [17, Theorem 5.4]).

Proposition 7.3. Suppose r 0 is a minimal Lagrangian connection on the compactoriented surface † satisfying �.†/ < 0. Then r 0 is spacelike.

Proof. Suppose r 0 were timelike and let g D Ric.r 0/. Then we obtainZ†

trg Ric.r 0/dAg D 2Z†

dAg D 2Area.†; g/ > 0

MINIMAL LAGRANGIAN CONNECTIONS 25

and hence Proposition 3.2 and Theorem 3.5 imply that

4��.†/ D supp2P.†/

infr2p

Z†

trg Ric.r/dAg > 0;

a contradiction. �

Without losing generality we henceforth assume that the torsion-free minimalLagrangian connection r on a compact oriented surface † with �.†/ < 0 isspacelike. We will show that the triple .g; ˇ; C / defined byr is uniquely determinedin terms of Œg� and .ˇ; C /.

Suppose .g; ˇ; C / with ˇ closed satisfy

Kg D �1C 2 jC j2g C ıgˇ:

Let g0 denote the hyperbolic metric in Œg� and write g D e2ug0, so that

e�2u.�1 ��g0u/ D �1C 2e�6ujC j2g0

C e�2uıg0ˇ:

We obtain

��g0u D 1C ıg0

ˇ � e2u C 2e�4ujC j2g0:

Omitting henceforth reference to g0 we will show:

Theorem 7.4. Let .†; g0/ be a compact hyperbolic Riemann surface. Supposeˇ 2 �1.†/ is closed and C is a cubic differential on †. Then the equation

(7.1) ��u D 1C ıˇ � e2u C 2e�4ujC j2

admits a unique solution u 2 C1.†/.

Using the Hodge decomposition theorem it follows from the closedness of ˇ thatwe may write ˇ D C dv for a real-valued function v 2 C1.†/ and a uniqueharmonic 1-form 2 �1.†/. Since is harmonic, it is co-closed, hence (7.1)becomes

�u D �1 � ıdv C e2u � 2e�4ujC j2 D �1C�v C e2u � 2e�4ujC j2:

Writing u0 WD u � v, we obtain

�u0 D �1C e2.u0Cv/� 2e�4.u

0Cv/jC j2:

Using the notation � D �e2v < 0 and � D e�4vjC j2, as well as renaming u WD u0,we see that (7.4) follows from:

Theorem 7.5. Let .†; g0/ be a compact hyperbolic Riemann surface. Suppose�; � 2 C1.†/ satisfy � < 0 and � > 0. Then the equation

(7.2) ��u D 1C �e2u C 2�e�4u

admits a unique solution u 2 C1.†/.

Remark 7.6. This theorem can also be proved using the technique of sub – andsupersolutions, see [17, Chapter 9]. Here we instead use techniques from thecalculus of variations.

26 T. METTLER

In order to prove this theorem we define an appropriate functional E�;� on theSobolev space W 1;2.†/. As usual, we say a function u 2 W 1;2.†/ is a weaksolution of (7.2) if for all � 2 C1.†/

(7.3) 0 D

Z†

�hdu; d�i C�1C �e2u C 2�e�4u

�� dA:

Note that this definition makes sense. Indeed, it follows from the Moser–Trudingerinequality that the exponential map sends the Sobolev space W 1;2.†/ into Lp.†/for every p <1, hence the right hand side of (7.3) is well defined.

Lemma 7.7. Suppose u 2 W 1;2.†/ is a critical point of the functional

E�;� W W1;2.†/! R; u 7!

1

2

Z†

jduj2 � 2u � �e2u C �e�4udA:

Then u 2 C1.†/ and u solves (7.2).

Proof. For u; v 2 W 1;2.†/ we define u;v.t/ D uC tv for t 2 R. We considerthe curve �u;v D E�;� ı u;v W R! R so that

(7.4) �u;v.t/ D1

2

Z†

jduj2 C 2thdu; dvi C t2jdvj2

� 2.uC tv/ � �e2.uCtv/ C �e�4.uCtv/dA:

The curve �u;v.t/ is differentiable in t with derivative

ddt�u;v.t/ D

Z†

hdu; dvi C t jdvj2 � v � v�e2.uCtv/ � 2v�e�4.uCtv/dA:

Note that this last expression is well-defined. Again, it follows from the Moser–Trudinger inequality that e2.uCtv/ 2 L2.†/ for all u; v 2 W 1;2.†/ and t 2R. Since W 1;2.†/ � L2.†/ it follows that ve2.uCtv/ is in L1.†/ by Hölder’sinequality and thus so is ve�4.uCtv/. In particular, assuming that u is a critical pointand setting t D 0 after differentiation gives

0 Dddt

ˇtD0

�u;v.t/ D

Z†

hdu; dvi � v � v�e2u � 2v�e�4udA:

Since C1.†/ � W 1;2.†/ it follows that u is a weak solution of (7.2). Since theright hand side of (7.2) is in Lp.†/ for all p <1, it follows from the Caldéron-Zygmund inequality that u 2 W 2;p.†/ for any p <1. Therefore, by the Sobolevembedding theorem, u is an element of the Hölder space C 1;˛.†/ for any ˛ < 1.Since the right hand side of (7.2) is Hölder continuous in u, it follows from Schaudertheory that u 2 C 2.†/, so that u is a classical solution of (7.2). Iteration of theSchauder estimates then gives that u 2 C1.†/. �

Since � > 0 we have E�;� > E�;0 where here 0 stands for the zero-function. Thefunctional E�;0 appears in the variational formulation of the equation for prescribedGauss curvature � of a metric g D e2ug0 on †. In particular, E�;0 is well-known tobe coercive and hence so is E�;� . In addition, we have:

Lemma 7.8. The functional E�;� is strictly convex on W 1;2.†/.

MINIMAL LAGRANGIAN CONNECTIONS 27

Proof. Let u; v 2 W 1;2.†/ be given. Using the notation of the previous lemma, weobserve that �u;v.t/ is twice differentiable in t with derivative

(7.5)d2

dt2�u;v.t/ D

Z†

jdvj2 � 2v2�e2.uCtv/ C 8v2�e�4.uCtv/dA:

Note again that by Sobolev embedding v2 2 L2.†/ for v 2 W 1;2.†/ and that bothe2.uCtv/ and e�4.uCtv/ are in L2.†/, hence the right hand side of the equation(7.5) is well-defined by Hölder’s inequality. In particular, computing the secondvariation gives

E 00�;� .u/Œv; v� Dd2

dt2

ˇtD0

E�;� .uC tv/

D

Z†

jdvj2dAC 2Z†

v2.4� � e6u�/e�4udA

> kdvk2L2.†/

;

where we have used that � > 0 and � < 0. Since for a non-zero constant functionv we obviously have E 00�;� .u/Œv; v� > 0 it follows that the quadratic form E 00�;� ispositive definite on W 1;2.†/. Hence, the claim is proved. �

Proof of Theorem 7.5. We have shown that E�;� is a continuous strictly convexcoercive functional on the reflexive Banach space W 1;2.†/, hence E�;� attainsa unique minimum on W 1;2.†/, see for instance [34]. Since we know that theminimum is smooth, Theorem 7.5 is proved. �

We define the area of a timelike/spacelike connection to be the area of o.†/ �.T �†; hr/. We have:

Theorem 7.9. Let r be a minimal Lagrangian connection on the compact orientedsurface † with �.†/ < 0. Then we have

Area.r/ D �2��.†/C 2kCk2g :

Proof. We have seen that the Gauss curvature of the metric o�hr D g D �Ric.r/defined by a minimal Lagrangian connection r on † satisfies

Kg D �1C 2 jC j2g C ıgˇ:

Integrating against dAg and using the Stokes and Gauss–Bonnet theorem gives

2��.†/ D �Area.r/C 2kCk2g ;

thus proving the claim. �

Remark 7.10. An obvious consequence of Theorem 7.9 is the area inequality

(7.6) Area.r/ > �2��.†/holding for minimal Lagrangian connections. Recall that if r is projectively flat,then the projective structure defined by r is properly convex. Labourie [26] associ-ated to every properly convex projective surface .†; p/ a unique minimal mappingfrom the universal cover Q† to the symmetric space SL.3;R/=SO.3/ which satisfiesthe very same area inequality, that is (7.6), see [27]. Moreover, it is shown in [27]that equality holds if and only if p is defined by the Levi-Civita connection of ahyperbolic metric.

28 T. METTLER

Definition 7.11. We call a minimal Lagrangian connection r area minimising ifr has area �2��.†/.

Remark 7.12. Theorem 7.9 shows that a minimal Lagrangian connection r is areaminimising if and only if the induced cubic differential vanishes identically. Wehave shown in Proposition 5.5 that in the projectively flat case – when ˇ vanishesidentically – this translates to r being the Levi-Civita connection of a hyperbolicmetric, a statement in agreement with [27].

Theorem 7.4 shows that the triple .g; ˇ; C / is uniquely determined in terms ofthe conformal equivalence class Œg�, the cubic differential C and the 1-form ˇ on†. Since C can locally be rescaled to be holomorphic, its zeros must be isolatedand hence ˇ is uniquely determined by C provided C does not vanish identically.Therefore, applying Corollary 5.3 shows:

Theorem A. Let † be a compact oriented surface with �.†/ < 0. Then we have:(i) there exists a one-to-one correspondence between area minimising Lag-

rangian connections on T† and pairs .Œg�; ˇ/ consisting of a conformalstructure Œg� and a closed 1-form ˇ on †;

(ii) there exists a one-to-one correspondence between non-area minimisingminimal Lagrangian connections on T† and pairs .Œg�; C / consisting ofa conformal structure Œg� and a non-trivial cubic differential C on † thatsatisfies @C D

�ˇ � i ?g ˇ

�˝ C for some closed 1-form ˇ.

7.2. Concluding remarks

Remark 7.13. We have proved that on a compact oriented surface † of negativeEuler characteristic we have a bijective correspondence between projectively flatspacelike minimal Lagrangian connections and pairs .Œg�; C / consisting of a con-formal structure Œg� and a holomorphic cubic differential C on †. By the workof Labourie [26] and Loftin [30], the latter set is also in bijective correspondencewith the properly convex projective structures on †. Since by Corollary 7.2 everyprojectively flat spacelike minimal Lagrangian connection defines a properly convexprojective structure, we conclude that every such projective structure arises from aunique projectively flat spacelike minimal Lagrangian connection.

Remark 7.14. Recall that the universal cover Q† of a properly convex projectivesurface .†; p/ is a convex subset of RP2. Pulling back the minimal Lagrangianconnection r 2 p to the universal cover gives a section of A! Q†, where now, bythe work of Libermann [28], the total space of the affine bundle A! Q† is containedin the submanifold of the para-Kähler manifold A0 � RP2 ˝ RP2� consistingof non-incident point-line pairs. In particular, we obtain a minimal Lagrangianimmersion Q†! A, recovering the result of Hildebrand [22, 23] in the case of twodimensions. Therefore, using the result of Loftin [30], one should be able to showthat every properly convex projective manifold arises from a minimal Lagrangianconnection.

Remark 7.15. The case of the 2-torus can be treated with similar techniques, exceptfor the possible occurrence of Lorentzian minimal Lagrangian connections. Thiswill be addressed elsewhere.

MINIMAL LAGRANGIAN CONNECTIONS 29

Remark 7.16. In higher dimensions, the class of totally geodesic Lagrangian con-nections contains the Levi-Civita connection of Einstein metrics of non-zero scalarcurvature. We also refer the reader to [7, 21] for a study of Einstein metrics inprojective geometry.

Remark 7.17. In [31], the author has introduced the notion of an extremal conformalstructure for a projective manifold .M; p/. In two-dimensions, the naive EinsteinAH structures of [17] appear to provide examples of projective surfaces admittingan extremal conformal structure. This may be taken up in future work.

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INSTITUT FÜR MATHEMATIK, GOETHE-UNIVERSITÄT FRANKFURT, FRANKFURT AM MAIN,GERMANY

Email address: mettler@math.uni-frankfurt.de