CUBO, A Mathematical Journal Vol.22, N◦03, (395–410). December 2020 http://dx.doi.org/10.4067/S0719-06462020000300395 Received: 31 July, 2020 | Accepted: 04 December, 2020 Toric, U(2), and LeBrun metrics Brian Weber Department of Mathematics, ShanghaiTech University, 319 Yueyang Road, Xuhui District, Shanghai, China, CN 201210. bjweber@shanghaitech.edu.cn ABSTRACT The LeBrun ansatz was designed for scalar-flat Kähler metrics with a continuous sym- metry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is Kähler, scalar-flat, or extremal Kähler. Second, toric Kähler metrics (such as the generalized Taub-NUTs) and U(2)-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the Le- Brun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples—the exceptional half-plane metric and the Page metric—in terms of the LeBrun ansatz. RESUMEN El ansatz de LeBrun fue diseñado para métricas Kähler escalares-planas con una simetŕıa continua; acá mostramos que es generalizable a clases mucho más amplias de métricas con una simetŕıa. Establecemos las condiciones para que una métrica sea (localmente) expresable con la forma de ansatz de LeBrun, las condiciones bajo las cuales su estructura compleja natural es integrable, y las condiciones que producen una métrica que es Kähler, escalar-plana, o Kähler extremal. En segundo lugar, métricas tóricas Kähler (tales como las Taub-NUT generalizadas) y métricas U(2)-invariantes (tales como la métrica de Fubini-Study o la de Page) son ciertamente expresables en el ansatz de LeBrun. Damos fórmulas generales para tales transiciones. Concluimos el art́ıculo con ejemplos, y encontramos expresiones para dos ejemplos—la métrica excep- cional del semiplano y la métrica de Page—en términos del ansatz de LeBrun. Keywords and Phrases: Differential geometry, Kähler geometry, canonical metrics, ansatz. 2020 AMS Mathematics Subject Classification: 53B21, 53B35. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000300395 396 Brian Weber CUBO 22, 3 (2020) 1 Introduction LeBrun [19] created an ansatz for scalar-flat Kähler metrics with a continuous symmetry. This was an expansion of the Gibbons-Hawking ansatz for Ricci-flat Kähler metrics with a symmetry, itself a version of the Kaluza ansatz [18] [6]. In the original construction Kaluza showed that if a Lorentzian 5-metric is endowed with a spacelike continuous symmetry, the Einstein equations will partially linearize, with the linear part being the Maxwell equations. The Gibbons-Hawking construction utilized this idea except in Euclidean signature and a dimension lower, where the Maxwell equations reduce to just the Laplace equation on a potential, and the “gravity” equations (the Ricci-flat equations) fully linearize. LeBrun’s ansatz, which also works for 4-dimensional Riemannian metrics with a circle sym- metry, partially linearizes the scalar-flat Kähler (SFK) equations. These SFK equations, normally exceedingly complicated and nonlinear, were shown to reduce to a pair of second order equations, one linear and the other quasilinear. We show that LeBrun’s ansatz is much more general than this original use, and is suitable for expressing interesting 4-metrics that are not scalar-flat, Kähler, or even have an integrable complex structure. We show the conditions under which a metric is expressible in terms of the LeBrun ansatz, and give the explicit transformations into the LeBrun ansatz from two toric Kähler ansätze, and from the U(2)-invariant ansatz. In the last section we use these translations to express several common metrics in the LeBrun ansatz. Finally we indicate how the LeBrun ansatz can be used, at least in principle, to create new metrics of special kinds, a subject we shall take up elsewhere. 2 The LeBrun ansatz We lay out the basic definitions in the LeBrun ansatz and determine when the ansatz possesses an integrable complex structure and when it possesses a closed Kähler 2-form. We end with some expressions for curvature quantities of such metrics, and state when such a metric is extremal Kähler. The reference for this section is [19]. 2.1 The ansatz The LeBrun ansatz is an S1-fibration π : M4 → N3 along with the metric g = weu ( dx2 + dy2 ) + w dz2 + w−1 (dτ + π∗A) 2 (2.1) CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 397 where (x,y,z) are local coordinates on N3, w = w(x,y,z) and u = u(x,y,z) are functions, and A is a 1-form A = Ax(x,y,z)dx + Ay(x,y,z)dy + Az(x,y,z)dz on N 3.1 The coordinate τ is defined after a choice of a transversal: after setting τ = 0 on this transversal, τ is pushed forward via the S 1-action. The field d dτ is invariant under rechoosing the transversal so it is globally defined, and it is Killing. The exterior derivative of A will be important. Because dπ∗A = π∗dA, it is immaterial whether we compute on M4 or N3. Letting B = dA we have B = Bx dy ∧ dz − By dx ∧ dz + Bz dx ∧ dy, where Bx = Ay,x − Ax,y, By = Ax,z − Az,x, Bz = Az,y − Ay,z. (2.2) In the spirit of Kaluza’s work, we may interpret A as a vector potential over 3-space and B = dA as the corresponding Maxwell field strength. It so closely resembles a magnetostatic field that we will sometimes call it the metric’s magnetic field. In all curvature computations A never appears; only its field B appears. A g-compatible almost-complex structure on (M4,g) is J(dx) = −dy, J(dz) = −w−1(dτ + π∗A), (2.3) which dualizes to J(∇x) = ∇y, J(∇z) = ∂ ∂τ (2.4) where the duality convention is J(η) , η ◦J for η ∈ ∧1 . The corresponding antisymmetric form is ω = g(J·, ·) = weudx ∧ dy + dz ∧ (dτ + π∗A) . (2.5) 2.2 The complex and symplectic structures As usual, the almost complex structure splits ∧1 C = ∧1 (M4) ⊗ C into holomorphic and antiholo- morphic bundles, where ∧1 C = ∧1,0 ⊕ ∧0,1 are the respective ± √ −1 eigenspaces of J. In bases, ∧ 1,0 = spanC { dx + √ −1dy, dz + √ −1w−1(dτ + π∗A) } , ∧ 0,1 = spanC { dx − √ −1dy, dz − √ −1w−1(dτ + π∗A) } . (2.6) Of the many ways to check the integrability of an almost-complex structure, the most conve- nient will be verifying that d : ∧0,1 → ∧1 C ∧ ∧0,1 . Lemma 2.1. The complex structure (2.3) is integrable if and only if wx = Bx and wy = By. (2.7) 1LeBrun denotes ω = dτ + π∗A, and interprets this as a connection. Following a different but very standard convention, we shall prefer using the symbol ω for the 2-form ω = g(J·, ·). 398 Brian Weber CUBO 22, 3 (2020) Proof. This comes from out of the proof of Proposition 1 of [19]. We compute on bases. Certainly d(dx − √ −1dy) = 0. Then d ( dz − √ −1w−1(dτ + π∗A) ) = w−1 ( dw ∧ ( dz − √ −1w−1(dτ + π∗A) ) − dw ∧ dz − √ −1B ) . (2.8) From (2.6), the first term is in ∧1 C ∧ ∧0,1 . The second and third terms become − dw ∧ dz − √ −1B = −(wx − √ −1By)dx ∧ dz − (wy + √ −1Bx)dy ∧ dz − √ −1Bzdx ∧ dy = 1 2 ( (wx − Bx) − √ −1(wy − By) ) dz ∧ (dx + √ −1dy) + 1 2 ( (wx + Bx)dz − √ −1(By + wy)dz − √ −1Bz(dx + √ −1dy) ) ∧ (dx − √ −1dy). (2.9) Because dx − √ −1dy ∈ ∧0,1 the second term on the right is in ∧ 1 C ∧ ∧ 0,1. But the first term is in ∧1 C ∧ ∧1,0 . We conclude J is integrable if and only if this term is zero, which is the same as (wx − Bx) − √ −1(wy − By) = 0. Lemma 2.2. We have dω = (−Bz + (weu)z) dz ∧ dx ∧ dy. In particular, the antisymmetric form ω of (2.5) is closed if and only if Bz = (we u)z. Proof. Using ω = dz ∧ (dτ + π∗A) + weudx ∧ dy and dπ∗A = π∗dA = π∗B, dω = −dz ∧ dπ∗A + (weu)zdz ∧ dx ∧ dy = (−Bz + (weu)z)dz ∧ dx ∧ dy, (2.10) from which the assertion follows. Theorem 2.1. The triple (g,J,ω) always has g(J·,J·) = g(·, ·). It is i) Hermitian if and only if Bx = wx and By = wy, ii) symplectic if and only if Bz = (we u)z, and iii) Kähler if and only if Bx = wx, By = wy, and Bz = (we u)z. Condition (iii) implies wxx + wyy + (we u)zz = 0. (2.11) CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 399 Proof. After Lemmas 2.2 and 2.1, we must only verify equation (2.11). But with B = dA, after assuming the relations in (iii) then equation (2.11) is just dB = 0. Remark. The metric is almost Kähler if (ii) holds but (i) does not. Remark. The original approach of LeBrun [19] was essentially the reverse of this. LeBrun solves (2.11) for w first, and then finds a 1-form A (which will have Dirac string singularities) whose field B satisfies (iii). This contrasts with our method which starts with a metric of the form (2.1), finds conditions on A and w that give it special traits, and from such traits derives equation (2.11). We have the following characterization of the LeBrun ansatz. Theorem 2.2. Let g be a metric on M4. Then g can be expressed locally via the LeBrun ansatz if and only if the following three conditions hold: i) M4 has a vector field v and an almost-complex structure J compatible with g so that, letting ω = g(J·, ·) be the associated antisymmetric form, then ω, g, and J are all v-invariant, ii) Given any simply connected domain Ω ⊂ M4, there is a function z : Ω → R with ivω = dz, and iii) The action of ∇z on J, when restricted to the rank-2 distribution P ⊂ ∧1 M4 that is null on span{v,Jv}, is zero. Remark. Regarding condition (iii), P is specifically the distribution P = {η ∈ ∧1 M4 such that η(v) = 0 and η(Jv) = 0}. Remark. Condition (iii) is certainly the most technical; it exists so that the first two terms in the ansatz can be written in the form f(x,y,z)(dx2 +dy2), instead of f1dx 2 +f2(dxdy+dydx)+ f3dy 2. Condition (iii) could also be written L∇z(J ∣ ∣ P ) = 0 where L is the Lie derivative. Proof. Supposing g can be expressed via the LeBrun ansatz, we simply set v = ∂ ∂t and let J be as in (2.3) or equivalently (2.4). The work above shows J and ω are v-invariant and ivω = dz. We compute L∇zJ ∣ ∣ P by (L∇zJ)(dx) = L∇z(Jdx) − JL∇zdx = L∇z(dy) − JL∇zdx. (2.12) The Cartan formula gives L∇zdx = di∇zdx = d〈dz,dx〉. But this inner product is zero, as is easily verified after computing the inverse matrix gij. Similarly L∇zdy = 0, so we have shown L∇zJ(dx) = 0. The same argument works for L∇zJ(dy), so we have shown that L∇z(J ∣ ∣ P ) = 0. For the converse we assume g, J, ω are v-invariant, and that ivω = dz for some function z. This allows us to perform a version of the Kähler reduction. Because z is itself v-invariant (due to the fact that Lvz = ivivω = 0), the function z passes to the quotient manifold N3 = M4/v 400 Brian Weber CUBO 22, 3 (2020) where the quotient is by the action of the Killing field v—this works if the orbits of v are closed; if not then a second Killing field must exist, and we can take an appropriate linear combination to find a Killing field with closed orbits. Pick a level-set Σ2z = {z = const} on which to place isothermal coordinates (x,y), and then extend (x,y) along trajectories of ∇z so the functions x, y are now defined on some region of N3. We show that (x,y) remains isothermal on all other nearby level-sets of z; this is a consequence of J|P being invariant under trajectories of ∇z. To see this, note that J|P restricts to the Hodge-star ∗2 on any level-set of z, and x, y are isothermal if and only if d ∗2 dx = d ∗2 dy = 0 and dx ∧ ∗dy = 0. By construction, d ∗2 dx = d ∗2 dy = 0 and dx ∧ ∗dy = 0 holds on one level-set of z; to see it is true on all nearby level-sets we compute L∇zd ∗2 dx = dL∇zJ|P dx = dJ|P L∇zdx = dJ|P dL∇zx = 0. (2.13) where we used the facts that d always commutes with L∇z, that by hypothesis L∇zJ|P = 0, and that by construction L∇zx = 0. Therefore d ∗2 dx remains zero on all level-sets. Similarly we compute L∇z (dx ∧ ∗2dy) = (L∇zdx) ∧ ∗2dy + dx ∧ (L∇z ∗2 dy) = dx ∧ ∗2 (L∇zdy) = 0 (2.14) where again we used L∇zdx = L∇zdy = 0 and L∇z∗2 = L∇zJ|P = 0. Now, because the functions x, y remain an isothermal system on any level-set of z, we may express the metric g3 on the quotient manifold N 3 in the form g3 = f1(x,y,z)dz 2 +f2(x,y,z) ( dx2 + dy2 ) . We define the functions w, eu by w , |dz|−2g3 = f1 weu , |dx|−2g3 = |dy| −2 g3 = f2. (2.15) The functions x and y pull back from N3 to M4, where we now have three coordinate functions x, y, and z. For the fourth coordinate τ, after choosing a transversal to v, we may set τ = 0 along this transversal, and push τ along trajectories of v—incidentally, this establishes ∂ ∂τ = v and J∇z = ∂ ∂τ . We now have coordinates (x,y,z,τ) on M4. From (2.15) we have w−1 = |dz|2 = |∇z|2 = |J∇z|2 = |∂/∂τ|2. We define functions C, Ax, Ay, and Az in terms of the complex structure J by −C (dτ + Axdx + Aydy + Azdz) = Jdz. (2.16) We can compute the value of C. Transvecting both sides of (2.16) with ∂ ∂τ gives −C = Jdz ( ∂ ∂τ ) = 〈 ∇z, J ∂ ∂τ 〉 = −|∇z|2 = −|dz|2 = −w−1. (2.17) Therefore C = w−1. Finally because the distribution {∇x,∇y} is perpendicular to the distri- bution {∇z, ∂/∂τ}, we arrive at the expression CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 401 g = weu ( dx2 + dy2 ) + wdz2 + w−1 (dτ + Axdz + Aydy + Azdz) 2 . (2.18) 2.3 Curvature quantities Proposition 2.1. Assume the metric (2.1) is Kähler, meaning (iii) of Theorem 2.1 holds. Then the Ricci curvature of g is Ric = −1 2 ( Hess u (·, ·) + Hess u (J·, J·) ) (2.19) Proof. The proof of Proposition 1 of [19] gives Ricci form and Ricci curvature ρ = − √ −1∂∂̄u, and Ric = ρ(·, J·) = −1 2 ( Hessu (·, ·) + Hessu (J·, J·) ) . (2.20) Proposition 2.2. Assume the metric (2.1) is Kähler, meaning (iii) of Theorem 2.1 holds. Then the scalar curvature s of g is s = − 1 weu (uxx + uyy + (e u)zz) . (2.21) Proof. This is computed in the proof of Proposition 1 of [19]. Proposition 2.3 (The extremal condition). Assume the metric (2.1) is Kähler. Then it is an extremal Kähler metric if constants m,b ∈ R exist so − 1 weu (uxx + uyy + (e u)zz) = mz + b. (2.22) Proof. If (2.22) holds then s = mz + b and so ∇s = m∇z and J∇s = m ∂ ∂τ ; thus J∇s is a Killing field. The proposition is established after recalling that a Kähler metric is extremal if and only if J∇s is Killing [7] [8]. Remark. Whether g is Kähler or not, its scalar curvature is s = − 1 weu ( ( uxx + uyy + (e u)zz ) + 1 w ( wxx + wyy + (we u)zz ) + 1 2w2 (B2x − (wx)2) + 1 2w2 (B2y − (wy)2) + e−u 2w2 (B2z − ( (weu)z )2 ) . (2.23) 402 Brian Weber CUBO 22, 3 (2020) 3 Expressing Toric Kähler metrics using the LeBrun ansatz The LeBrun ansatz operates on 4-manifolds with one symmetry. On Kähler 4-manifolds with two holomorphic symmetries, there are more specialized ansätze. Letting X 1, X 2 be commuting holomorphic Killing fields (recall that “holomorphic” means LX iJ = 0, just as Killing means LX ig = 0), then (M4,g,J,X 1,X 2) can be considered a toric Kähler 4-manifold. This situation has been studied in [17] [1] [13] [14] [2] [9] and many other works. Certainly a toric Kähler metric can be translated into the LeBrun ansatz once a distinguished Killing field is chosen. We do this here. 3.1 The two toric ansätze There are two standard presentations for toric Kähler 4-manifolds. These were originally explored by Guillemin [17], who also discovered that they are equivalent via a Legendre transform. The LeBrun ansatz is a mixture of the two. The first of the two presentations is the symplectic ansatz. If {X 1,X 2} are independent commuting holomorphic Killing fields, we can use the Arnold-Liouville construction [3] to produce the so-called action-angle coordinates on M4. To execute this construction, one defines action variables (up to a constant) by ∇ϕi = −JX i or equivalently by dϕi = iX iω, and defines angle variables, denoted θ1, θ2, by choosing a transversal and then pushing forward the action of the fields X 1, X 2. In these coordinates, the ansatz demands the metric be expressed g = Uijdϕ i ⊗ dϕj + Uijdθi ⊗ dθj (3.1) where U = U(ϕ1,ϕ2) is a convex function of the action variables. The matrix (Uij) is defined by Uij , ∂ 2 U ∂ϕiϕj , and we define (Uij) , (Uij) −1. The map M4 → R2 given by p 7→ (ϕ1(p),ϕ2(p)) sends M4 to a region Σ2 ⊂ R2; this is sometimes called the Arnold-Liouville reduction or, by abuse of terminology, the moment map. If M4 is compact then its image Σ2 is a compact polygon in R2. This polygon encodes the topology of M4, via the Delzant gluing rules [11]. If M4 is non-compact, then Σ2 need not be a polygon nor even be topologically closed. The second ansatz, the holomorphic ansatz, also begins with the fields {X 1,X 2}. Again we may produce corresponding coordinates θ1, θ2 after choosing a transversal. Because X 1, X 2 are not only symplectomorphic but holomorphic, the variables θi are actually pluriharmonic, meaning d(Jdθi) = 0. The Poincaré lemma then guarantees functions ξ1, ξ2 exist (at least locally) so that dξi = Jdθi, and we have two holomorphic functions fi = ξi+ √ −1θi which constitute a holomorphic chart (f1,f2) : Ω → C2 on some subdomain Ω ⊆ M4. The Kähler form on this chart, as usual, can be expressed ω = √ −1∂∂̄V for some pseudoconvex function V . Because V is θ1-θ2 invariant, it is CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 403 convex instead of just pseudoconvex. The metric is then g = V ijdξi ⊗ dξi + V ijdθi ⊗ dθj (3.2) where (V ij) is the matrix with components V ij , ∂ 2 V ∂ξi∂ξj . We might consider the map p 7→ (ξ1(p), ξ2(p)) for p ∈ M4, just as we considered the moment map p 7→ (ϕ1(p),ϕ2(p)). But it is much less interesting than the moment map. If M4 is compact then its image is all of R2. In particular there is no way to read off the topology of M4 from its image. A duality relationship exists between the symplectic system (ϕ1,θ1,ϕ 2,θ2) with its symplectic potential U and the holomorphic system (ξ1,θ1,ξ2,θ2) with its Kähler potential V . As shown in [17], they are Legendre transforms of each other: ξi = ∂U ∂ϕi , ϕi = ∂V ∂ξi , and U(ϕi) + V (ξi) = ∑ i ϕiξi. (3.3) 3.2 Translation to the LeBrun Ansatz It is now possible to relate these two systems to the LeBrun ansatz, which is a mixed symplectic- holomorphic system. We define the LeBrun variable τ to be the angle variable θ1 corresponding to X 1, and y the angle variable θ2 corresponding to X 2. Let z be the symplectic variable corresponding to the angle τ, meaning z = ϕ1, and x the holomorphic variable corresponding the angle variable y, meaning x = ξ2. Then we create the LeBrun functions w and u, and determine the 1-form A. We record the change of frame from the symplectic frame { ∂ ∂ϕ1 , ∂ ∂θ1 , ∂ ∂ϕ2 , ∂ ∂θ2 } to the LeBrun frame { ∂ ∂z , ∂ ∂τ , ∂ ∂x , ∂ ∂y } . One easily computes ∂ ∂ϕ1 = ∂ ∂z + U21 ∂ ∂x dϕ1 = dz ∂ ∂θ1 = ∂ ∂τ dθ1 = dτ ∂ ∂ϕ2 = U22 ∂ ∂x dϕ2 = −U21 U22 dz + 1 U22 dx ∂ ∂θ1 = ∂ ∂y dθ2 = dy. (3.4) Upon substituting the symplectic frame components into the LeBrun metric (2.1), we find the functions w, u and the components Ax, Ay, and Az to be w = 1/U11, u = log ( U11U22 − (U12)2 ) Ax = 0, Ay = U12 U11 , Az = 0. (3.5) We express this in the form of a proposition. 404 Brian Weber CUBO 22, 3 (2020) Proposition 3.1. Assume (M4,J,g,X 1,X 2) is a toric Kähler manifold. Let (ϕ1,θ1,ϕ2,θ2) be symplectic coordinates and (ξ1,θ1,ξ2,θ2) holomorphic coordinates on M 4. There exists a convex function U(ϕ1,ϕ2) on Σ2, where Σ2 is the image of the moment map (ϕ1,ϕ2) : M4 → R2, so that g = Uijdϕ i ⊗ dϕj + Uijdθi ⊗ dθj (3.6) where Uij = ∂ 2 U ∂ϕiϕj and (Uij) = (Uij) −1. There also exists a convex function V = V (ξ1,ξ2) on R 2 so that g = V ijdξi ⊗ dξj + V ijdθi ⊗ dθj (3.7) where V ij = ∂ 2 V ∂ξiξj . These systems are related via the Legendre transform: ϕi = ∂V ∂ξi , ξi = ∂U ∂ϕi , U(ϕ1,ϕ2) + V (ξ1,ξ2) = ϕ 1ξ1 + ϕ 2ξ2. (3.8) The metric (M4,g,J,X 1,X 2) can be expressed in the LeBrun ansatz after setting ( z, τ, x, y ) = ( ϕ1, θ1, ξ2, θ2 ) . (3.9) A LeBrun ansatz expression of g is obtained by setting u = log detUij = log ( U11U22 − (U12)2 ) , w = 1 U11 , and A = Aydy = U12 U11 dy (3.10) (the components Ax and Az are zero). The components of the magnetic 2-form are Bx = −Ay,z, By = 0, and Bz = Ay,x. 3.3 Variation of LeBrun structures In our construction of Section 3.2 we began by setting τ = θ1, but we could have chosen τ = θ2 or indeed any linear combination of the cyclic variables. Up to scale a toric metric automatically has a 1-parameter family of distinct LeBrun structures. If α ∈ [0,π/2] is a constant and X 1, X 2 are symplectomorphic Killing fields, then for each α we may select the field X = cos(α)X 1 + sin(α)X 2. (3.11) Then, referring to the construction of Section 3.2, the corresponding angle variable is τ = cos(α)θ1 +sin(α)θ2 with conjugate momentum variable z = cos(α)ϕ 1 +sin(α)ϕ2. The holomorphic variables are then x = − sin(α)ξ1 + cos(α)ξ2 and y = − sin(α)θ1 + cos(α)θ2. This allows for a “tuning” or selection of a distinguished 1-parameter symmetry field form which the LeBrun ansatz metric can be constructed. The variable y remains cyclic (that is, its field CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 405 remains a symmetry direction), and u, w will remain functions of x and z. These functions will change with α, so we may write u = uα(x,z) and w = wα(x,z). We remark that a third auxiliary function u̇α , d dα uα exists. If the uα solve the LeBrun equation (uα)xx + (e uα)zz = 0 then u̇α will solve the linearized equation (u̇α)xx + (u̇αe uα)zz = 0. Under some conditions uα will be positive, and setting w = u̇α we have an entirely new LeBrun metric. 4 Expressing U(2)-invariant metrics in the LeBrun ansatz The usual ansatz for U(2)-invariant metrics is g = Adr2 + B (η1) 2 + C ( (η2) 2 + (η3) 2 ) (4.1) where {η1,η2,η3} is a standard left-invariant coframe on S3, and A, B, C are functions of the radial variable r. If (ψ,ϕ,θ) are Euler coordinates on on S3, the usual frame transitions are η1 = 1 2 (dψ + cos(θ)dϕ) η2 = 1 2 (sin(θ) cos(ψ)dϕ − sin(ψ)dθ) η3 = 1 2 (sin(θ) sin(ψ)dϕ + cos(ψ)dθ) . (4.2) From this we deduce (η2) 2 + (η3) 2 = 1 4 ( dθ2 + sin2(θ)dϕ2 ) , so in Euler coordinates g = Adr2 + B 4 (dψ + cos(θ)dϕ) 2 + C 4 ( dθ2 + sin2(θ)dϕ2 ) (4.3) This is already close to LeBrun ansatz form. To place it precisely in LeBrun ansatz form we make the change of variables x = log cot θ 2 , y = ϕ, z = 1 2 ∫ √ AB dr, τ = ψ. (4.4) This gives dθ2 + sin2(θ)dϕ2 = sech2(x)(dx2 + dy2), and the metric now reads g = 4 B dz2 + B 4 (dτ + tanh(x)dy) 2 + C 4 sech2(x) ( dx2 + dy2 ) . (4.5) Reading off the LeBrun ansatz quantities from (2.1), we have w = 4 B , u = log ( BC 16 sech2(x) ) Ax = 0, Ay = tanh(x), Az = 0 (4.6) where B and C are now functions of the new variable z, via the transition from r to z given in (4.4). Because U(2) has a rank 2 toral subgroup, any U(2)-invariant metric is also T2-invariant— if the metric is Kähler then it is toric. One can see directly that the metric (4.5) has no τ- or y-dependency so has T2 symmetry. 406 Brian Weber CUBO 22, 3 (2020) 5 Examples We give two examples of our method. The exceptional half-plane metric from [21] was originally written in a toric ansatz, and the Page metric on CP2♯CP2 was originally written in the U(2) ansatz. We use our methods to express both in the LeBrun ansatz. In the last section we outline methods for creating new metrics that are Einstein, half-conformally flat, or Bach-flat. 5.1 The exceptional half-plane metric on C2. This toric SFK metric on C2 appears in [21]. It has one translational and one rotational field. In rectangular coordinates (x1,y1,x2,y2) on C 2, these fields are X 1 = ∂ ∂y1 and X 2 = −y2 ∂∂x2 +x2 ∂ ∂y2 , which are clearly translational and rotational, respectively. Let U = U(ϕ1,ϕ2) be the symplectic potential U = 1 2 ( (ϕ2)2 1 + 2Mϕ1 + ϕ1 log(ϕ1) + M(ϕ1)2 ) (5.1) where M ≥ 0 is a constant. The case M = 0 produces the flat metric. When M > 0, the resulting metric is the exceptional half-plane metric; the fact that (5.1) is the correct symplectic potential for the exceptional half-plane metric can be verified directly from equations (6-1) and (6-3) of [21]. The Kähler potential V is difficult to write explicitly, as it involves inverting a function with transcendental and algebraic parts. However it is possible to find LeBrun coordinates, which in terms of the symplectic coordinates are x = ϕ2 1 + 2Mϕ1 , y = θ2, z = ϕ 1, τ = θ1. (5.2) The LeBrun functions w and u are w = M + 1 2z , u = log (2z) (5.3) and the vector potential and field strength are A = 2Mxdy, which is Ax = 0, Ay = 2Mx, Az = 0, B = 2Mdx ∧ dy, which is Bx = 0, By = 0, Bz = 2M. (5.4) We notice that u = log(2z) gives what LeBrun called the hyperbolic ansatz in section 4 of [19]. If M = 0 this is the flat metric, which LeBrun wrote down on p. 233 of [19] (unfortunately LeBrun’s equations are mostly unnumbered). The exceptional half-plane metric in LeBrun ansatz form is g = (1 + 2Mz)(dx2 + dy2) + 1 + 2Mz 2z dz2 + 2z 1 + 2Mz (dτ + 2Mxdy)2. (5.5) CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 407 5.2 The Page metric The Page metric was originally developed in [20], and can be found explicitly in (3.25) of [16] (unfortunately its expression in the appendix of [15] has a typo). Methods for building Ricci-flat metrics, including the Page metric, can be found [4]; see also 9.125 of [5]. This metric exists on CP2♯CP 2; it is Einstein, Hermitian, and Bach-flat, but not half-conformally flat. It is conformal to an extremal Kähler metric, which Calabi [7] [8] independently wrote down; see [10] for the specific conformal transformation, or [12] for a more general theory of conformal transformations between extremal Kähler and Einstein metrics on 4-manifolds. From [16] the Page metric is g = 3(1 + ν2) Λ [ 1 − ν cos2(r) 3 − ν2 − ν2(1 + ν2) cos2(r) dr2+ + 3 − ν2 − ν2(1 + ν2) cos2(r) (3 + ν2)2(1 − ν cos2(r)) sin2(r)η21 + 4 1 − ν2 cos2(r) 3 + 6ν2 − ν4 ( η22 + η 2 3 ) ] . (5.6) The method of Section 4 gives its expression in the LeBrun ansatz: g = weu ( dx2 + dy2 ) + wdz2 + 1 w (dτ + tanh(x)dy) 2 , where w = F(z) G(z) and weu = 1 3Λ(1 + ν2)(3 + 6ν2 − ν4) H(z) sech2(x) (5.7) and F , G, H are the polynomials F(z) = 27(1 + ν2 − ν4 − ν6) + 36(4ν2 + 4ν4 + ν6)Λz − 12(9ν2 + 6ν4 + ν6)Λ2z2 G(z) = 27(1 + ν2 − ν4 − ν6) + 3(−9 + 9ν2 + 11ν4 + 15ν6)Λz − 24(3ν2 + 3ν4 − ν6)Λ2z2 + 4(9ν2 + 6ν3 + ν6)Λ3z3 H(z) = 9(1 + ν2 − ν4 − ν6) + 12(3ν2 + 16ν4 + ν6)Λz − 4(9ν2 + 6ν4 + ν6)Λ2z2. (5.8) The domain for (x,z) is x ∈ R and z ∈ [ 0, 3(1+ν2) Λ(3+ν2) ] . 5.3 New metrics Creation of special metrics, namely Einstein, half-conformally flat, or Bach-flat metrics are of considerable importance in differential geometry. One may regard the metric g, if expressed in the LeBrun ansatz, as a dynamic variable with five unknowns w, u, Bx, By, Bz which are each functions of the coordinates (x,y,z). These values can be specified independently, subject to the single requirement that Bx,x +By,y +Bz,z = 0 which is equivalent to the definition of B from (2.2), 408 Brian Weber CUBO 22, 3 (2020) which is that B = dA for a 1-form A. In a sense, there are four completely independent variables that may be chosen, with the choice of a fifth being partially constrained. Letting W+ be the self-dual part of the Weyl tensor, one might consider the condition W+ = 0. Because the operator W+ : ∧+ → ∧+ has three eigenvalues which are subject to the condition that they sum to zero, the condition W+ = 0 imposes two differential identities on our five variables. With the fifth constraint discussed above, we arrive at an underdetermined system, which surely has a large solutions space. There remain many obstacles, both technical and theoretical, to fully understanding this system. Similar comments hold for systems like Rı ◦ c = 0 and B = 0 where Rı ◦ c is the trace-free Ricci tensor and B is the Bach tensor. This subject will be taken up elsewhere. CUBO 22, 3 (2020) Toric, U(2), and LeBrun metrics 409 References [1] M. Abreu, “Kähler geometry of toric varieties and extremal metrics”, International Journal of Mathematics, vol. 9, pp. 641–651, 1998. [2] M. Abreu and R. Sena-Dias, “Scalar-flat Kähler metrics on non-compact symplectic toric 4- manifolds”, Annals of Global Analysis and Geometry, vol. 41, no. 2, pp. 209–239, 2012. [3] V. Arnold, Mathematical methods of classical mechanics, Springer Science & Business Media, vol. 60, 2013. [4] L. Bérard-Bergery: “Sur de nouvelles variétés riemanniennes d’Einstein”, Publications de l’Institut Élie Cartan, vol. 6, 1982. [5] A. Besse: Einstein manifolds. Springer Science & Business Media, 2007. [6] J. Bourguignon, “A mathematician’s visit to Kaluza-Klein theory”, Presented at Conference on Differential Geometry and Partial Differential Equations, Torino, Italy, Rend. Semin. Mat. Torino Fasc., pp. 143-163, 1989. [7] E. Calabi: Extremal Kähler metrics. In Seminar on differential geometry, Princeton University Press, vol. 102, pp. 259–290, 1982. [8] E. Calabi, Extremal Kähler metrics II. In Differential geometry and complex analysis, Springer, Berlin, Heidelberg, pp. 95-114, 1985. [9] D. Calderbank, L. David, and P. Gauduchon, “The Guillemin formula and Kähler metrics on toric symplectic manifolds”, Journal of Symplectic Geometry, vol. 4, no. 1, pp. 767–784, 2002. [10] T. Chave and G. Valent, “Compact extremal versus compact Einstein metrics”, Classical and Quantum Gravity, vol. 13, no. 8, pp. 2097–2108, 1996. [11] T. Delzant, “Hamiltoniens périodiques et images convexes de l’application moment”, Bulletin de la Société Mathématique de France, vol. 116, pp. 315–339, 1988. [12] A. Derdzinski, “Self-dual Kähler manifolds and Einstein manifolds of dimension four”, Com- positio Mathematica, vol. 49, no. 3, pp. 405–433, 1983. [13] S. Donaldson, “A generalized Joyce construction for a family of nonlinear partial differential equations”, Journal of Gökova Geometry/Topology Conferences, vol. 3, 2009. [14] S. Donaldson, “Constant scalar curvature metrics on toric surfaces”, Geometric and Functional Analysis, vol. 19, no. 1, pp. 83–136, 2009. 410 Brian Weber CUBO 22, 3 (2020) [15] T. Eguchi, P. Gilkey, and A. Hanson. “Gravitation, gauge theories and differential geometry”, Physics reports, vol. 66, no. 6, pp. 213–393, 1980. [16] G. Gibbons and S Hawking, “Classification of gravitational instanton symmetries”, Commu- nications in Mathematical Physics, vol. 66, no. 3, pp. 291–310, 1979. [17] V. Guillemin, “Kähler structures on toric varieties”, Journal of Differential Geometry, vol. 40, pp. 285–309, 1994. [18] T. Kaluza, “Zum Unitatsproblem der Physik”, Sitzungsber. d. Berl. Akad., pp. 966–972, 1921. [19] C. Lebrun, “Explicit self-dual metrics on CP2# · · · #CP2”, Journal of Differential Geometry, vol. 34, no. 1, pp. 223–253, 1991. [20] D. Page. “A compact rotating gravitational instanton”, Physics Letters B, vol. 79, no. 3, pp. 235–238, 1978. [21] B. Weber, “Generalized Kähler Taub-NUT metrics and two exceptional instantons”, arXiv:1602.06178 (to appear in Communications in Analysis and Geometry). Introduction The LeBrun ansatz The ansatz The complex and symplectic structures Curvature quantities Expressing Toric Kähler metrics using the LeBrun ansatz The two toric ansätze Translation to the LeBrun Ansatz Variation of LeBrun structures Expressing U(2)-invariant metrics in the LeBrun ansatz Examples The exceptional half-plane metric on C2. The Page metric New metrics