j. differential geometry

56 (2000) 189-234

THE SPACE OF KÄHLER METRICS

XIUXIONG CHEN

Abstract

Donaldson conjectured [16] that the space of Kähler metrics is geodesically convex by smooth geodesics and that it is a metric space. Following Donaldson’s program, we verify the second part of Donaldson’s conjecture completely and verify his first part partially. We also prove that the constant scalar curvature metric is unique in each Kähler class if the first Chern class is either strictly negative or 0. Furthermore, if C_{1} ≤ 0, the constant scalar curvature metrics: realizes the global minimum of the Mabuchi K energy functional; thus it provides a new obstruction for the existence of constant curvature metrics: if the infimum of the K energy (taken over all metrics in a fixed Kähler class) is not bounded from below, then there does not exist a constant curvature metric. This extends the work of Mabuchi and Bando [3]: they showed that K energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class.

1. Introduction to the problem

1.1 Brief introduction to the classical problems in Kähler geometry

Let V be a Kähler manifold. E. Calabi conjectured in 1954 that any (1,1) form which represents C_{1}(V) (the first Chern class) is the Ricci form of some Kähler metric on V. Yau [44], in 1976, proved this Calabi conjecture. Aubin [1] and Yau proved independently the existence of a Kähler-Einstein metric on a Kähler manifold with negative first Chern class (also a conjecture of E. Calabi). G. Tian [38], in 1987, proved the existence of Kähler-Einstein metric in a canonical Kähler class on complex surfaces if the first Chern class is positive and the group of

Received April 16, 1999, and, in revised form, August 27, 1999. Research was supported partially by NSF postdoctoral fellowship.

automorphisms is reductive. For further references on this subject, see [39] and [40]. An important conjecture by Yau [45] relates the existence of Kähler-Einstein metrics to stability in the sense of Hilbert schemes and geometric invariant theory.

Kähler-Einstein metrics can be treated as a special kind of extremal Kähler metric. The question of extremal Kähler metrics was first raised by E. Calabi in his paper [9]: he considered the L^{2} norm of curvature as a functional from a given Kähler class; a critical point of this functional is called an “extremal Kähler metric.” He showed that any extremal Kähler metric must be symmetric under a maximal compact subgroup of the holomorphic transformation group. Using this structure theorem of Calabi, Marc Levine [30] was able to construct a Kähler surface on which there is no extremal Kähler metric. In 1992, D. Burns and P. de Bartolomeis [8] also produced an example of non-existence of extremal Kähler metrics; their example suggests some new obstruction for the existence of extremal metrics which is related to some borderline semistability of hermitian vector bundles. LeBrun [26] also demonstrated that the existence of critical Kähler metrics might be tied up with the stability of corresponding vector bundles. Donaldson [16] thought that Yau’s conjecture [45] should extend over to the general extremal Kähler metrics. For further references on the subject of extremal metrics, please; see [27], [26], [19] and references therein.

Futaki [18] in 1983 introduced an analytic invariant for any Kähler manifold with positive first Chern class. The vanishing of this invariant is a necessary condition for the existence of a Kähler-Einstein metric on the manifold. Later, Futaki and Calabi [10] generalized the invariant to any compact Kähler manifold. This generalized Futaki invariant, i.e., Calabi-Futaki invariant, is an analytic obstruction to the existence of a constant scalar curvature metric on a Kähler manifold. In the same paper, Calabi also showed that constant scalar curvature metrics and extremal Kähler metrics with non-constant scalar curvature do not coexist in a single Kähler class.

For uniqueness, the known results are as follows:

1) in the 1950s, E. Calabi showed the uniqueness of Kähler-Einstein metric if C_{1} ≤ 0.

2) in 1987, Mabuchi and S. Bando [3] showed the uniqueness of Kähler-Einstein metric up to holomorphic transformation if the first Chern class is positive. Recently, Tian and X.H. Zhu [43] proved the uniqueness of Kähler-Ricci Soliton with respect to a fixed holomorphic vector field on any Kähler manifold with positive first Chern class. Although very little was known about the uniqueness of general extremal Kähler metrics, most experts in Kähler geometry expect that the extremal Kähler metric is unique in each Kähler class up to holomorphic transformation. In [12] (also see [11] for further references), we demonstrated two degenerate extremal Kähler metrics in the same Kähler class with different energy levels and different symmetry groups: one example is due to Calabi, the other is due to the author. To my knowledge, it appears that this is the only non-uniqueness example known today.

Main results. Mabuchi ([31])^{1} in 1987 defined a Riemannian metric on the space of Kähler metrics, under which it becomes (formally) a non-positive curved infinite dimensional symmetric space. Apparently unaware of Mabuchi’s work, Semmes [35] and Donaldson [16] rediscover this same metric again from different angles. In [35], S. Semmes first pointed out that the geodesic equation is a homogeneous complex Monge-Ampère equation on a manifold of one dimension higher. In [16], Donaldson further conjectured that the space is geodesically convex and is a genuine metric space. We prove that it is at least convex by C^{1,1} geodesics,^{2} and from which we conclude that the space is indeed a metric space, thus verifying the second part of Donaldson’s conjecture. Moreover, this C^{1,1} geodesic realizes the absolute minimum of length over all possible paths connecting the two end points; thus the metric aforementioned is a genuine one. Using these results, we are able to show that the constant curvature metric is unique in each Kähler class if C_{1} < 0 or C_{1} = 0. Furthermore, if C_{1} ≤ 0, we show that constant scalar metric (if it exists) realizes the global minimum of K energy, which gives an affirmative answer to a question raised by Gang Tian [42] in this special case. This last statement also extends the work of Mabuchi and Bando [3]: they showed that K energy bounded from below is a necessary condition for the existence of Kähler-Einstein metrics in the first Chern class. Tian [40] showed that in Kähler manifold with positive first Chern class and no non-trivial holomorphic fields, the Kähler-Einstein metric exists if and only if the K energy is proper (he actually uses an equivalent functional instead of the K energy).^{3} One would like to ask: Is this still true for constant scalar curvature metrics?^{4}

1Around the same time of Mabuchi’s work, J. P. Bourguignon has worked on something similar in a related subject [6].

2Here we mean the mixed second derivatives are uniformly bounded. See Theorem 3 in Section 3 for details.

3The sufficient part of this result was proved in [14].

Organization. In Section 2, we first summarize the different approaches taken by Mabuchi, Semmes and Donaldson in the space of Kähler metrics; then we introduce the Riemannian metric on this infinite dimensional space and prove that it has non-positive sectional curvature in the formal sense. Then we introduce Donaldson’s two conjectures and reduce the first conjecture to the existence problem for the complex homogeneous Monge-Ampère equation with Dirichlet boundary data. Readers are alerted that material in Sections 2.3-2.5 is essentially a representation of Donaldson’s work [16], included here for the convenience of readers. In Section 3, we prove that this geodesic (CHMA) equation always has a C^{1,1} solution. In Section 4, we prove that a continuous solution to the geodesic (CHMA) equation in some appropriate weak sense is unique. In Section 5, we show that the geodesic distance defined by the length of C^{1,1} geodesics satisfies the triangle inequality. Using this, we prove the space of Kähler metrics is a metric space. In Section 6 we show that the extremal Kähler metric is unique in each Kähler class if either C_{1}(V) < 0 or C_{1}(V) = 0.

Acknowledgments. The author is very grateful to Simon Donaldson who introduced him to this problem. He is also grateful for the constant encouragement and support of E. Calabi, L. Simon and R. Schoen during this work. He also wants to thank Professor E. Stein for his help in Soblev functions and embedding theorems. The author wishes to thank S. Semmes for pointing out some important references in this problem. Thanks also go to W.Y. Ding and his colleagues, and P. Guan for pointing out some errors in an earlier version of this paper.

2. Space of Kähler metrics

2.1 Mabuchi and S. Semmes’ ideas

Shortly after introducing the K energy, Mabuchi [31] defined a Riemannian metric on the space of Kähler metrics. Besides showing formally it is a locally symmetric space with non-positive sectional curvature, he also pointed out that the K energy is formally convex in this infinite dimensional space (in the sense that the Hessian is semi-positive definite).

4Tian informed us that he [41] has conjectured that constant scalar curvature metrics exist if and only if the K energy is proper.

Perhaps, this is his original motivation for introducing such a metric. Unaware of Mabuchi’s work, in [35], S. Semmes studied the geometry of solutions of the complex Homogeneous Monge-Ampère equation (CHMA). He observed that in some special domain Ω×D where Ω is an n-dimensional domain in C^{n} and D is a domain in the complex plane, the solution to CHMA is some sort of geodesic equation if the data is rotationally symmetric when restricted to D. He then considered the space of pluri-subharmonic functions in Ω and defined a Riemannian metric in this space according to this geodesic equation. It turns out that this space becomes a non-positively curved (locally) symmetric space in some formal sense. Unlike the real homogeneous Monge-Ampère equation (RHMA) whose solution always has proper geometric meaning, the solution of a CHMA equation doesn’t have a preferred geometric interpretation. Without a proper geometry interpretation, it is very hard to work on this subject. Of course, great progress has been made since the famous work of L. Caffarelli, L. Nirenberg and J. Spruck [24] and later their joint work with J. Kohn [23], e.g., L. Lempert [29], E. Bedford and B.A. Taylor [4]; P. Lelong [28] and important work of Krylov [22] and Evans [25]. This is by no means a complete list of papers on complex Monge-Ampère equations since the author is quite new to this important field. For a complete and updated list of references, please; see S. Kolodziej [21]. Donaldson’s recent work certainly makes Mabuchi and Semmes’s original work more interesting.

2.2 Brief summary of Donaldson’s theory on space of Kähler metrics

S. K. Donaldson [16] outlined a strategy to relate this geometry of infinite dimensional space to the existence problems in Kähler geometry. In particular, he explains how one can use this extra structure on the infinite dimensional space to solve the problem of the existence and uniqueness of extremal Kähler metrics. In general, the latter are intractable problems from traditional means. He regards the space of Kähler metrics in a fixed Kähler class as an infinite dimensional symplectic manifold with the automorphism group SDiff(V) (symplectic diffeomorphism group of V into itself). In [15], he pointed out that scalar curvature is the moment map μ from this infinite dimensional symplectic manifold to the dual space of the Lie algebra of its automorphism group^{5} . Thus, to find an extremal Kähler metric in a fixed Kähler class in classical Kähler geometry could be re-interpreted as to find a pre-image of 0 of the moment map μ in this symplectic setting. This acute observation sheds new light into the otherwise intractable problem of the existence of extremal Kähler metrics on a Kähler manifold; at least conceptually, the picture looks much clearer. He then proposed several conjectures whose ultimate resolution will lead to a better understanding of extremal Kähler metrics, and for that matter, better understanding of Kähler geometry as well. The most fundamental one among his conjectures is the so called geodesic conjecture: any two Kähler metrics in the same class are connected by a smooth geodesic. A second conjecture by him is that this space of Kähler metrics is a metric space under this metric. If the geodesic conjecture is true, this second conjecture will be a direct consequence (since this space of Kähler metrics in a fixed Kähler class is non-positively curved in the formal sense.). He went on to show that the uniqueness of extremal Kähler metrics is a consequence of this geodesic conjecture as well.

5The moment map point of view here was also observed by A. Fujiki [17].

2.3 Riemannian metrics on the infinite dimensional space.

Consider the space of Kähler potentials in a fixed Kähler class as:

H = \{φ ∈ C^{∞}(V) : ω_{φ} = ω_{0} + \sqrt{−1}∂\overline{∂}φ > 0 on V \}.

We now introduce an L^{2} metric in this space (cf. Mabuchi [31]). Clearly, the tangent space T H is C^{∞}(V). Each Kähler potential ϕ ∈ H defines a measure d μ_{ϕ} = \frac{1}{n!}ω_{ϕ}^{n}. Now we define a Riemannian metric on the infinite dimensional manifold H using the L^{2} norm provided by this measure. We define the length of any vector ψ ∈ T_{φ}H as

∥ψ∥_{φ}^{2} = ∫_{V} ψ^{2} d μ_{φ}.

For a path φ(t) ∈ H(0 ≤ t ≤ 1), the length is given by

∫_{0}^{1} \sqrt{∫_{V} φ′(t)^{2}d μ_{φ(t)}} d t

and the geodesic equation is

φ(t)′′ − \frac{1}{2}|∇φ′(t)|_{φ(t)}^{2} = 0, (1)

where the derivative and norm in the 2nd term of the left-hand side are taken with respect to the metric ω_{φ(t)}.

This geodesic equation shows us how to define a connection on the tangent bundle of H. The notation is simplest if one thinks of such a connection as a way of differentiating vector fields along paths. Thus, if ϕ(t) is any path in H and ψ(t) is a field of tangent vectors along the path (that is, a function on V ×[0, 1]), we define the covariant derivative along the path to be

D_{t}ψ = \frac{∂ψ}{∂t} − \frac{1}{2}(∇ψ, ∇ϕ′)_{ϕ}.

This connection is torsion-free because in the canonical “coordinate chart”, which represents H as an open subset of C^{∞}(V), the “Christoffel symbol”

Γ : C^{∞}(V) × C^{∞}(V) → C^{∞}(V)

at ϕ is just

Γ(ψ_{1}, ψ_{2}) = −\frac{1}{2}(∇ψ_{1}, ∇ψ_{2})_{ϕ}

which is symmetric in ψ_{1}, ψ_{2}. The connection is metric-compatible because

\frac{1}{2} \frac{d}{dt}∥ψ∥_{ϕ}^{2} = \frac{d}{dt} ∫_{V} ψ^{2}dμ_{ϕ}

= ∫_{V} \frac{∂ψ}{∂t} ψ + \frac{1}{2}ψ^{2}Δ(ϕ′)dμ_{ϕ}

= ∫_{V} \frac{∂ψ}{∂t} ψ − \frac{1}{4}(∇(ψ^{2}), ∇ϕ′)_{ϕ} dμ_{ϕ}

= ∫_{V} (\frac{∂ψ}{∂t} − \frac{1}{2}(∇ψ, ∇ϕ′)_{ϕ} ) ψdμ_{ϕ}

= ⟨D_{t}ψ, ψ⟩.

Here △ is a complex Laplacian operator. The main theorem proved in [31] (and later reproved in [35] and [16]) is:

Theorem A. The Riemannian manifold H is an infinite dimensional symmetric space; it admits a Levi-Civita connection whose curvature is covariant constant. At a point ϕ ∈ H the curvature is given by

R_{ϕ}(δ_{1}ϕ, δ_{2}ϕ)δ_{3}ϕ = −\frac{1}{4}\{\{δ_{1}ϕ, δ_{2}ϕ\}_{ϕ}, δ_{3}ϕ\}_{ϕ},

where { , }_{ϕ} is the Poisson bracket on C^{∞}(V) of the symplectic form ω_{ϕ}; and δ_{1}ϕ, δ_{2}ϕ ∈ T_{ϕ}H.

Recall that in infinite dimensions the usual argument gives the uniqueness of a Levi-Civita [i.e., torsion-free, metric-compatible] connection, but not the existence in general. The formula for the curvature of H entails that the sectional curvature is non-positive, given by

K_{ϕ}(δ_{1}ϕ, δ_{2}ϕ) = −\frac{1}{4}∥\{δ_{1}ϕ, δ_{2}ϕ\}_{ϕ}∥_{ϕ}^{2}.

Different proofs of this theorem have appeared in [31], [35] and [16]. We will skip the proof here; interested readers are referred to these papers for the proof.

The expression for the curvature tensor in terms of Poisson brackets shows that R is invariant under the action of the symplectic-morphism group. Since the connection on T H is induced from an SDiff-connection, it follows that R is covariant constant, and hence H is indeed an infinitedimensional symmetric space.

2.4 Splitting of H

There is obviously a decomposition of the tangent space:

T_{ϕ}H = {ψ : ∫_{V} ψdμ_{ϕ} = 0} ⊕ R.

We claim that this corresponds to a Riemannian decomposition

H = H_{0} × R.

We are interested in seeing this Riemannian splitting more explicitly, partly because we see the appearance of a functional I on the space of Kähler potentials, which is well-known in the literature; see [2], [40] for example. The decomposition of the tangent space of H gives a 1-form α on H with

α_{ϕ}(ψ) = ∫_{V} ψdμ_{ϕ},

and it is straightforward to verify that this 1-form is closed. Indeed

(dα)_{ϕ}(ψ, ψ̃) = ∫_{V} (ψ̃Δψ − ψΔψ̃) = 0.

This means that there is a function I : H → R with I(0) = 0 and dI = α, and it is this function which gives rise to the corresponding Riemannian decomposition. We call a Kähler potential ϕ normalized if I(ϕ) = 0. Then any Kähler metric has a unique normalized potential, and the restriction of our metric on H to I^{−1}(0) endows the space H_{0} of Kähler metrics with a Riemannian structure; this is independent of the choice of the base point ω_{0} and clearly makes H_{0} into a symmetric space. The functional I can be written more explicitly by integrating α along lines in H to give the formula

I(ϕ) = ∑_{p=0}^{n} \frac{1}{(p + 1)!(n − p)!} ∫_{V} ω_{0}^{n−p}(∂\overline{∂}ϕ)^{p} ϕ.

2.5 Donaldson’ Conjectures

We will now study the geodesic equation in H in more detail, and interpret the solutions geometrically. Suppose ϕ_{t}, t ∈ [0, 1], is a path in H. We can view this as a function on V × [0, 1] and in turn as a function on V × [0, 1] × S^{1}, with trivial dependence on the S^{1} factor; that is, we define

Φ(v, t, e^{is}) = ϕ_{t}(v).

We regard the cylinder R = [0, 1] × S^{1} as a Riemann surface with boundary in the standard way—so t + is is a local complex coordinate. Let Ω_{0} be the pull-back of ω_{0} to V × R under the projection map and put Ω_{Φ} = Ω_{0} + ∂\overline{∂}Φ, a (1, 1)-form on V × R. Then we have:

Proposition 1. The path ϕ_{t} satisfies the geodesic equation (1) if and only if Ω_{Φ}^{n+1} = 0 on V × R.

Proof. Denote the metrics defined by ω_{0}, ω_{ϕ} as g, g′. Then

\frac{1}{n!}ω_{ϕ}^{n} = det g′; \frac{1}{n!}ω_{0}^{n} = det g.

Thus the geodesic equation is equivalent to the following (if det g′ ≠ 0)

(ϕ′′ − \frac{1}{2} | ∇ϕ′ |_{g′}^{2}) det g′ = 0,

which may be given as

det ( \begin{matrix}g′ & ( \frac{∂ϕ′}{∂z_{1}} ; \frac{∂ϕ′}{∂z_{2}} ; ⋮ ; \frac{∂ϕ′}{∂z_{n}} ) \\ ( \frac{∂ϕ′}{∂\overline{z_{1}}} \frac{∂ϕ′}{∂\overline{z_{2}}} ⋯ \frac{∂ϕ′}{∂\overline{z_{n}}} ) & ϕ′′ \end{matrix} ) = 0.

Let w = t + \sqrt{−1}s. Then t = Re(w). The above equation could be rewritten as

det ( \begin{matrix}(g + \frac{∂^{2}ϕ}{∂z_{α}∂\overline{z}_{β}})_{n n} & ( \frac{∂^{2}ϕ}{∂z_{1}∂\overline{w}} ; \frac{∂^{2}ϕ}{∂z_{2}∂\overline{w}} ; ⋮ ; \frac{∂^{2}ϕ}{∂z_{n}∂\overline{w}} ) \\ ( \frac{∂^{2}ϕ}{∂\overline{z_{1}}∂w} \frac{∂^{2}ϕ}{∂\overline{z_{2}}∂w} ⋯ \frac{∂^{2}ϕ}{∂\overline{z_{n}}∂w} ) & \frac{∂^{2}ϕ}{∂w∂\overline{w}} \end{matrix} ) = 0.

This is just Ω_{Φ}^{n+1} = 0. The proposition is then proved. q.e.d.

Given boundary data — a real valued function ρ ∈ C^{∞}(∂(V × R)), we consider the set of functions Φ on V × R which agree with ρ on the boundary. Then we define the variation of I_{ρ} on this set by

δI_{ρ} = \frac{1}{(n + 1)!} ∫_{V ×R} δΦ Ω_{Φ}^{n+1},

where the variation δΦ vanishes on the boundary by hypothesis. This boundary condition means that we can show easily that this formula defines a functional I_{ρ}. To prove this, one only needs to show that the second derivatives of I_{ρ} with respect to two infinitesimal variations δ_{1}Φ and δ_{2}Φ are symmetric. The second derivatives are:

\frac{1}{2} · \frac{1}{(n + 1)!} ∫_{V} δ_{1}Φ △ δ_{2}Φ Ω_{Φ}^{n+1},

which are clearly symmetric in δ_{1}Φ and δ_{2}Φ. Here △ is the Laplacian operator of Ω_{Φ} on V × R.

This functional I_{ρ} reduces to the energy functional on paths, by an integration by parts, in the case when R is the cylinder and we restrict to S^{1}-invariant data. Suppose ϕ(t)(0 ≤ t ≤ 1) is a path in H, and δϕ represents the infinestimal variation of ϕ while keeping the value of ϕ fixed when t = 0, 1. Thus, the variation of I_{ρ} in the δϕ direction is (follow notation in the proof of the previous proposition):

δI_{ρ} =\frac{1}{(n + 1)!} ∫_{V ×R} δϕ Ω_{Φ}^{n+1}
=\frac{1}{(n + 1)!} ∫_{t=0}^{1} ∫_{V} δϕ(ϕ′′ − \frac{1}{2} | ∇ϕ′ |_{g′}^{2}) det g′d t.

On the other hand, the variation of energy functional along this path is:

δE = ∫_{t=0}^{1} ∫_{V} δϕ (ϕ′′ − \frac{1}{2} | ∇ϕ′ |_{g′}^{2}) det g′d t,

where E = ∫_{t=0}^{1} ∫_{V} ϕ′(t)^{2} det g′d t. Thus, in case when R is the cylinder and we restrict to S^{1}-invariant data, I_{ρ} equals to the energy functional on the path up to a constant multiple.

The following is the first conjecture by Donaldson in [16]:

Conjecture 1 (Donaldson). Let ℝ be a compact Riemann surface with boundary and ρ : V ×∂R → ℝ be a function such that ω_{0}−\sqrt{−1} \overline{∂}∂ρ is a strictly positive (1, 1) form on each slice V × {z} for each fixed z ∈ ∂R. Let S_{ρ} be the set of functions Φ on V × R equal to ρ over the boundary and such that ω_{0} − \sqrt{−1} \overline{∂}∂Φ is strictly positive on every slice V × {w}, w ∈ R. Then there is a unique solution of the Monge-Ampère equation (Ω_{0} − \sqrt{−1} \overline{∂}∂Φ)^{n+1} = 0 in S_{ρ}, and this solution realizes the absolute minimum of the functional I_{ρ}.

This question is a version of the Dirichlet problem for the complete degenerate Monge-Ampère equation, a topic around which there is a substantial literature; see [2], [20] for example. Note that regularity questions are very important in this theory, since the equation is not elliptic.

In the case of the geodesic problem, when the functional can be rewritten as the energy of a path; if these infima are strictly positive, for all choices of fixed, distinct, end points, they make H into a metric space, in the usual fashion. In this connection, Donaldson proposes the following conjecture (after verifying that it will be satisfied by a smooth geodesic):

Conjecture 2 (Donaldson). If ϕ ∈ H_{0} is normalized and ϕ̃_{t}, t ∈ [0, 1] is any path from 0 to ϕ in H, then

∫_{0}^{1} ∫_{V} (\frac{dϕ̃}{dt})^{2} dμϕ̃_{t}dt ≥ M^{−1} (max(∫_{ϕ>0} ϕdμ_{ϕ}, − ∫_{ϕ<0} ϕdμ_{0}))^{2} . (2)

The restriction to normalized potentials ϕ is not important since we know that H splits as a product, and we could immediately write down a corresponding inequality, involving I(ϕ), for any ϕ ∈ H. If this conjecture and the geodesic conjecture are proved, then H is a metric space. We want to use the continuity method to treat this existence problem of geodesics between any two points in H.

3. Existence of C^{1,1} solutions

Let V be a n− dimensional Kähler manifold without boundary, R be a Riemann surface with boundary. The case we are concerned the most is when R is a cylinder. Suppose g = g_{α\overline{β}}dz_{α}d \overline{z_{β}}(1 ≤ α, β ≤ n) is a given Kähler metric on V. Then g̃ = g_{α\overline{β}}dz_{α}d \overline{z}_{β} + dw \overline{dw} is a Kähler metric on V × R, and φ̃ = φ − |w|^{2}. For convenience, we still denote g̃ as g, and φ̃ as φ when there is no confusion. Also, let z_{n+1} = w. Then z = (z_{1}, z_{2}, …, z_{n}, z_{n+1}) is a point on V × R and z′ = (z_{1}, z_{2}, …, z_{n}) is a point in V. Let φ(z) = φ(z′, w) be a function on V × R such that g + ∂_{z′} \overline{∂_{z′}}φ(z′, w) is a Kähler metric on V for each w ∈ R. We want to solve the degenerate Monge-Ampère equation:

det (g + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} )_{(n+1)(n+1)} = 0 in V × R;

and φ = φ_{0} in ∂(V × R).

(3)

We want to use the continuity method to solve this equation. Consider the continuity equation 0 ≤ t ≤ 1:

det (g + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} ) = t det (g + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} ) , in V × R;

and φ = φ_{0} in ∂(V × R).

(4)

Suppose φ_{0} is a solution to (4) at t = 1 such that

∑_{α,β=1}^{n+1} (g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α} ∂\overline{z}_{β}} ) dz_{α}d \overline{z}_{β}

is a strictly positive Kähler metric on V × R^{6} . Let

f = det (g + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} ) (det g)^{−1} > 0.

Then equation (4) can be rewritten in a better form

det (g + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} ) = t · f · det(g) in V × R;

and φ = φ_{0} in ∂(V × R).

(5)

Clearly, φ_{0} is the unique solution to this equation at t = 1. Since the equation is elliptic, this equation can be uniquely solved for t sufficiently closed to 1 (the kernel of the linearized operator is zero for any t > 0). Let t_{0} be such that (5) has a unique smooth solution for every t ∈ (t_{0}, 1]. We want to show that t_{0} = 0 in this section. Observe that equation (5) is elliptic for every t > 0. Hence, the solution will be as smooth as the boundary data once we show that 2nd derivatives of φ are uniformly bounded. Let h be a super harmonic function on V × R with respect to g such that △_{g}h + n + 1 = 0. and h = φ_{0} in ∂(V × R). Then for any solution of equation (5) for t < 1, we have a C^{0} bound on the solution:

Lemma 1. If φ is a solution of equation (5) for 0 < t < 1, then φ has the following a priori C^{0} estimate due to the maximum principle:

φ_{0} ≤ φ ≤ h, in V × R.

For a C^{2} estimate, we follow Yau’s famous work on Calabi’s conjecture. Essentially, we reduce it to a boundary estimate since we have a C^{0} estimate:

6By definition, for any φ_{0} ∈ H, ∑_{α,β=1}^{n+1}(g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}})dz_{α}d \overline{z}_{β} is a strictly positive Kähler metric on each V − slice V × {w}. Let Ψ be a strictly convex function of w which vanishes on ∂R. Then for large enough constants m, ∑_{α,β=1}^{n+1}(g_{α\overline{β}} + \frac{∂^{2}(φ_{0} + mΨ)}{∂z_{α}∂\overline{z}_{β}})dz_{α}d \overline{z}_{β} is a strictly positive Kähler metric on V × R.

Lemma 2 (Yau). If φ is a solution of equation (5) for 0 < t < 1, then φ has the following a priori C^{2} estimate:

△′(e^{−Cφ}(n + 1 + △φ))

≥ e^{−Cφ}(△ ln f − (n + 1)^{2} inf_{i≠l}(R_{i\overline{i}l\overline{l}}))

− Ce^{−Cφ}(n + 1)(n + 1 + △φ)

+ (C + inf_{i≠l}(R_{i\overline{i}l\overline{l}}))e^{−Cφ}(n + 1 + △φ)^{1+ \frac{1}{n}} (tf)^{−1},

where C + inf_{i≠l}(R_{i\overline{i}l\overline{l}}) > 1, △ is the Laplacian operator with respect to g,

while △′ is the Laplacian operator with respect to g′ = g+ \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} d z_{α} \overline{d z_{β}} and R_{i\overline{i}l\overline{l}} is the Riemannian curvature of g.

From the a priori estimate in Lemma 2, either e^{−Cφ}(n + 1 + △φ) is uniformly bounded in V × R or it achieves its maximum value at ∂(V × R). Lemma 1 asserts that φ is uniformly bounded from above and below, then

Corollary 1. There exists a constant C which depends only on (V × R, g) such that

max_{V ×R} (n + 1 + △φ) ≤ C(1 + max_{∂(V ×R)} (n + 1 + △φ)).

Theorem 1. If φ is a solution of equation (5) for 0 < t < 1, then there exists a constant C which depends only on (V × R, g) such that:

(6)  max_{V ×R} (n + 1 + △φ) ≤ C max_{V ×R} (|∇φ|_{g}^{2} + 1).

In light of Corollary 1, we only need to prove inequality (6) on the boundary, i.e.,

max_{∂(V ×R)} (n + 1 + △φ) ≤ C max_{V ×R} (|∇φ|_{g}^{2} + 1).

We will prove this inequality in the next subsection.

Theorem 2. If φ_{i}(i = 1, 2, …) are solutions of equation (5) for 0 < t_{i} < 1, and the inequality (6) holds uniformly for all these solutions {φ_{i}, i ∈ N}, then there exists a constant C_{1} independent of i such that

max_{V ×R} (n + 1 + △φ) ≤ C max_{V ×R} (|∇φ|_{g}^{2} + 1) < C_{1}.

This is proved via a blowing up argument. We will show this in Subsection 3.2.

Remark 1. By now it is a standard estimate of Monge-Ampère equations, that if

max_{V ×R} (n + 1 + △φ) ≤ C max_{V ×R)} (|∇φ|_{g}^{2} + 1) < C_{1}

then equation (5) for t_{1}, t_{2}, … is a sequence of uniform elliptic equations. The higher derivative of the solution φ_{i} has a uniform bound as long as

lim inf_{i→∞} t_{i} > 0.

Theorem 3. There exists a C^{1,1}(V ×R) function which solves equation (3) weakly. In other words, for any two points φ_{0}, φ_{1} ∈ H, there exists a geodesic path φ(t) : [0, 1] → \overline{H} and a uniform constant C such that the following holds:

0 ≤ (g_{i\overline{j}} + \frac{∂^{2}φ}{∂z_{i}∂\overline{z_{j}}} )_{(n+1)(n+1)} ≤ C \binom{}{g_{i\overline{j}}}_{(n+1)(n+1)} .

Here z_{1}, z_{2} …, z_{n} are local coordinates in V and t = Re (z_{n+1}). And g̃ = g_{α\overline{β}}dz_{α}d \overline{z}_{β} + dw \overline{dw} is a fixed product metric on V × R.

Following notation in Theorem 2, we want to show that t_{0} = lim inf_{i→∞} t_{i} = 0. Otherwise, assume t_{0} > 0. Then equation (5) has a unique smooth solution for 1 ≥ t > t_{0}. By Theorem 2 we have a uniform upper bound for △φ + (n + 1) for all t_{i} > t_{0} > 0. Then equation (5) implies that g′_{i} = g + \frac{∂^{2}φ_{i}}{∂z_{α}∂\overline{z}_{β}} d z_{α} \overline{d z_{β}} is bounded uniformly from below by a uniform positive constant (this positive lower bound approaches 0 when t → 0). Thus, from equation (5), we obtain uniform higher derivative estimates for solutions φ_{i}. Therefore these solutions converge to a regular solution at t_{0} > 0. Again, since equation (5) at t_{0} is an elliptic equation and the kernel of the linearized operator is zero, it can then be solved for any t sufficiently close to t_{0}. But this contradicts the definition of t_{0}. Thus t_{0} = 0. We can choose a subsequence of t_{i} → 0 such that the φ_{i} converge weakly in C^{1,1}(V × R) where Ω is a relative compact subset of V × R. Again via the maximum principle, we can show this limit is unique and defines a weak solution of equation (3).

3.1 Boundary estimate

We want to estimate △φ at any point in the boundary ∂(V × R) = V × ∂R. Let p be a generic point in ∂(V × R). Now choose a small neighborhood U of p in V × R (this will be a half geodesic ball since p ∈ ∂(V × R)) and a local coordinate chart such that g_{α\overline{β}}(p) = δ_{α\overline{β}} and p = (z = 0)

\frac{1}{2}δ_{α\overline{β}} ≤ g_{α\overline{β}}(q) ≤ 2δ_{α\overline{β}}, ∀q ∈ U.

Since ∑_{α,β=1}^{n+1} (g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} ) d z_{α}d \overline{z}_{β} is a positive Kähler metric on V ×

R, there exists a constant ϵ > 0 such that

g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} > 2 ϵ · g_{α\overline{β}}, in V × R.

In the neighborhood U of p, we have

g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} > ϵ · δ_{α\overline{β}} in V × R. (7)

We have the trivial estimates in ∂(V × R):

\frac{∂(φ − φ_{0})}{∂z_{α}} = 0,  \frac{∂^{2}(φ − φ_{0})}{∂z_{α}∂\overline{z}_{β}} = 0,  ∀ 1 ≤ α, β ≤ n.

In order to estimate △φ = ∑_{α,β=1}^{n+1} g^{α\overline{β}} \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} in ∂(V × R), we only need

to estimate \frac{∂^{2}(φ−φ_{0})}{∂z_{α}∂\overline{z}_{β}} when either α or β is n + 1. We will estimate

\frac{∂^{2}(φ−φ_{0})}{∂z_{α}∂\overline{z}_{n+1}}(α ≤ n) first, then use equation (5) to derive as estimate for

\frac{∂^{2}(φ−φ_{0})}{∂z_{n+1}∂\overline{z}_{n+1}} .

Now we set up some conventions:

z_{α} = x_{α} + \sqrt{−1} y_{α}, ∀ 1 ≤ α ≤ n; z_{n+1} = x + \sqrt{−1} y,

where R near ∂R is given by x ≥ 0.

Lemma 3. There exists a constant C which depends only on (V × R, g) such that

|\frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{n+1}} (p)| ≤ C(max_{V ×R} |∇φ|_{g} + 1).

Proof of Theorem 1. At point p, equation (5) reduces to

det (δ_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} ) = t · f.

In other words,

\frac{∂^{2}φ}{∂z_{n+1}∂\overline{z}_{n+1}} = t · f − \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{n+1}} · \frac{∂^{2}φ}{∂\overline{z}_{α}∂z_{n+1}} .

Lemma 3 then implies that

|\frac{∂^{2}φ}{∂z_{n+1}∂\overline{z}_{n+1}} | ≤ C(max_{V ×R} |∇φ|_{g}^{2} + 1).

Thus,

|△φ(p)| = | ∑_{α,β=1}^{n+1} g^{α\overline{β}} \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} (p)| ≤ C(max_{V ×R} |∇φ|_{g}^{2} + 1).

Since p is a generic point in ∂(V × R), Theorem 2 holds true. q.e.d.

Let D be any constant linear 1st order operator near the boundary ( for instance D = ±\frac{∂}{∂x_{α}} , ±\frac{∂}{∂y_{α}} for any 1 ≤ α ≤ n). Notice D is just defined locally. Define a new operator L as ( ϕ is any test function):

Lϕ = ∑_{α,β=1}^{n+1} g′^{α\overline{β}} \frac{∂^{2}ϕ}{∂z_{α}∂\overline{z}_{β}} ,

where (g′^{α\overline{β}}) = (g′_{α\overline{β}})^{−1} = (g_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} )^{−1} . Differentiating both side of equation (5) by D, we get

L Dφ = D ln f + ∑_{α,β=1}^{n+1} g′^{α\overline{β}}Dg_{α\overline{β}}.

Thus there exists a constant C which depends only on (V × R, g) such that

LD(φ − φ_{0}) ≤ C (1 + ∑_{α=1}^{n+1} g′^{α\overline{α}}) . (8)

We will now employ a barrier function of the form

ν = (φ − φ_{0}) + s (h − φ_{0}) − N · x^{2} (9)

near the boundary point, and s, N are positive constants to be determined. We may take δ small enough so that x is small in Ω_{δ} = (V × R) ∩ B_{δ}(0). The main essence of the proof is:

Lemma 4. For N sufficiently large and s, δ sufficiently small, we have

L ν ≤ −\frac{ϵ}{4} (1 + ∑_{α=1}^{n+1} g′^{α\overline{α}}) in Ω_{δ}, ν ≥ 0 on ∂Ω_{δ}.

Proof. Since g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} ≥ ϵδ_{α\overline{β}}, we have

L(φ − φ_{0}) = ∑_{α,β=1}^{n+1} g′^{α\overline{β}} [(g_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} ) − (g_{α\overline{β}} + \frac{∂^{2}φ_{0}}{∂z_{α}∂\overline{z}_{β}} )]

≤ n + 1 − ϵ ∑_{α=1}^{n+1} g′^{α\overline{α}}

and

L(h − φ_{0}) ≤ C_{1} (1 + ∑_{α=1}^{n+1} g′^{α\overline{α}})

for some constant C_{1}. Furthermore, L x^{2} = 2g′^{(n+1)\overline{n+1}}. Thus

Lν = L(φ − φ_{0}) + s · L(h − φ_{0}) − 2 · N · g′^{(n+1)\overline{n+1}}
≤ n + 1 − ϵ ∑_{α=1}^{n+1} g′^{α\overline{α}} + sC_{1} + sC_{1} ∑_{α=1}^{n+1} g′^{α\overline{α}}
− 2Ng′^{(n+1)\overline{n+1}}.

Suppose 0 < λ_{1} ≤ λ_{2} ≤ ⋯ ≤ λ_{n+1} are eigenvalues of (g′_{α\overline{β}})_{(n+1)(n+1)}. Then

∑_{α=1}^{n+1} g′^{α\overline{α}} = ∑_{α=1}^{n+1} λ_{α}^{−1}, g′^{(n+1)\overline{n+1}} ≥ λ_{n}^{−1}.

Thus,

\frac{ϵ}{4} ∑_{α=1}^{n+1} g′^{α\overline{α}} + Ng′^{(n+1)\overline{n+1}} ≥ \frac{ϵ}{4} ∑_{α=1}^{n} λ_{α}^{−1} + (N + \frac{ϵ}{4})λ_{n+1}^{−1}
≥ (n + 1)\frac{ϵ}{4}N^{\frac{1}{n+1}} (λ_{1} · λ_{2} ⋯ λ_{n+1})^{− \frac{1}{n+1}}
= C_{2}N^{\frac{1}{n+1}} .

Choose N large enough so that

−C_{2}N^{\frac{1}{n+1}} + (n + 1) + sC_{1} < −\frac{ϵ}{4}.

Choose s small enough so that s · C_{1} ≤ \frac{ϵ}{4}. Then

Lν ≤ −\frac{ϵ}{4} (1 + ∑_{α=1}^{n+1} g′^{α\overline{α}}) .

From now on we fix N. Observe that △(h − φ_{0}) < −2ϵ. Then there exists a constant C_{0} which depends only on g such that h − φ_{0} > C_{0} x near ∂(V × R). Choose δ small enough so that

s(h − φ_{0}) − Nx^{2} ≥ (sC_{0} − Nδ)x ≥ 0.

Then ν ≥ 0 in ∂Ω_{δ}. q.e.d.

Proof of Lemma 3. Let M = max(|∇φ|_{g} + 1). Choose A ≫ B ≫ C, C_{1}. In addition, choose A, B as big multiples of M. Notice that |Dφ| ≤ 2M in Ω_{δ}. For δ fixed as in Lemma 4, we have Bδ^{2} − |D(φ − φ_{0})| > 0. Consider w = A ν + B |z|^{2} + D(φ − φ_{0}). Then w ≥ 0 in ∂Ω_{δ} and w(0) = 0. Moreover,

Lw ≤ (−\frac{ϵA}{4} + 2B + C) (1 + ∑_{α=1}^{n+1} g′^{α\overline{α}}) < 0.

The Maximum Principle implies that w ≥ 0 in Ω_{δ}. Since w(0) = 0, we have \frac{∂w}{∂x} ≥ 0. In other words,

\frac{∂}{∂x}Dφ(0) < C_{3} · M

for some uniform constant C_{3}. Since D is any 1st order constant operator near ∂(V × R), replacing D with −D, we get

−\frac{∂}{∂x}Dφ(0) < C_{3} · M.

On the other hand, since ∂R is given by x = 0 in our special case, we then have the trivial estimate:

\frac{∂}{∂y} D(φ − φ_{0})(0) = 0.

Therefore,

|\frac{∂}{∂z_{n+1}} Dφ(0)| < C_{3} · M

Lemma 3 follows from here directly. q.e.d.

3.2 Blowing up analysis

Lemma 5. Any bounded weakly sub-harmonic function on the two dimensional plane is a constant.

This is a standard fact in geometric analysis. We will omit the proof here. Note that the lemma is false if the dimension is no less than 3.

The essence of blowing up analysis is to use a “microscope” to analyze what happens in a small neighborhood via rescaling. Hence it doesn’t make any difference what the global structure of the background metric is, or what the metric is. Under rescaling, everything become Euclidean anyway. We might as well view the manifold as a domain in Euclidean space. We will use the variable x to denote the position in V × R.

Proof of Theorem 2. Suppose \frac{1}{ϵ_{i}} = max_{V ×R} |∇φ_{i}|_{g} → ∞. We want to

draw a contradiction from this statement.

Suppose |∇φ_{i}|_{g}(x_{i}) = \frac{1}{ϵ_{i}} . By Theorem 1, we have max_{V ×R} △φ_{i} ≤ \frac{1}{ϵ_{i}^{2}} .

Choose a convergent subsequence of x_{i} such that x_{i} → \underline{x}. Choose a tiny neighborhood B_{δ}(\underline{x}) of \underline{x} so that g_{α\overline{β}}(\underline{x}) = δ_{α\overline{β}} and g is essentially an identity matrix in B_{δ}(\underline{x}). For simplicity, let us pretend that g is a Euclidean metric in B_{δ}(\underline{x}). There are two cases to consider: the first case is when \underline{x} ∈ ∂(V × R) and the 2nd case is when \underline{x} is in the interior of V × R.

We define the blowing up sequence as

φ̃_{i}(x) = φ_{i}(x_{i} + ϵ_{i}x), ∀x ∈ B_{\frac{δ}{ϵ_{i}}} (0).

Then |∇φ̃_{i}(0)| = 1 and

max_{B_{\frac{δ}{ϵ_{i}}} (0)} |∇φ̃_{i}| ≤ 1, and max_{B_{\frac{δ}{ϵ_{i}}} (0)} |△φ̃_{i}| ≤ C.

Observe φ_{0} ≤ φ_{i} ≤ h ( ∀i). Rescale φ_{0} and h accordingly:

φ̃_{0}(x) = φ_{0}(x_{i} + ϵ_{i}x),  h̃(x) = h(x_{i} + ϵ_{i}x), ∀ x ∈ B_{\frac{δ}{ϵ_{i}}} (0).

Thus lim_{i→∞} φ̃_{0}(x) = φ_{0}(\underline{x}) and lim_{i→∞} h̃(x) = h(\underline{x}). Moreover,

φ̃_{0} ≤ φ̃_{i} ≤ h̃, ∀i = 1, 2, …. (10)

There exist a subsequence of φ̃_{i} and a limit function φ̃ in C^{n+1} (or half plane in case \underline{x} is in the boundary) such that in any fixed ball B_{l}(0) (or half ball if \underline{x} is in the boundary) we have φ̃_{i} → φ̃ in C^{1,η} in the ball B_{l}(0) (or half ball) for any 0 < η < 1. This implies

|∇φ̃(0)| = 1. (11)

In addition, inequality (10) holds in the limit:

φ_{0}(\underline{x}) ≤ φ̃(x) ≤ h(\underline{x}), ∀ x. (12)

Case 1. Suppose \underline{x} ∈ ∂(V × R). Then h(\underline{x}) = φ_{0}(\underline{x}). Inequality (12) implies that φ̃ is a constant function in its domain. In particular, we have |∇φ̃(x)| ≡ 0. This contradicts our assertion (11). Thus the theorem is proved in this case.

Case 2. Suppose \underline{x} is in the interior of V × R. Then φ̃(x) is a well defined C^{1,η} and bounded function in C^{n+1}. We claim that this function is weakly sub-harmonic on any complex line through the origin. If this claim is true, then Lemma 5 says it must be constant for any complex line through the origin. Therefore, the function itself must be a constant as well. Thus |∇φ̃| ≡ 0. It again contradicts our assertion (11). Thus the theorem is proved also, provided we can prove this claim.

Without loss of generality, we consider the complex line T to be

z_{2} = z_{3} = ⋯ = z_{n+1} = 0.

Observe that (near \underline{x}) the following holds

0 < (δ_{α\overline{β}} + \frac{∂^{2}φ_{i}}{∂z_{α}∂\overline{z}_{β}} )_{(n+1)(n+1)} < \frac{C}{ϵ_{i}^{2}} (δ_{α\overline{β}})_{(n+1)(n+1)}, ∀ i.

After rescaling, we have

0 < ϵ_{i}^{2} · (δ_{α\overline{β}})_{(n+1)(n+1)} + (\frac{∂^{2}φ̃_{i}}{∂z_{α}∂\overline{z}_{β}} )_{(n+1)(n+1)} < C · (δ_{α\overline{β}})_{(n+1)(n+1)}.

Restricting this to a complex line T, we have

0 < ϵ_{i}^{2} + \frac{∂^{2}φ̃_{i}}{∂z_{1}∂\overline{z}_{1}} < C. (13)

Thus one can choose a subsequence of φ̃_{i} which converges C^{1,η}(0 < η < 1) locally in T to some function ψ. Since the convergence is in C^{1,η}, we have ψ = φ̃|_{T}; i.e., ψ is the restriction of φ̃ to this complex line T. By taking a weak limit in inequality (13), the φ̃_{i}|_{T} weakly converge to ψ in the H_{loc}^{2,p} topology for any p > 1. Hence, ψ is a weakly sub-harmonic function by taking a weak limit in inequality (13). Therefore ψ = φ̃|_{T} is a constant by Lemma 5. Our claim is then proved. q.e.d.

4. Uniqueness of weak C^{0} geodesic

Notation follows from previous section.

Definition 1. A function φ is generalized-pluri-subharmonic in V ×

R if ∑_{α,β=1}^{n+1} (g_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} ) dz_{α}d \overline{z}_{β} defines a strictly positive Kähler

metric on V × R.

Definition 2. A continuous function φ in V × R is a weak C^{0} solution to the degenerate Monge-Ampère equation (3) with prescribed boundary data φ_{0} if the following statement is true: ∀ ϵ > 0, there exists a pluri subharmonic function φ̃ in V × R such that |φ − φ̃| < ϵ and φ̃ solves equation (5) with some positive function 0 < f < ϵ at t = 1, and with the same boundary data φ_{0}.

Clearly, the solution we obtain through the continuity method is a weak C^{0} solution of equation (3).

Theorem 4. Suppose φ_{1}, φ_{2} are two C^{0} weak solutions to the degenerate Monge-Ampère equation with prescribed boundary conditions h_{1}, h_{2}. Then

max_{V ×R} |φ_{1} − φ_{2}| ≤ max_{∂(V ×R)} |h_{1} − h_{2}|.

Corollary 2. The solution to the degenerate Monge-Ampère equation is unique as soon as the boundary data is fixed.

Proof. Suppose ϕ_{1}, ϕ_{2} are two approximate generalized-pluri-subharmonic solutions of φ_{1}, φ_{2} in the sense of Definition 2. More precisely,

det (g + \frac{∂^{2}ϕ_{i}}{∂z_{α}∂\overline{z}_{β}} ) = f_{i} · det(g) > 0 in V × R;

and ϕ_{i} = h_{i} in ∂(V × R), i = 1, 2

such that max_{V ×R}( |φ_{1} − ϕ_{1}| + f_{1}) and max_{V ×R}( |φ_{2} − ϕ_{2}| + f_{2}) can be made

as small as we wanted.

∀ϵ > 0, we want to show

max_{V ×R} (φ_{1} − φ_{2}) ≤ max_{V ×R} (h_{1} − h_{2}) + 2ϵ.

Choose f_{1} such that 0 < f_{1} < ϵ and max_{V ×R} |φ_{1} − ϕ_{1}| < ϵ. Choose f_{2}

such that 0 < f_{2} ≤ \frac{1}{2} min_{V ×R} f_{1} < ϵ and max_{V ×R} |φ_{2} − ϕ_{2}| < ϵ. Then ϕ_{1} is a

sub-solution to ϕ_{2} (thus ϕ_{1} < ϕ_{2}) if h_{1} = h_{2}. In general, we have

max_{V ×R} (ϕ_{1} − ϕ_{2}) ≤ max_{∂(V ×R)} (h_{1} − h_{2}).

Thus

max_{V ×R} (φ_{1} − φ_{2}) = max_{V ×R} (φ_{1} − ϕ_{1}) + max_{V ×R} (ϕ_{1} − ϕ_{2}) + max_{V ×R} (ϕ_{2} − φ_{2})
≤ ϵ + max_{∂(V ×R)} (h_{1} − h_{2}) + ϵ
= max_{∂(V ×R)} (h_{1} − h_{2}) + 2ϵ.

Change the role of φ_{1} and φ_{2}, we obtain

max_{V ×R} (φ_{2} − φ_{1}) ≤ max_{∂(V ×R)} (h_{2} − h_{1}) + 2ϵ.

Hence

max_{V ×R} |φ_{1} − φ_{2}| ≤ max_{∂(V ×R)} |h_{1} − h_{2}| + 2ϵ.

Let ϵ → 0, we obtain the desired result. q.e.d.

5. The space of Kähler metric is a metric space—Triangle inequality

In this section, we want to prove that the space of Kähler metric is a metric space, and the C^{1,1} geodesic between any two points realizes the global minimal length over all possible paths. To prove this claim, one inevitably needs to take derivatives of lengths for a family of C^{1,1} geodesics. However, the length for a C^{1,1} geodesic is just barely defined (the integrand is in L^{p} space). In general, one cannot take derivatives. Therefore, we must find ways to circumvent this difficulty.

Definition 3. A path φ(t)(0 < t < 1) in the space of Kähler metrics is a convex path if φ(t) is a generalized-pluri-subharmonic function in V × (I × S^{1}) (cf. Definition 1).

Suppose vol(t)(0 ≤ t ≤ 1) is a family of strictly positive volume form in V such that

∫_{V} vol(t) = ∫_{V} det g.

The notion of ϵ-approximate geodesic is defined with respect to such a volume form:

Definition 4. A convex path φ(t) in the space of Kähler metrics is called an ϵ-approximate geodesic if the following holds:

(φ′′ − |∇φ′|_{g(t)}^{2}) det g(t) = ϵ · vol(t),

where g(t)_{α\overline{β}} = g_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} (1 ≤ α, β ≤ n).

Remark 2. The definition is really independent of these volume forms since we only care what happens when ϵ is really small. For convenience, sometimes we choose vol(t) ≡ det g (a volume form independent of t).

Lemma 6. Suppose φ(t)(0 ≤ t ≤ 1) is an ϵ-approximate geodesic.

Define the energy element as E(t) = ∫_{V} φ′(t)^{2}d g(t). Then

max_{t} |\frac{d E}{d t} | ≤ 2 ϵ · max_{V ×I} |φ′(t)| · M,

where M = ∫_{V} det g is the total volume of V which depends only on the

Kähler class.

Proof.

|\frac{d E}{d t} | = |∫_{V} (2φ′′φ′ + φ′^{2}△_{g(t)}φ′) d g(t)|

= 2 |∫_{V} φ′(φ′′ − \frac{1}{2}|∇φ′|_{g(t)}^{2}) det g(t)|

= 2 |∫_{V} φ′ ϵ vol(t)| ≤ 2 ϵ · max_{V ×I} |φ′(t)| · M. q.e.d..

Proposition 2. Suppose φ(t) is a C^{1,1} geodesic in H from 0 to φ and I(φ) = 0. Then the following inequality holds

∫_{0}^{1} \sqrt{∫_{V} φ′^{2}dμ_{φ_{t}}}dt ≥ M^{−1} (max(∫_{φ>0} φdμ_{φ}, − ∫_{φ<0} φdμ_{0})) .

In other words, the length of any C^{1,1} geodesic is strictly positive.

Proof. As in Definition 4, suppose φ(ϵ, t) is a ϵ-approximate geodesic between 0 and φ. (We will drop the dependence of ϵ in this proof since no confusion shall arise from this omission). First of all, from the definition of ϵ-approximate geodesic, we have

φ′′ − \frac{1}{2}|∇φ′|_{g(t)}^{2} > 0.

In particular, we have φ′′(t) ≥ 0. Thus

φ′(0) ≤ φ ≤ φ′(1). (14)

Consider f(t) = I(tφ), t ∈ [0, 1]. Then f′(t) = ∫_{V} φ dμ_{tφ} and

f′′(t) = ∫_{V} φ △_{g(tφ)} φ dμ_{tφ} ≤ 0.

Thus, we have f′(0) ≥ \frac{f(1)−f(0)}{1−0} ≥ f′(1). In other words, we have

∫_{V} φ d μ_{0} ≥ I(φ) ≥ ∫_{V} φ d μ_{φ}.

Since we assume I(φ) = 0, and φ is not identically zero, it must take both positive and negative values. Then the length (or energy) of the geodesic is given by

E = ∫_{V} φ′^{2}dμ_{φ_{t}},

for any t ∈ [0, 1]. In particular, taking t = 1,

\sqrt{E(1)} ≥ M^{−1/2} ∫_{V} |φ′(1)|dμ_{φ} > M^{−1/2} ∫_{φ′(1)>0} φ′(1)dμ_{φ},

where M is the volume of V (which is of course the same for all metrics in H). It follows from inequality (14) that

∫_{φ′(1)>0} φ′ dμ_{φ} ≥ ∫_{φ>0} φ dμ_{φ},

where the last term is strictly positive by the remarks above, and depends only on φ and not on the geodesic. A similar argument gives

\sqrt{E(0)} > −M^{−1/2} ∫_{φ<0} φ dμ_{0}.

The previous lemma implies that for any t_{1}, t_{2} ∈ [0, 1], we have

|E(t_{1}) − E(t_{2})| < C · ϵ

for some constant C independent of ϵ. Thus

\sqrt{E(t)} ≥ M^{−1/2} max (∫_{φ>0} φdμ_{φ}, − ∫_{φ<0} φdμ_{0}) − C · ϵ.

Now integrate from t = 0 to 1 and let ϵ → 0. Then

∫_{0}^{1} \sqrt{∫_{V} φ′^{2}dμ_{φ}} d t ≥ M^{−1/2} max (∫_{φ>0} φdμ_{φ}, − ∫_{φ<0} φdμ_{0}) .

This proposition is proved. q.e.d.

Remark 3. This proposition verifies Donaldson’s second conjecture. Unfortunately, it does not imply H is a metric space automatically since the geodesic is not sufficiently differentiable. On the other hand, one can easily verify that the C^{1,1} geodesic minimizes length over all possible convex curves between the two end points. To show that it minimizes length over all possible curves, not just convex ones, we need to prove that the triangle inequality is satisfied by the geodesic distance (see Definition 5 below).

Definition 5. Let φ_{1}, φ_{2} be two distinct points in the space of metrics. According to Theorem 3 and Corollary 2, there exists a unique geodesic connecting these two points. Define the geodesic distance as the length of this geodesic, denoted it by d(φ_{1}, φ_{2}).

Theorem 5. Suppose C : φ(s) : [0, 1] → H is a smooth curve in H. Suppose p is a base point of H. For any s, the geodesic distance from p to φ(s) is no greater than the sum of the geodesic distance from p to φ(0) and the length from φ(0) to φ(s) along this curve C. In particular, if C : φ(s) : [0, 1] → H is a geodesic, then the geodesic distance satisfies

d(0, φ(1)) ≤ d(0, φ(0)) + d(φ(0), φ(1)).

Lemma 7 (Geodesic approximation lemma). Suppose C_{i} : φ_{i}(s) : [0, 1] → H(i = 1, 2) are two smooth curves in H. For ϵ_{0} small enough, there exist two parameter smooth families of curves C(s, ϵ) : ϕ(t, s, ϵ) : [0, 1] × [0, 1] × (0, ϵ_{0}](0 ≤ t, s ≤ 1, 0 < ϵ ≤ ϵ_{0}) such that the following properties hold:

1. For any fixed s and ϵ, C(s, ϵ) is an ϵ-approximate geodesic from φ_{1}(s) to φ_{2}(s). More precisely, ϕ(z, t, s, ϵ) solves the corresponding Monge-Ampère equation:

det (g + \frac{∂^{2}ϕ}{∂z_{α}∂\overline{z}_{β}} ) = ϵ · det(g) in V × R;
and ϕ(z′, 0, s, ϵ) = φ_{1}(z′, s), ϕ(z′, 1, s, ϵ) = φ_{2}(z′, s).

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Here we follow notations in Section 3, and z_{n+1} = t+\sqrt{−1} θ where the dependence of ϕ on θ is trivial.

2. There exists a uniform constant C (which depends only on φ_{1}, φ_{2}) such that

|ϕ| + |\frac{∂ϕ}{∂s}| + |\frac{∂ϕ}{∂t}| < C;  0 ≤ \frac{∂^{2}ϕ}{∂t^{2}} < C,  \frac{∂^{2}ϕ}{∂s^{2}} < C.

3. For fixed s, let ϵ → 0 : the convex curve C(s, ϵ) converges to the unique geodesic between φ_{1}(s) and φ_{2}(s) in the weak C^{1,1} topology.

4. Define the energy element along C(s, ϵ) by

E(t, s, ϵ) = ∫_{V} |\frac{∂ϕ}{∂t}|^{2} d g(t, s, ϵ),

where g(t, s, ϵ) is the corresponding Kähler metric define by ϕ(t, s, ϵ). Then there exists a uniform constant C such that

max_{t,s} |\frac{∂ E}{∂ t} | ≤ ϵ · C · M.

In other words, the energy/length element converges to a constant along each convex curve if ϵ → 0.

Proof. Everything follows from Theorems 3, 4 and Lemma 6 except the bound on |\frac{∂ϕ}{∂s}| and an upper bound on \frac{∂^{2}ϕ}{∂s^{2}} which follow from the maximum principle directly since

L (\frac{∂ϕ}{∂s}) = 0

and

L (\frac{∂^{2}ϕ}{∂s^{2}}) = tr_{g′} \{Hess\frac{∂ϕ}{∂s} · Hess\frac{∂ϕ}{∂s}\} ≥ 0. q.e.d.

Proof of Theorem 5. Apply the geodesic approximation lemma in the special case that φ_{1}(s) ≡ p. We follow notations in the previous lemma. For ϵ_{0} small enough, there exist two parameter smooth families of curves C(s, ϵ) : ϕ(t, s, ϵ) : [0, 1]×[0, 1]×(0, ϵ_{0}](0 ≤ t, s ≤ 1, 0 < ϵ ≤ ϵ_{0}) such that

det (g + \frac{∂^{2}ϕ}{∂z_{α}∂\overline{z}_{β}} ) = ϵ · det(g), in V × R;
and ϕ(z′, 0, s, ϵ) = 0, ϕ(z′, 1, s, ϵ) = φ(z′, s).

Denote the length of the curve ϕ(t, s, ϵ) from p to φ(s) by L(s, ϵ), denote the geodesic distance between p and φ(s) by L(s), and denote the length from φ(0) to φ(s) along curve C by l(s). Clearly,

l(s) = ∫_{0}^{s} \sqrt{∫_{V} |\frac{∂φ}{∂τ}|^{2}d g(τ)} d τ,

where g(τ) is the Kähler metric defined by φ(τ), and

L(s, ϵ) = ∫_{0}^{1} \sqrt{E(t, s, ϵ)} d t = ∫_{0}^{1} \sqrt{∫_{V} |\frac{∂ϕ}{∂t}|^{2}d g(t, s, ϵ)} d t,

and lim_{ϵ→0} L(s, ϵ) = L(s). Define F (s, ϵ) = L(s, ϵ) + l(s) and F (s) = L(s)+l(s). What we need to prove is : F (1) ≥ F (0). This will be done if we can show that F′(s) ≥ 0, ∀ s ∈ [0, 1]. The last statement would be straightforward if the deformation of geodesics is C^{1}. Since we do not have that, we need to take derivatives of F (s, ϵ) for ϵ > 0 instead. Notice \frac{∂ϕ}{∂s} = 0 at t = 0 in the following deduction:

\frac{d L(s, ϵ)}{d s} = ∫_{0}^{1} \frac{1}{2} E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} (2\frac{∂ϕ}{∂t} \frac{∂^{2}ϕ}{∂t∂s}

+ (\frac{∂ϕ}{∂t})^{2} △_{g(t,s,ϵ)} \frac{∂ϕ}{∂s}) dg(t, s, ϵ) d t

= ∫_{0}^{1} E(t, s, ϵ)^{− \frac{1}{2}} \{\frac{∂}{∂t} (∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ))

− ∫_{V} \frac{∂ϕ}{∂s} (\frac{∂^{2}ϕ}{∂t^{2}} − \frac{1}{2} |∇\frac{∂ϕ}{∂t}|^{2}) d g(t, s, ϵ)\} d t

= \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ))\}|_{0}^{1}

− ∫_{0}^{1} \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂s} (\frac{∂^{2}ϕ}{∂t^{2}} − \frac{1}{2} |∇\frac{∂ϕ}{∂t}|^{2}) d g(t, s, ϵ)\} d t

+ ∫_{0}^{1}\{E(t, s, ϵ)^{− \frac{3}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ)

· ∫_{V} \frac{∂ϕ}{∂t} (\frac{∂^{2}ϕ}{∂t^{2}} − \frac{1}{2} |∇\frac{∂ϕ}{∂t}|^{2}) d g(t, s, ϵ)\} d t

= ∫_{V} \frac{∂ϕ(1, s, ϵ)}{∂t} \frac{d φ}{d s} d g(s) · \{∫_{V} |\frac{∂ϕ(1, s, ϵ)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}}

− ∫_{0}^{1} \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂s} ϵ · det g\} d t

+ ∫_{0}^{1}\{E(t, s, ϵ)^{− \frac{3}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ) · ∫_{V} \frac{∂ϕ}{∂t} ϵ · det g\} d t.

Observe that by the Schwartz inequality, we have

\frac{d l(s)}{d s} = \sqrt{∫_{V} |\frac{∂φ}{∂s}|^{2} d g(s)}

≥ − ∫_{V} \frac{∂ϕ(1, s, ϵ)}{∂t} \frac{d φ}{d s} d g(s) · \{∫_{V} |\frac{∂ϕ(1, s, ϵ)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}} .

Observe that F (s, ϵ) = L(s, ϵ) + l(s). Thus

\frac{d F (s, ϵ)}{d s} ≥ − ∫_{0}^{1} \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂s} ϵ · det g\} d t

+ ∫_{0}^{1}\{E(t, s, ϵ)^{− \frac{3}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ)

· ∫_{V} \frac{∂ϕ}{∂t} ϵ · det\} d t.

Integrating from 0 to s ∈ (0, 1], we obtain

F (s, ϵ) − F (0, ϵ) ≥ − ∫_{0}^{s} ∫_{0}^{1} \{E(t, τ, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂τ} ϵ · det g\} d t d τ

+ ∫_{0}^{s} ∫_{0}^{1}\{E(t, τ, ϵ)^{− \frac{3}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂τ} d g(t, τ, ϵ)

· ∫_{V} \frac{∂ϕ}{∂t} ϵ · det g\} d t d τ

≥ − Cϵ

for some large constant C depending only on (V × R, g) and the initial curve C : φ(s) : [0, 1] → H. Now taking the limit as ϵ → 0, we have F (s) ≥ F (0). In other words, the geodesic distance from p to φ(s) is no greater than the sum of the geodesic distance from p to φ(0) and the length from φ(0) to φ(s) along this curve C. q.e.d.

Corollary 3. The geodesic distance between any two metrics realizes the absolute minimum of the lengths over all possible paths.

Proof. For any smooth curve C : φ(s) : [0, 1] → H, we want to show that the geodesic distance between the two end points φ(0) and φ(1) is no greater than the length of C. However, this follows directly from Theorem 5 by taking p = φ(1) and s = 1. q.e.d.

Theorem 6. For any two Kähler potentials φ_{1}, φ_{2}, the minimal length d(φ_{1}, φ_{2}) over all possible paths which connect these two Kähler potentials is strictly positive, as long as φ_{1} ≠ φ_{2}. In other words, (H, d) is a metric space. Moreover, the distance function is at least C^{1}.

Proof. Immediately from Corollary 3 and Proposition 2, we see that (H, d) is a metric space. Now we want to prove the differentiability

220 xiuxiong chen

of the distance function. From the proof of Theorem 5, we have

\frac{d L(s, ϵ)}{d s} = ∫_{V} \frac{∂ϕ(1, s, ϵ)}{∂t} \frac{d φ}{d s} d g(s)

· \{∫_{V} |\frac{∂ϕ(1, s, ϵ)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}}

− ∫_{0}^{1} \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} \frac{∂ϕ}{∂s} ϵ · det g\} d t

+ ∫_{0}^{1}\{E(t, s, ϵ)^{− \frac{3}{2}} ∫_{V} \frac{∂ϕ}{∂t} \frac{∂ϕ}{∂s} d g(t, s, ϵ)

· ∫_{V} \frac{∂ϕ}{∂t} ϵ · det g\} d t.

Integrating this from s_{1} to s_{2} and dividing it by s_{2} − s_{1}, gives

|\frac{L(s_{2}, ϵ) − L(s_{1}, ϵ)}{s_{2} − s_{1}} − \frac{1}{s_{2} − s_{1}} ∫_{s_{1}}^{s_{2}} ∫_{V} \frac{∂ϕ(1, s, ϵ)}{∂t} \frac{d φ}{d s} d g(s)

· \{∫_{V} |\frac{∂ϕ(1, s, ϵ)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}} d s|

≤ \frac{1}{s_{2} − s_{1}} ∫_{s_{1}}^{s_{2}} ∫_{0}^{1} \{E(t, s, ϵ)^{− \frac{1}{2}} ∫_{V} |\frac{∂ϕ}{∂s}| ϵ · det g\} d t d s

+ \frac{1}{s_{2} − s_{1}} ∫_{s_{1}}^{s_{2}} ∫_{0}^{1}\{E(t, s, ϵ)^{− \frac{3}{2}} ∫_{V} |\frac{∂ϕ}{∂t}| |\frac{∂ϕ}{∂s}| d g(t, s, ϵ)

· ∫_{V} |\frac{∂ϕ}{∂t}| ϵ · det g\} d t d s

≤ Cϵ. Let ϵ → 0, and s_{2} → s_{1}. Then we have

lim_{s_{2}→s_{1}} \frac{L(s_{2}) − L(s_{1})}{s_{2} − s_{1}} = lim_{s_{2}→s_{1}} \frac{1}{s_{2} − s_{1}} ∫_{s_{1}}^{s_{2}} ∫_{V} \frac{∂ϕ(1, s)}{∂t} \frac{d φ}{d s} d g(s)

· \{∫_{V} |\frac{∂ϕ(1, s)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}} d s

= ∫_{V} \frac{∂ϕ(1, s)}{∂t} \frac{d φ}{d s} d g(s)

· \{∫_{V} |\frac{∂ϕ(1, s)}{∂t}|^{2} d g(s)\}^{− \frac{1}{2}} .

The distance function L is therefore a differentiable function. q.e.d.

6. Application: Uniqueness of extremal Kähler metrics if C_{1}(V) < 0 and C_{1}(V) = 0

In this section, we want to show that if C_{1}(V) < 0, or if C_{1}(V) = 0, then the extremal Kähler metric is unique in any Kähler class. Furthermore, if C_{1}(V) ≤ 0, the extremal Kähler metric (if it exists) realizes the global minimum of the K energy functional in any Kähler class, thus giving an affirmative answer to a question raised by Tian Gang in this special case.

6.1 Uniqueness of c.s.c metric when C_{1}(V) = 0 and the lower bound of K energy for C_{1}(V) ≤ 0

We should now introduce an important operator— the Lichernowicz operator D. For any function h, Dh = h_{,αβ}dz^{α} ⊗ dz^{β}. If Dh = 0, then ↑ \overline{∂}h = g^{α\overline{β}} \frac{∂h}{∂\overline{β}} \frac{∂}{∂z_{α}} is a holomorphic vector field. Now let us introduce the K energy. Like I_{ρ}, I, it is again defined by its derivatives and one should check it is well defined by verifying the second derivatives is symmetric (we will leave this to the reader). Let R be the scalar curvature of the metric g = g_{0} + \sqrt{−1}∂\overline{∂}φ and \underline{R} be the average scalar curvature in the cohomology class. Let ψ ∈ T_{φ}H. Then the variation of K energy of g in the direction ψ is:

δ_{ψ}E = − ∫_{V} (R − \underline{R}) · ψ det g.

Along any smooth geodesic φ(t) ∈ H, S. Donaldson shows

\frac{d^{2}E}{d t^{2}} = ∫_{V} |Dφ′(t)|_{g}^{2} det g.

Using this, Donaldson shows that the constant curvature metric is unique in each Kähler class if the smooth geodesic conjecture is true. Now we want to prove the uniqueness of constant curvature metrics in each Kähler class when C_{1}(V) < 0 or C_{1}(V) = 0, despite the fact we have not proved the smooth geodesic conjecture yet.

Theorem 7. If either C_{1}(V) < 0 or C_{1}(V) = 0, then the constant curvature metric (if it exists) in any Kähler class must be unique.

Proof. Notation follows from Section 5. Suppose φ(t) is an ϵapproximate geodesic. Then

det g (φ′′ − \frac{1}{2}|∇φ′|_{g}^{2}) = ϵ · det h,

where h is a given metric in the Kähler class such that Ric(h) < −ch if C_{1}(V) < 0 and Ric(h) ≡ 0 if C_{1}(V) = 0. Let f = φ′′ − \frac{1}{2}|∇φ′|_{g}^{2} ≥ 0. Then

(16) ∇ ln \frac{det g}{det h} = −∇ ln f

and

(17) \frac{d}{d t} (∫_{V} φ′(t) det g) = ∫_{V} f · det g = ϵ · ∫_{V} det h.

Let E denote the K energy functional. Then

\frac{dE}{d t} = − ∫_{V} (R − \underline{R}) · ψ det g.

A direct calculation yields

\frac{d^{2}E}{d t^{2}} = ∫_{V} |Dφ′(t)|_{g}^{2} det g

− ∫_{V} (φ′′ − \frac{1}{2}|∇φ′|_{g}^{2}) · R det g + ϵ · \underline{R} · ∫_{V} det h,

(18) where we already use Equation (17). Now the second term on the righthand side of the above equation is:

− ∫_{V} R·f det g = ∫_{V} Δ_{g} ln det g · f · det g
= ∫_{V} Δ_{g} \frac{ln det g}{ln det h} · f · det g + ∫_{V} Δ_{g} ln det h · f · det g
= − ∫_{V} ∇_{g} \frac{ln det g}{ln det h} · ∇ ln f det g − ∫_{V} tr_{g}(Ric(h)) f det g
= ∫_{V} |∇f|_{g}^{2} \frac{1}{f} det g − ∫_{V} tr_{g}(Ric(h)) f det g.

Integrating from t = 0 to 1 gives

∫_{V ×I} |Dφ′|_{g}^{2} det g d t + ∫_{V ×I} \frac{|∇f|^{2}}{f} det g d t

− ∫_{V ×I} tr_{g}(Ric(h)) f det g d t

= \frac{dE}{d t} |_{0}^{1} − ϵ\underline{R} · ∫_{V} det h d t.

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If φ(0) and φ(1) are both constant scalar curvature metrics, then \frac{dE}{d t} |_{0}^{1}= 0 and

∫_{V ×I} |Dφ′|_{g}^{2} det g d t + ∫_{V ×I} \frac{|∇f|_{g}^{2}}{f} det g d t

− ∫_{V ×I} tr_{g}(Ric(h)) f det g d t
= −ϵ\underline{R} · ∫_{V} det h d t.

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Observing that f det g = ϵ · det h, then

∫_{V ×I} \frac{|Dφ′|_{g}^{2}}{f} det h + ∫_{V ×I} |∇ ln f|_{g}^{2} det h − ∫_{V ×I} tr_{g}(Ric(h)) det h

= −\underline{R} ∫_{V} det h.

If C_{1}(V) = 0, then \underline{R} = 0. Consequently

∫_{V ×I} \frac{|Dφ′|_{g}^{2}}{f} det hd t + ∫_{V ×I} |∇ ln f|_{g}^{2} det h d t = 0.

This easily implies that Dφ(t) ≡ 0 and ↑ \overline{∂}φ′(t) is a holomorphic vector field. Since C_{1} = 0, the only holomorphic vector field is constant vector field. Thus φ′(t) is constant in the V direction. In other words, φ′(t) is a functional of t only. Hence, there exists at most one constant scalar curvature metric in each Kähler class when C_{1} = 0. We postpone the proof of the case C_{1} < 0 to the next subsection.

Theorem 8. If C_{1}(V) ≤ 0, then a constant scalar curvature metric, if it exists, realizes the global minimum of the K energy functional in each Kähler class. In other words, if K energy doesn’t have a lower bound, then there exists no constant curvature metric in that cohomology class.

Proof. Suppose φ_{0} ∈ H is a metric of constant curvature. Then

\frac{dE}{d t} |_{φ_{0}} = − ∫_{V} (R − \underline{R}) · ψ det g = 0.

For any metric φ(1), let φ(t)(0 ≤ t ≤ 1) be a path in H which connects φ(0) and φ(1). In addition, let us assume this is an ϵ-approximate geodesic where ϵ > 0 may be chosen arbitrarily small. From equation (17), we have

\frac{d^{2}E}{d t^{2}} = ∫_{V} |Dφ′(t)|_{g}^{2} det g − ∫_{V} (φ′′ − \frac{1}{2}|∇φ′|_{g}^{2}) · R det g
+ ϵ · \underline{R} · ∫_{V} det h
= ∫_{V} |Dφ′(t)|_{g}^{2} det g + ∫_{V} |∇f|_{g}^{2} \frac{1}{f} det g
− ∫_{V} tr_{g}(Ric(h)) f det g + ϵ · \underline{R} · ∫_{V} det h
> − Cϵ.

The last inequality holds since the average of the scalar curvature is a topological invariant. Thus

E(t) − E(0) ≥ −Cϵ\frac{t^{2}}{2} , ∀t ∈ [0, 1].

In particular, this holds for t = 1

E(φ(1)) − E(φ(0)) = E(1) − E(0) ≥ −\frac{C · ϵ}{2}.

Let ϵ → 0, we have

E(φ(1)) ≥ E(φ(0)).

The theorem is proved since φ(1) is arbitrary.

6.2 Uniqueness of c.s.c. metric when C_{1} < 0

Now we turn our attention to the case C_{1} < 0. By our initial assumption, Ric(h) < −ch for some positive constant c > 0. Thus

∫_{V ×I} \frac{|Dφ′|_{g}^{2}}{f} det h + ∫_{V ×I} |∇ ln f|_{g}^{2} det h + c · ∫_{V ×I} tr_{g}(h) det h

≤ C(= −\underline{R} ∫_{V} det h).

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We want to show that in the limit as ϵ → 0, we still have Dφ′(t) = 0 in some weak sense. Let us first get an integral estimate on f^{\frac{q}{2−q}} (1 < q < 2) with respect to the measure det h d t :

∫_{V ×I} f^{\frac{q}{2−q}} det h d t ≤ C · ∫_{V ×I} f det h d t

≤ C · ∫_{V ×I} \{f · \frac{det g}{det h}\}^{\frac{1}{n}} · \{\frac{det h}{det g}\}^{\frac{1}{n}} det h d t

≤ ϵ^{\frac{1}{n}} ∫_{V ×I} \{\frac{det h}{det g}\}^{\frac{1}{n}} det h d t

≤ C · ϵ^{\frac{1}{n}} ∫_{V ×I} tr_{g}(h) det h d t → 0.

Let X =↑ \overline{∂}φ′(t) = g^{α\overline{β}} \frac{∂φ′}{∂\overline{z_{β}}} \frac{∂}{∂z_{α}} . We want to show that X is uniformly in L^{2} with respect to the measure h + d t^{2}.

∫_{V ×I} |X|_{h}^{2} det h d t = ∫_{V ×I} ∑_{α,β} h_{α\overline{β}}X^{α}\overline{X^{β}} det h d t

= ∫_{V ×I} ∑_{α,β,γ,δ} h_{α\overline{β}}g^{α\overline{γ}} \frac{∂φ′}{∂\overline{z_{γ}}} \overline{\{g^{β\overline{δ}} \frac{∂φ′}{∂\overline{z_{δ}}} \}} det h d t
= ∫_{V ×I} ∑_{α,β,γ,δ} h_{α\overline{β}}g^{α\overline{γ}}g^{δ\overline{β}} \frac{∂φ′}{∂\overline{z_{γ}}} \frac{∂φ′}{∂z_{δ}} det h d t
≤ ∫_{V ×I} tr_{g}(h)|∇φ′|_{g}^{2} det h
≤ C · ∫_{V ×I} tr_{g}(h) det h d t ≤ C.

The second to last inequality holds since f = φ′′ − \frac{1}{2}|∇φ′|_{g}^{2} ≥ 0 and φ′′ < C. Thus X ∈ L^{2}(V × I) has a uniform upper bound for the L^{2} norm.

Consider |Dφ′|_{g} as a function in L^{2}(V × I). First of all, it has a weak limit in L^{2}(V × I); secondly, its L^{q}(1 < q < 2) norm tends to 0 as ϵ → 0.

∫_{V ×I} |Dφ′|_{g}^{q} det h d t = · ∫_{V ×I} \frac{|Dφ′|_{g}^{q}}{f^{l}} · f^{l} det h d t
≤ · (∫_{V ×I} \frac{|Dφ′|_{g}^{sq}}{f^{ls}} det h d t)^{\frac{1}{s}}
· (∫_{V ×I} f^{lτ} det h d t)^{\frac{1}{τ}}

(where \frac{1}{s} + \frac{1}{τ} = 1).

Now l is some number which we should choose appropriately:

ls = 1;  qs = 2;  \frac{1}{s} + \frac{1}{τ} = 1.

Since for any q < 2, we have

s = \frac{2}{q};  l = \frac{q}{2};  τ = \frac{2}{2 − q} ,

the above inequality reduces to

∫_{V ×I} |Dφ′|_{g}^{q} det h d t

≤ C · (∫_{V ×I} \frac{|Dφ′|_{g}^{2}}{f} det h d t)^{\frac{2}{q}} · (∫_{V ×I} f^{\frac{q}{2−q}} det h d t)^{\frac{(2−q)}{2}} → 0.

For any vector Y ∈ T^{1,0}(V ×I) (i.e., Y = ∑_{i=1}^{n} Y^{i} \frac{∂}{∂z_{i}} where z_{1}, z_{2}, …, z_{n}

are all of the coordinate functions in a local chart in V. We use \frac{∂Y}{∂\overline{z}} to

denote the vector valued (0,1) form ∑_{i,j=1}^{n} \frac{∂Y^{i}}{∂\overline{z_{j}}} \frac{∂}{∂z_{i}} ⊗ d \overline{z_{j}}.) For a scalar

function ψ in V × I, let \frac{∂ψ}{∂\overline{z}} = ∑_{j=1}^{n} \frac{∂ψ}{∂\overline{z_{j}}} d \overline{z_{j}}. Now the norms of \frac{∂Y}{∂\overline{z}} and \frac{∂ψ}{∂\overline{z}} in terms of the metric h are:

|\frac{∂Y}{∂\overline{z}} |_{h}^{2} = ∑_{α,β,r,δ=1}^{n} h_{α\overline{r}}h^{\overline{β}δ} \frac{∂Y^{α}}{∂\overline{z_{β}}} \overline{(\frac{∂Y^{r}}{∂\overline{z_{δ}}} )} (22)

and

|\frac{∂ψ}{∂\overline{z}} |_{h}^{2} = ∑_{α,β=1}^{n} h^{α\overline{β}} \frac{∂ψ}{∂z_{α}} \frac{∂ψ}{∂\overline{z_{β}}} . (23)

We claim the following inequality holds (for some uniform constant C):

|\frac{∂X}{∂\overline{z}} |_{h} ≤ \sqrt{∑_{α,β,r,δ=1}^{n} h_{α\overline{r}}h^{\overline{β}δ} \frac{∂X^{α}}{∂\overline{z_{β}}} \overline{(\frac{∂X^{r}}{∂\overline{z_{δ}}} )}} ≤ C \sqrt{tr_{g}(h)} |Dφ′|_{g}. (24)

This could be proved by choosing preferred coordinates, where h_{i\overline{j}} = δ_{i\overline{j}}(1 ≤ i, j ≤ n) while g_{i\overline{j}} = λ_{i}δ_{i\overline{j}}(1 ≤ i, j ≤ n) at an arbitrary point O. Here λ_{i} are eigenvalues of the metric g in terms of the metric h. These λ_{i}’s are uniformly bounded from above since g is so. We want to verify the above inequality at this point O.

|\frac{∂X}{∂\overline{z}} |_{h}^{2} = ∑_{α,β,a,b=1}^{n} \frac{∂X^{α}}{∂z\overline{β}} \frac{∂X\overline{a}}{∂z_{b}} h_{α\overline{a}}h^{\overline{β} b}

= ∑_{α,β,a,b,c,d=1}^{n} h_{α\overline{a}}h^{\overline{β} b}g^{α\overline{c}}φ′_{,\overline{c}\overline{β}}φ′_{,d b}g^{\overline{a}d}

= ∑_{α,β=1}^{n} δ_{α\overline{a}}δ^{\overline{β} b} \frac{1}{λ_{α}} δ^{α\overline{c}}φ′_{,\overline{c}\overline{β}}φ′_{,d b} \frac{1}{λ_{a}} δ^{\overline{a}d}

= ∑_{α,β=1}^{n} \frac{1}{λ_{α}^{2}} φ′_{,\overline{α}\overline{β}}φ′_{,α β}

≤ ( ∑_{α,β=1}^{n} \frac{λ_{β}}{λ_{α}} ) ∑_{α,β=1}^{n} \frac{1}{λ_{α}λ_{β}} φ′_{,\overline{α}\overline{β}}φ′_{,α β}
≤ C · tr_{g}(h) · |Dφ′|_{g}^{2}.

Here C is a uniform constant. From inequality (21) and the fact that g is bounded from above, it follows that

∫_{V ×I} |∇ log \frac{det g}{det h}|_{h}^{2} = ∫_{V ×I} | ∇ log f |_{h}^{2}
≤ C · ∫_{V ×I} | ∇ log f |_{g}^{2} det gd t ≤ C,

and

∫_{V ×I} (\frac{det h}{det g})^{\frac{1}{n}} det h ≤ ∫_{V ×I} tr_{g}(h) det h ≤ C.

From now on, all of the norms, inner products, and integrations are taken with respect to the the metric h + d t^{2} unless otherwise specified. Now define a new vector field Y by

Y = X · \frac{det g}{det h}.

Then

|Y |_{h} = |X|_{h} \frac{det g}{det h} ≤ C.

In other words, Y has a uniform L^{∞} bound. This implies that Y · ∂ ln \frac{det g}{det h}/∂\overline{z} has a uniform L^{q} bound for any 1 < q < 2. Moreover, for any 1 < q < 2, we have

∫_{V ×I} |\frac{∂Y}{∂\overline{z}} − Y · \frac{∂ ln \frac{det g}{det h}}{∂\overline{z}} |_{h}^{q} = ∫_{V ×I} (|\frac{∂X}{∂\overline{z}} |_{h} \frac{det g}{det h})^{q}

≤ ∫_{V ×I} (\sqrt{tr_{g}(h)}\frac{det g}{det h})^{q} |Dφ′|_{g}^{q}

≤ ∫_{V ×I} C |Dφ′|_{g}^{q} → 0.

This immediately implies that \frac{∂Y}{∂\overline{z}} is uniformly bounded in L^{q} for any 1 < q < 2.

Now, all of these quantities, X, Y, \frac{∂Y}{∂\overline{z}} , and \frac{det g}{det h}, … are geometric quantities which depend on ϵ. Since their respective Soblev norms are uniformly controlled, we can take weak limits of these quantities in some appropriate sense. Denote the corresponding weak limits (when ϵ → 0) as X, Y, \frac{det g}{det h}, …. Then X(ϵ) ⇀ X weakly in L^{2}(V × I), Y (ϵ) ⇀ Y weakly in L^{∞}(V × I) and \frac{det g}{det h}(ϵ) ⇀ \frac{det g}{det h} weakly in L^{∞}(V × I), ….

Consider u = ln \frac{det h}{det g}. For simplicity, assume u > 0 (otherwise u > −c for some positive constant). Then the following two equations hold in the limit

\frac{∂Y}{∂\overline{z}} + Y · \frac{∂u}{∂\overline{z}} = 0,  and  Y = X e^{−u}

in the sense of L^{q}(V × I) for any 1 < q < 2. Moreover, we have the following estimates:

∫_{V ×I} e^{\frac{1}{n} u} ≤ C;  ∫_{V ×I} |\frac{∂u}{∂\overline{z}}|^{2} ≤ C;  and ∫_{V ×I} |X|^{2} ≤ C.

Now define a new sequence of vectors X_{,k}(k = 1, 2, …) by X_{,k} =

Y ∑_{i=0}^{k} \frac{u^{i}}{i!} . This is well defined since u is in L^{p}(V × I) for any p > 1.

Then

\begin{aligned}|X_{,k}| & = & |Y | ∑_{i=0}^{k} \frac{u^{i}}{i!} \\ & ≤ & (|X|e^{−u})e^{u} ≤ |X|. \end{aligned}

Equality holds in the last inequality whenever e^{−u} ≠ 0. Thus

∫_{V ×I} |X_{,k}|^{2} ≤ ∫_{V ×I} |X|^{2} ≤ C.

By definition, it is clear ∥X_{,k}∥_{L^{2}(V ×I)} ≤ ∥X_{,m}∥_{L^{2}(V ×I)} whenever k ≤ m. Thus, there exists a positive number A ≤ ∥X∥_{L^{2}(V ×I)} such that

lim_{k→∞} ∥X_{,k}∥_{L^{2}(V ×I)} = A. For m > k, we have

∥X_{,m}∥_{L^{2}(V ×I)}^{2} = ∫_{V ×I} |X_{,m}|^{2}
= ∫_{V ×I} |Y |^{2}(∑_{i=0}^{m} \frac{u^{i}}{i!} )^{2}

≥ ∫_{V ×I} |Y |^{2} ((∑_{i=0}^{k} \frac{u^{i}}{i!} )^{2} + ( ∑_{i=k+1}^{m} \frac{u^{i}}{i!} )^{2})
= ∫_{V ×I} (|X_{,k}|^{2} + |X_{,m} − X_{,k}|^{2})
= ∥X_{,k}∥_{L^{2}(V ×I)}^{2} + ∥X_{,m} − X_{,k}∥_{L^{2}(V ×I)}^{2}.

Taking limits as m, k → ∞, we have ∥X_{,m} − X_{,k}∥_{L^{2}(V ×I)}^{2} → 0. Thus X_{,k} (k = 1, 2, …) is a Cauchy sequence in L^{2}(V × I) and there exists a strong limit X_{,∞} in L^{2}(V × I). By definition, we know that X_{,∞} = X almost everywhere in V × I^{7} . We want to show that X_{,∞} is weakly holomorphic in the V direction. A straightforward calculation yields

\frac{∂X_{,k}}{∂\overline{z}} = \frac{∂Y}{∂\overline{z}} ∑_{i=0}^{k} \frac{u^{i}}{i!} + Y \frac{∂}{∂\overline{z}} (∑_{i=0}^{k} \frac{u^{i}}{i!} )

= −Y \frac{∂u}{∂\overline{z}} ∑_{i=0}^{k} \frac{u^{i}}{i!} + Y (∑_{i=0}^{k−1} \frac{u^{i}}{i!} ) \frac{∂u}{∂\overline{z}}

= −(X_{,k} − X_{,k−1})\frac{∂u}{∂\overline{z}}.

We want to show that \frac{∂X_{,∞}}{∂\overline{z}} = 0 in the sense of distributions. We just need to show it in any open set U × I where U is a coordinate chart in V. Let (z_{1}, z_{2}, …, z_{n}) be coordinate variables in U. Then, for any vector valued smooth function ψ = (ψ^{1}, ψ^{2}, …, ψ^{n}) which vanishes on ∂(U × I), and for any 1 ≤ j ≤ n. We have

|∫_{V ×I} X_{,k} · \frac{∂\overline{ψ}}{∂\overline{z_{j}}} | = |− ∫_{V ×I} \frac{∂X_{,k}}{∂\overline{z_{j}}} · \overline{ψ}|

= |∫_{V ×I} (X_{,k} − X_{,k−1})\frac{∂u}{∂\overline{z_{j}}} \overline{ψ}|

≤ C · ∥X_{,k} − X_{,k−1}∥_{L^{2}(V ×I)} · \sqrt{∫_{V ×I} |∇u|^{2}}

≤ C∥X_{,k} − X_{,k−1}∥_{L^{2}(V ×I)}.

Taking the limit as k → ∞ yields

∫_{V ×I} X_{,∞} · \frac{∂\overline{ψ}}{∂\overline{z_{j}}} = 0, for any j = 1, 2, …, n,

and for any smooth vector valued function ψ = (ψ^{1}, ψ^{2}, …, ψ^{n}) which vanishes on ∂(U × I). Thus, X_{,∞} is a weak holomorphic vector field in the V direction for almost all t. Now recall that

∫_{V ×I} |X_{,∞}|_{h}^{2} det h d t < C.

7It is easy to prove that X_{,∞} = X in the sense of L^{q}(V × I) for any (1 < q < 2).

This implies that X_{,∞} is in L^{2}(V × {t}) for almost all t ∈ [0, 1]. Since X_{,∞} is weakly holomorphic in V × {t} for all t, X_{,∞} must be holomorphic for those t where X_{,∞} is in L^{2}(V × {t}). However, there is no holomorphic vector field on V since C_{1} < 0. Thus X_{,∞} ≡ 0 for all of those t where X_{,∞} is in L^{2}(V ×{t}). This shows that X_{,∞} = 0 in V ×I. Thus X = 0 since X = X_{,∞} in the sense of L^{q}(V ×I) for any 1 < q < 2. Recall

\frac{∂φ′(t)}{∂z_{α}} = ∑_{β=1}^{n} g_{α\overline{β}}X\overline{β} = ∑_{β=1}^{n} g_{α\overline{β}}X_{,∞}\overline{β} = 0.

In other words, φ′(t) is trivial in the V direction and it is a function of t only for all t ∈ [0, 1]. Therefore, φ(0) and φ(1) differ only by a constant in the V direction, and hence represent the same metric in each Kähler class.

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Princeton University