j. differential geometry 66 (2004) 211-232

CONVERGENCE OF THE KÄHLER–RICCI FLOW ON NONCOMPACT KÄHLER MANIFOLDS

Albert Chau

Abstract

We study the Kähler–Ricci flow on noncompact Kähler manifolds and provide conditions under which the flow has a long time solution converging to a complete negative Kähler–Einstein metric. We also study the complex parabolic Monge–Ampère equation.

1. Introduction

In [15] Yau provided necessary and sufficient conditions for a compact Kähler manifold to admit a Kähler–Einstein metric with either zero or negative scalar curvature. Under these conditions, Cao ([2]) was able to prove that a Kähler metric converges to a Kähler–Einstein metric under the normalized Kähler–Ricci flow. It would be interesting to know the extent to which this result can be generalized to complete noncompact Kähler manifolds. The main results generalizing Yau’s work in this direction appear in ([4],[5], [12]) in the case of negative scalar curvature, and in ([13],[14]) in the case of zero scalar curvature. In this paper we determine sufficient conditions under which a complete noncompact Kähler manifold (M, g_{ij̄}) converges to a complete Kähler–Einstein manifold with negative scalar curvature under the Kähler–Ricci flow:

\frac{dg̃_{ij̄}}{dt} = −R̃_{ij̄} + g̃_{ij̄}, (1.1)

g̃_{ij̄}(x, 0) = g_{ij̄}.

Received 09/19/2002.

We also study the following type of complex parabolic Monge–Ampère equations on (Mg_{ij̄}):

(1.2) \frac{du}{dt} = log \frac{det(g_{kl̄}+ u_{kl̄})}{det(g_{kl̄})} − u − f,

u(x, 0) = 0,

for appropriate choices of the function f. Our main result is:

Theorem 1.1. Assume (M, g_{ij̄}) is a complete noncompact Kähler manifold with bounded curvature and that R_{ij̄}+g_{ij̄} = f_{ij̄} for some smooth bounded function f on M. Then (1.2) has a long time smooth solution u(x, t) which converges, as t → ∞, on every compact subset of M, to a smooth limit u(x, ∞). Moreover, g̃_{ij̄}(x, t) := g_{ij̄}(x) + u_{ij̄}(x, t) provides a long time smooth solution to (1.1) and converges, as t → ∞, on every compact subset of M, to a complete Kähler–Einstein metric g̃_{ij̄}(x, ∞) where g̃_{ij̄}(x, ∞) has negative scalar curvature, is equivalent to g_{ij̄}, and has all covariant derivatives of its curvature tensor bounded.

Analogous analytic results are obtained in ([4], [5], [12]) for the corresponding complex elliptic Monge–Ampère equation, and examples of complete noncompact Kähler–Einstein manifolds of negative scalar curvature are provided there. Their examples are either pseudocovex domains in ℂ^{n}, or the complement of a divisor in a compact projective variety.

Using Shi’s basic theory for (1.1), established in ([9], [11]), we will derive short time existence and basic theory for (1.2). In doing this we will see how the smoothing effect of the Ricci flow allows us to avoid various Hölder type conditions typically required when dealing with the complex elliptic Monge–Ampère equation. We will then follow the approach in [2]; to adapt the a priori estimates for the complex elliptic Monge–Ampère equation to the parabolic case. We will also formulate higher-order a priori estimates for (1.2) which are necessary in the noncompact setting. These higher-order estimates correspond to the curvature estimates formulated by Shi [11]. Most of our estimates will be proved by the parabolic maximum principles of Shi ([9], [11]) for noncompact manifolds.

The paper is organized as follows: in Section 2 we establish some analytic preliminaries as well as state Shi’s short time existence result for (1.1). In Section 3 we derive (1.2) and show its equivalence to (1.1).

Short time existence for (1.2) is proved in Section 4 and a priori estimates for (1.2) are proved in Section 5. Finally, we prove Theorem 1.1 in Section 6.

Acknowledgments. The author would like to thank his Ph.D. thesis advisor M. Kuranishi for his advice and his enduring support and encouragement, R. Hamilton and J. Loftin for many helpful discussions and S.T. Yau for overall support in preparing this paper.

2. Preliminaries

2.1. Parabolic Hölder spaces. The natural setting for the Monge– Ampère equation on a general Kähler manifold (M, g_{ij̄}) is the elliptic Hölder space defined relative to the metric (see [4], [12], [15]). We will use these Hölder spaces, as well as their parabolic versions, to apply standard Schauder estimates and to provide appropriate norms to state our convergence results. In this section we define the parabolic and elliptic Hölder spaces of a noncompact Kähler manifold.

Definition 2.1. Let m be a positive integer and α ∈ (0, 1). A complex n-dimensional Kähler manifold (M, g_{ij̄}) is said to have bounded geometry of order m + α if there are numbers r_{1}, r_{2}, k_{1}, k_{2}, C > 0 such that for every p ∈ M:

1) There is a neighborhood U_{p} of p and a holomorphic covering map ξ_{p} : V_{p} → U_{p} where V_{p} ⊂ ℂ^{n}, B_{r_{1}}(0) ⊆ V_{p} ⊆ B_{r_{2}}(0) and ξ_{p}(0) = p.

2) k_{1}δ_{ab̄} ≤ ξ_{p}^{∗}g_{ab̄} ≤ k_{2}δ_{ab̄} on V_{p}.

3) For all a, b we have ∥ξ_{p}^{∗}g_{ab̄}∥_{p,m+α} ≤ C where ∥ · ∥_{p,m+α} is the standard C^{m+α} Hölder norm on V_{p} ∈ ℂ^{n}.

(M, g_{ij̄}) is said to have bounded geometry of order ∞ if it has bounded geometry of order m + α for every m. Let (M, g_{ij̄}) have bounded geometry of order m + α and let [0, T) be an arbitrary time interval. For some choice of maps ξ_{p} as in Definition 2.1, consider the following norm for any smooth function u on M × [0, T):

∥u∥_{m+α,m/2+α/2} := sup_{p∈M}{∥ξ_{p}^{∗}u∥_{p,m+α,m/2+α/2}}, (2.1)

where ξ_{p}^{∗}u is the pull back of u to V_{p} and ∥ · ∥_{p,m+α,m/2+α/2} is the standard parabolic Hölder norm on V_{p} × [0, T). The following definition is independent of the choice of ξ′_{p}s.

Definition 2.2. : Let (M, g_{ij̄}) be a complete Kähler manifold with bounded geometry of order m + α. With respect to (2.1), we define the parabolic Hölder spaces C^{m+α,m/2+α/2}(M × [0, T)) to be the closure of the set of all smooth functions u(x, t) : M × [0, T) ⟼ ℝ for which (2.1) is finite. Also, given (M, g_{ij̄}) above, one can define the elliptic Hölder spaces C^{m+α}(M) in an obvious way.

C^{m+α,m/2+α/2}(M × [0, T)) with the norm (2.1) for some choice of maps ξ_{p}, and C^{m+α}(M) with the analogous elliptic norm, are easily checked to be Banach spaces.

2.2. Short time existence for Kähler–Ricci flow. The Ricci flow was first introduced by Hamilton [6] to evolve a real Riemannian metric with positive Ricci curvature to one of constant positive curvature on a compact three manifold. In [6] Hamilton showed that on an arbitrary smooth compact Riemanian manifold, the Ricci flow always has a short time solution. Under the hypothesis of bounded sectional curvature, Shi [9] extended this result to complete noncompact Riemannian manifolds and later applied this result to the case of Kähler manifolds ([10] [11]). We will use the following version of Shi’s short time existence theorem for the Kähler–Ricci flow.

Theorem 2.3. Let (M, g_{ij̄}) be an n-dimensional complete noncompact Kähler manifold with Riemannian curvature tensor Rm satisfying

sup_{x∈M}∥Rm(x)∥_{g} ≤ K,

where K is a positive constant. Then for some constant T > 0 depending only on n and K, there is a smooth short time solution g̃_{ij̄}(x, t) to the following Kähler–Ricci flow equation on M:

\frac{dg̃_{ij̄}}{dt} = −R̃_{ij̄} − g̃_{ij̄}, (2.2)

g̃_{ij̄}(x, 0) = g_{ij̄},

for t ∈ [0, T). Moreover, for all t ∈ [0, T), g̃_{ij̄}(x, t) is a Kähler metric on M, equivalent to g_{ij̄}, and satisfies the following estimates for the covariant derivatives of its curvature tensor:

sup_{x∈M}∥ ˜∇^{m}R̃m(x, t)∥2 ; g̃ ≤ C_{n,m,k}(1/t)^{m}, (2.3)

where the covariant differentiation and the norm is with respect to the metric g̃_{ij̄}(x, t) and C_{n,m,k} is a constant depending only on n, m and k.

Remark 2.4. (2.2) is a normalization of the Kähler–Ricci flow equation treated in [9] and [11].

2.3. Maximum principles. We now state some maximum principles for parabolic equations on a noncompact Riemannian manifold.

Lemma 2.5. For t ∈ [0, T), Let g_{ij̄}(x, t) be a family of complete Riemannian metrics on a noncompact manifold M such that:

1) g_{ij̄}(x, t) varies smoothly in t.

2) For all t, g_{ij̄}(x, t) is equivalent to g_{ij̄}(x, 0).

3) For all t, g_{ij̄}(x, t) has bounded curvature. Suppose f(x, t) is a smooth bounded function on M × [0, T) such that:

1) f(x, 0) ≥ 0,

2) df/dt = Δ_{t}f + Q(f, x, t),

3) Q(f, x, t) ≥ 0 whenever f ≤ 0, where Δ_{t} denotes the Laplacian of the metric g_{ij̄}(x, t). Then f(x, t) ≥ 0 on M × [0, T).

Proof. See [9] and [11]. q.e.d.

Lemma 2.6. Let (M, g_{ij̄}(x, t)) satisfy the hypothesis of Lemma 2.5. Suppose f(x, t) is a smooth bounded function on M × [0, T) such that:

1) df/dt = Δ_{t}f + Q(f, x, t),

2) Q(f, x, t) ≤ −f^{α} + K, where K and α are positive constants. Then f(x, t) ≤ C_{K,α} on M×[0, T) for some constant C_{K,α} depending only on K, α and sup_{x∈M} |f(x, 0)|.

Proof. Let C = max{sup_{x∈M} |f(x, 0)|, K^{1/α}}, and consider the function ψ := C − f. Then ψ satisfies the first and second condition of the Lemma 2.5. Moreover, we have

(2.4)  dψ/dt = −df/dt
= −Δ_{t}f − Q(f, x, t)
= Δ_{t}ψ − Q(f, x, t)
≥ Δ_{t}ψ + f^{α} − K.

It follows that when ψ ≤ 0 we have f ≥ C and thus −Q(f, x, t) ≥ f^{α} − K ≥ C^{α} − K ≥ 0. Thus the evolution of ψ satisfies the third condition of Lemma 2.5 and ψ remains positive for all t. From this, and the the definition of ψ, the lemma is readily seen to be true. q.e.d.

At times, we will use versions of the maximum principle which are slightly different from Lemma 2.5 and Lemma 2.6, but which can also be derived from Lemma 2.5 by an argument similar to that above. We will omit these derivations.

3. The parabolic Monge–Ampère equation

Let (M, g_{ij̄}) be a complete noncompact Kähler manifold satisfying the following condition:

g_{ij̄} + R_{ij̄} = f_{ij̄}, (3.1)

for some smooth function f on M. We would then like to find a smooth function u on M such that the tensor

g̃_{ij̄}:= g_{ij̄} + u_{ij̄}, (3.2)

is a complete negative Kähler–Einstein metric on M. We will refer to such a function u as a negative Kähler–Einstein potential for (M, g_{ij̄}). We now derive the parabolic equation (1.2) on M and show that its stationary solutions are precisely given by negative Kähler–Einstein potentials for (M, g_{ij̄}).

Recall that g̃_{ij̄} is Kähler–Einstein if its Ricci tensor satisfies

R̃_{ij̄} = ρg̃_{ij̄}, (3.3)

for some constant ρ, which in our case, we assume to be −1. Also recall that on any Kähler manifold (M, g_{ij̄}) the Ricci tensor R_{ij̄} is given locally by

R_{ij̄} = −[log(det g_{kl̄})]_{ij̄}. (3.4)

It follows from (3.1), (3.3) and (3.4) that u is a negative Kähler–Einstein potential for (M, g_{ij̄}) iff u satisfies

(3.5) 0 = [log \frac{det(g_{kl̄}+ u_{kl̄})}{det(g_{kl̄})} − u − f]_{ij̄}

on M, and thus, a sufficient condition for u to be a negative Kähler– Einstein potential for (M, g_{ij̄}) is that

(3.6) 0 = log \frac{det(g_{kl̄}+ u_{kl̄})}{det(g_{kl̄})} − u − f.

on M. We point out that necessity in (3.6) holds (up to a constant) in the the case that M is compact, while in the case M is noncompact it does not. This results from the possible existence of nontrivial pluriharmonic functions on a noncompact Kähler manifold. (3.6) is a special case of the complex Monge–Ampère equation and has been solved on compact Kähler manifolds in [15] and [1] and on noncompact Kähler manifolds in [4] and [12] under special conditions.

It follows, from (3.3)–(3.6), that the stationary solutions to (1.2) are negative Kähler–Einstein potentials, and if u is a smooth solution to (1.2), then g̃_{ij̄} := g_{ij̄} + u_{ij̄} defines a family of Kähler matrics on M evolving by (1.1). We refer to [2] for a treatment of (1.2) in the case of compact manifolds.

4. Short time existence

We now prove short time existence for (1.2).

Lemma 4.1. Let (M, g_{ij̄}) be a complete noncompact Kähler manifold such that:

1) sup_{x∈M} ∥Rm(x)∥_{g} ≤ K for some constant K > 0.

2) R_{ij̄} + g_{ij̄} = f_{ij̄} for some smooth bounded function f on M. Then for some constant T > 0, depending only on n and K, (1.2) has a smooth solution u(x, t) for t ∈ [0, T). Moreover, for every t ∈ [0, T) we have:

1) The Kähler metric ĝ_{ij̄}:= g_{ij̄}(x) + u_{ij̄}(x, t) is equivalent to g_{ij̄} and has bounded geometry of infinite order.

2) R̂_{ij̄}+ ĝ_{ij̄} = f̂_{ij̄} where R̂_{ij̄} is the Ricci curvature tensor of ĝ_{ij̄} and f̂ is a smooth bounded function on M.

Proof. Starting with (M, g_{ij̄}), let g̃_{ij̄}(x, t) and T be as in Theorem 2.3. We will show that g̃_{ij̄}(x, t) gives us a smooth solution u(x, t) to (1.2) on [0, T). To show this we note first that ∀x ∈ M the following ODE in t has a smooth solution u(x, t) for t ∈ [0, T):

(4.1) \frac{d}{dt}u(x, t) = log \frac{det(g̃_{kl̄})(x, t)}{det(g_{kl̄})(x)} − u(x, t) − f(x),

u(x, 0) = 0.

By the equivalence of the metrics g̃_{ij̄}(x, t) and g_{ij̄} for all t ∈ [0, T), the logarithmic term in (4.1) is seen to be a smooth bounded function on M × [0, T) thus making (4.1) an ODE with smooth bounded terms. Consider now the difference tensor S_{ij̄}:= (g̃_{ij̄}(x, t) − g_{ij̄}(x)) − u_{ij̄}(x, t). It is easily checked that S_{ij̄} satisfies the following evolution equation:

\frac{dS_{ij̄}}{dt} = −S_{ij̄}, (4.2)

S_{ij̄}(x, 0) := 0,

on M × [0, T). Since the solution to (4.2) is unique, the zero solution, we conclude that g̃_{ij̄}(x, t) = g_{ij̄}(x) + u_{ij̄}(x, t). Finally, by substituting g_{ij̄}(x) + u_{ij̄}(x, t) for g̃_{ij̄}(x, t) in (4.1), we see that u(x, t) is in fact a smooth solution to (1.2) on M × [0, T).

Now fix an arbitrary T′ ∈ [0, T) and let ĝ_{ij̄}:= g̃_{ij̄}(x, T′). We want to show now that R̂_{ij̄} + ĝ_{ij̄} = f̂_{ij̄} where R̂_{ij̄} is the Ricci tensor of ĝ_{ij̄} and f̂ is some smooth bounded function on M. By taking applying the ∂ ¯∂ operator to (4.1) we get

(u_{t}(x, T′))_{ij̄} = −R̃_{ij̄}(x, T′) − g̃_{ij̄}(x, T′). (4.3)

Also, by differentiating (1.2) in t we get

\frac{du_{t}}{dt} = Δ̃u_{t} − u_{t}, (4.4)

u_{t}(x, 0) = −f(x).

where Δ̃ is the Laplace operator for the metric g̃_{ij̄}(x, t). Applying the maximum principle to (4.4), we see that u_{t}(x, t) begins as a bounded function on M and remains uniformly bounded on M ∀t ∈ [0, T). By (4.3) we see that u_{t}(x, T′) is precisely the function f̂ we are looking for.

To complete the proof of Lemma 4.1 we need to show that ĝ_{ij̄} has bounded geometry of infinite order. In [13] the authors prove that on a noncompact Kähler manifold, one has bounded geometry of order 2 + α provided one has bounded curvature and gradient of scalar curvature. Their proof can be extended to show that one has bounded geometry of infinite order provided one has all covariant derivatives of curvature bounded. Thus since ĝ_{ij̄} has all covariant derivatives of its curvature bounded by Theorem (2.3), we see that ĝ_{ij̄} in fact has bounded geometry of order infinity. This completes the proof of Lemma 4.1. q.e.d.

5. A priori estimates

Assume (M, g_{ij̄}) satisfies the hypothesis of Theorem 1.1 with the additional assumption of bounded geometry of order ∞. By the previous sections we know that on some time interval [0, T), (1.2) has a smooth solution u(x, t), and g̃_{ij̄} := g_{ij̄} + u_{ij̄} solves (1.1). We will show that u stays bounded in every Hölder norm on (M, g_{ij̄}) independent of T. This will be done by establishing several a priori estimates.

5.1. Estimates for u and u_{t}. Differentiating (1.2) in t we get

\frac{du_{t}}{dt} = Δ̃u_{t} − u_{t}, (5.1)

u_{t}(x, 0) = −f.

Applying the maximum principle to (5.1), we get that sup_{x∈M} |u_{t}(x, t)| starts off bounded and continues to decay exponentially:

sup_{x∈M}|u_{t}(x, t)| ≤ Ke^{−t}, (5.2)

for some constant K independent of T. From this we may bound sup_{(x,t)∈M×[0,T)} |u(x, t)|, independent of T, by the following estimate:

|u(x, t)| = |∫_{0}^{t} u_{s}(x, s)ds| ≤ K ∫_{0}^{t} e^{−s}ds ≤ C, (5.3)

where C is some constant independent of T.

5.2. Estimates for Δu. Consider the quantity

A = log(n + Δu) − ku, (5.4)

defined on M × [0, T), where we choose the constant k later. Clearly a bound on |A| implies a bound on |Δu|. We will bound A from above using the maximum principle. The bound from below will follow directly from some simple inequalities derived from (1.2). We begin by computing \frac{dA}{dt} and Δ̃A separately as

(5.5) \frac{dA}{dt} = \frac{1}{(n + Δu)} \frac{dΔu}{dt} − k \frac{du}{dt},
−Δ̃A ≤ −(k − C)g^{λμ̄}g_{λμ̄} − \frac{1}{(n + Δu)} (Δu + Δf + Δ\frac{du}{dt}) + nk,

where C depends only on the curvature of the initial metric g_{ij̄}. We estimate some of the terms on the right side of (5.5).

(5.6) (n + Δu) ≥ [\frac{det g̃_{ij̄}}{det g_{ij̄}} ]^{\frac{1}{n}} = e^{\frac{1}{n} (u+f+ \frac{du}{dt})},

g^{λμ̄}g_{λμ̄} ≥ \frac{1}{n − 1} [(n + Δu)e^{\frac{1}{n} (−u−f− \frac{du}{dt})} (\frac{1}{n − 1})^{1−n}]^{\frac{1}{n−1}} .

The details of (5.5) and (5.6) can be found in [1]. Note that by (5.2), (5.3) and (1.2), (5.6) already gives a bound on inf_{(x,t)∈M×[0,T)} A(x, t) independent of T. Substituting estimates (5.6) back into (5.5) gives the following:

(5.7) \frac{dA}{dt} ≤ Δ̃A + \frac{1}{(n + Δu)} \frac{dΔu}{dt} − k \frac{du}{dt} − (k − C)g^{λμ̄}g_{λμ̄}

− \frac{1}{(n + Δu)} (Δu + Δf + Δ\frac{du}{dt}) + nk

≤ Δ̃A − (k − C)g^{λμ̄}g_{λμ̄} − \frac{1}{(n + Δu)}(Δu + Δf) + nk − k \frac{du}{dt}

≤ Δ̃A − (k − C)g^{λμ̄}g_{λμ̄} − \frac{1}{(n + Δu)}(−n + Δf) + nk − k \frac{du}{dt}

≤ Δ̃A + e^{− \frac{1}{n} (u+f+ \frac{du}{dt})}|n + Δf| + nk − k \frac{du}{dt}

− \frac{k − C}{n − 1} [e^{^{\frac{1}{n}}(−u−f− \frac{du}{dt})} (\frac{1}{n − 1})^{1−n}]^{\frac{1}{n−1}} (n + Δu)^{\frac{1}{n−1}} .

Choosing k sufficiently large, we have

(5.8)  \frac{dA}{dt} ≤ Δ̃A − N_{1}(n + Δu)^{\frac{1}{n−1}} + N_{2}
≤ Δ̃A − N_{1}e^{\frac{A+ku}{n−1}} + N_{2}
≤ Δ̃A − N_{3}e^{\frac{A}{n−1}} + N_{2},

where the constants N_{i} are positive and independent of T. The bound on A from above now follows from the maximum principle. From this and our previous bound on A from below we obtain a bound on sup_{(x,t)∈M×[0,T)}|Δu(x, t)| independent of T. We now show that these bounds imply the equivalence of the metrics g_{ij̄} and g̃_{ij̄} where the factor of equivalence is independent of T.

Consider any point q ∈ M and any time t_{0} ∈ [0, T). Then at this point in time consider orthonormal holomorphic local coordinates z_{i} at q such that:

1) g_{ij̄} = δ_{ij},

2) g̃_{ij̄}(q, t) = g_{ij̄}(q) + u_{ij̄}(q, t) = 0; for i ≠ j. By our estimates above we have

(5.9)  g̃_{iī}(q, t) ≥ 0; ∀i,
∑_{i} g̃_{iī}(q, t) = (n + Δu)(q, t) ≤ C_{1},

∏_{i} g̃_{iī}(q, t) = \frac{det(g̃_{kl̄}(q, t))}{det(g_{kl̄}(q))} = e^{(u+f+ \frac{du}{dt})} ≥ C_{2},

where the constants C_{i} are independent of q and T. We see that these inequalities provide us with the following equivalences:

K_{1} ≤ g̃_{iī}(q, t) ≤ K_{2}; ∀i, (5.10)

where the constants K_{i} are independent of q and T. This provides, for all t ∈ [0, T), the uniform equivalence of the metrics g_{ij̄} and g̃_{ij̄} := g_{ij̄} +u_{ij̄}, where the uniformity is independent of T.

5.3. Estimates for third derivatives of mixed type. Consider the quantity

Q = g^{αβ̄}g^{λμ̄}g^{νγ̄}u_{,αμ̄ν}u_{,β̄λγ̄}, (5.11)

defined on M × [0, T), where the covariant differentiation is in the original metric g_{ij̄} and g^{ab̄} represents the inverse of the time dependent metric g̃_{αβ̄}. By the previous subsection, this norm is equivalent to that using the original metric g^{αβ̄}. We will apply the maximum principle to the evolution of Q. We begin by noting the following expansions, the details of which can be found in [1]:

\frac{dQ}{dt} = −g^{cd̄}(2g^{αδ̄}g^{γβ̄}g^{ab̄} + g^{αβ̄}g^{aδ̄}g^{γb̄})u_{,cb̄α}u_{,d̄aβ̄}(u_{t})_{γδ̄} (5.12)

+ g^{αβ̄}g^{ab̄}g^{cd̄}[u_{,d̄aβ̄}(u_{t})_{,cb̄α} + u_{,cb̄α}(u_{t})_{,d̄aβ̄}] ,

Δ̃Q = g^{λμ̄}g^{αβ̄}g^{ab̄}g^{cd̄}[(u_{,cb̄αμ̄} − u_{,b̄γμ̄}u_{,cδ̄α}g^{γδ̄})(u_{,d̄aβ̄λ} − u_{,d̄δ̄β̄}u_{,aγλ}g^{γδ̄})

+ (u_{,cb̄αλ} − u_{,ρb̄λ}u_{,cν̄α}g^{ρν̄} − u_{,cν̄λ}u_{,αb̄ρ}g^{ρν̄}) ×

· (u_{,d̄aβ̄μ̄} − u_{,ρaμ̄}u_{,d̄ν̄β̄}g^{ρν̄} − u_{,d̄ν̄μ̄}u_{,β̄aρ}g^{ρν̄})]

− g^{cd̄}(2g^{αδ̄}g^{γβ̄}g^{ab̄} + g^{αβ̄}g^{aδ̄}g^{γb̄})u_{,cb̄α}u_{d̄aβ̄}[(u_{t} + u + f)_{γδ̄} − R_{γδ̄}]

+ g^{αβ̄}g^{ab̄}g^{cd̄}[u_{,d̄aβ̄}(u_{t} + u + f)_{,cb̄α} + u_{,cb̄α}(u_{t} + u + f)_{,d̄aβ̄}]

+ g^{λμ̄}g^{αβ̄}g^{ab̄}g^{cd̄}[u_{,d̄aβ̄}(R_{cλb̄}^{ν}u_{,νμ̄α} + Rρ̄ ; b̄μ̄αu_{,cρ̄λ} + R_{cμ̄α}^{ν}u_{,νb̄λ})

+ u_{,cb̄α}(R_{d̄μ̄c}^{ν}u_{,νλβ̄} + Rρ̄ ; aλβ̄u_{,d̄ρ̄μ̄} + R_{d̄λβ̄}^{ν}u_{,νcμ̄})]

+ g^{αβ̄}g^{cd̄}[u_{,d̄aβ̄}(g^{λμ̄}R_{cμ̄α,λ}^{a} − g^{ab̄}R_{cb̄,α})

· u_{,cb̄α}(g^{λμ̄}R_{d̄λβ̄,μ̄}^{a} − g^{ab̄}R_{d̄c,β̄})].

We note that in \frac{dQ}{dt} − Δ̃Q, all terms involving \frac{du}{dt} cancel, and all terms containing fourth derivatives of u can be collected as above to give negative terms, while all other terms are contractions involving second and/or third mixed derivatives of u and f, the metric g̃_{ij̄} and the curvature tensor of the initial metric g_{ij̄}. Thus by our previous estimates on the second derivatives of u we have

\frac{dQ}{dt} ≤ Δ̃Q + C_{1}Q + C_{2}\sqrt{Q}, (5.13)

where the constants C_{i} are positive and independent of T. The evolution of Q does not admit to the standard maximum principle argument since C_{1} is positive. To fix this we consider the modified quantity

Q′ = Q + hΔu, (5.14)

where we determine the constant h later. Certainly, a bound on Q′ will provide a bounding on Q. We proceed by computing the evolution of Δu. The following estimate can be found in [1]:

(5.15)  Δ̃Δu = g^{αμ̄}g^{γβ̄}u_{,γμ̄}^{,ν}u_{,αβ̄ν} + Δ\frac{du}{dt} + Δu + Δf + E

≥ BQ + Δ\frac{du}{dt} + Δu + Δf + E,

where the constant B and the term E is bounded on M × [0, T) independent T. Equivalently, we write this as an evolution for Δu as

\frac{dΔu}{dt} ≤ Δ̃(Δu) − BQ − Δu − Δf − E. (5.16)

This provides the following estimate for the evolution of Q′:

(5.17)  \frac{dQ′}{dt} ≤ Δ̃Q′ + C(Q + \sqrt{Q}) − BhQ − hΔu − hδf − hE
≤ Δ̃Q′ + (C − hB)Q + C\sqrt{Q} − hΔu − hδf − hE.

Choosing the constant h such that (C − hB) is negative, we can apply the maximum principle to the above equation to conclude that sup_{(x, t) ∈ M × [0, T)} Q′(x, t), and thus also sup_{(x,t)∈M×[0,T)} Q(x, t), is bounded independent of T.

5.4. Estimates for fourth derivatives of mixed type. Consider the quantity

Q = g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν}, (5.18)

defined on M × [0, T), where the covariant differentiation is in the original metric g_{ij̄} and g^{ab̄} represents the inverse of the time dependent metric g̃_{αβ̄}. We begin by computing the evolution of Q:

\frac{dQ}{dt} = g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}(u_{t})_{,ασ̄λρ̄}u_{,β̄δμ̄ν} + g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}(u_{t})_{,β̄δμ̄ν} (5.19)

+ g_{t}^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν} + g^{αβ̄}g_{t}^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν} + g^{αβ̄}g^{δσ̄}g_{t}^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν} + g^{αβ̄}g^{δσ̄}g^{λμ̄}g_{t}^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν}.

Expanding the term g_{t}^{αβ̄} gives

(5.20)  g_{t}^{αβ̄} = −g^{αl̄}g^{kβ̄}u_{tl̄k}

= −g^{αl̄}g^{kβ̄} [log \frac{det(g_{ij̄} + u_{ij̄})}{det(g_{ij̄})} − u − f]_{,l̄k}

= −g^{αl̄}g^{kβ̄}[g^{ij̄}u_{,ij̄l̄}− u_{,l̄}− f_{,l̄}]_{,k}

= −g^{αl̄}g^{kβ̄}[−g^{ir̄}g^{sj̄}u_{,r̄sk}u_{,ij̄l̄}+ g^{ij̄}u_{,ij̄l̄k} − u_{,l̄k} − f_{,l̄k}],

while the term (u_{t})_{,ασ̄λρ̄} gives

(u_{t})_{,ασ̄λρ̄} = [log \frac{det(g_{ij̄} + u_{ij̄})}{det(g_{ij̄})} − u − f]_{,ασ̄λρ̄} (5.21)

= [g^{ij̄}u_{,ij̄α} − u_{,α} − f_{,α}]_{,σ̄λρ̄} = [−g^{il̄}g^{kj̄}u_{,l̄kσ̄}u_{,ij̄α} + g^{ij̄}u_{,ij̄ασ̄} − u_{,ασ̄} − f_{,ασ̄}]_{,λρ̄} = [g^{ir̄}g^{sl̄}g^{kj̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} + g^{il̄}g^{kr̄}g^{sj̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α}

− g^{il̄}g^{kj̄}u_{,l̄kσ̄λ}u_{,ij̄α} − g^{il̄}g^{kj̄}u_{,l̄kσ̄}u_{,ij̄αλ} − g^{ir̄}g^{sj̄}u_{,r̄sλ}u_{,ij̄ασ̄} + g^{ij̄}u_{,ij̄ασ̄λ} − u_{,ασ̄λ} − f_{,ασ̄λ}]ρ̄

= −g^{ip̄}g^{qr̄}g^{sl̄}g^{kj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} − g^{ir̄}g^{sp̄}g^{ql̄}g^{kj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} − g^{ir̄}g^{sl̄}g^{kp̄}g^{qj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} + g^{ir̄}g^{sl̄}g^{kj̄}u_{,r̄sλρ̄}u_{,l̄kσ̄}u_{,ij̄α} + g^{ir̄}g^{sl̄}g^{kj̄}u_{,r̄sλ}u_{,l̄kσ̄ρ̄}u_{,ij̄α} + g^{ir̄}g^{sl̄}g^{kj̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄αρ̄} − g^{ip̄}g^{ql̄}g^{kr̄}g^{sj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} − g^{il̄}g^{kp̄}g^{qr̄}g^{sj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} − g^{il̄}g^{kr̄}g^{sp̄}g^{qj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} + g^{il̄}g^{kr̄}g^{sj̄}u_{,r̄sλρ̄}u_{,l̄kσ̄}u_{,ij̄α}

+ g^{il̄}g^{kr̄}g^{sj̄}u_{,r̄sλ}u_{,l̄kσ̄ρ̄}u_{,ij̄α} + g^{il̄}g^{kr̄}g^{sj̄}u_{,r̄sλ}u_{,l̄kσ̄}u_{,ij̄αρ̄}
+ g^{ip̄}g^{ql̄}g^{kj̄}u_{,p̄qρ̄}u_{r̄sλ}u_{,l̄kσ̄}u_{,ij̄α} + g^{il̄}g^{kp̄}g^{qj̄}u_{,p̄qρ̄}u_{,l̄kσ̄λ}u_{,ij̄α}
− g^{il̄}g^{kj̄}u_{,l̄kσ̄λρ̄}u_{,ij̄α} − g^{il̄}g^{kj̄}u_{,l̄kσ̄λ}u_{,ij̄αρ̄}
+ g^{ip̄}g^{ql̄}g^{kj̄}u_{,p̄qρ̄}u_{,l̄kσ̄}u_{,ij̄αλ} + g^{il̄}g^{kp̄}g^{qj̄}u_{,p̄qρ̄}u_{,l̄kσ̄}u_{,ij̄αλ}
− g^{il̄}g^{kj̄}u_{,l̄kσ̄}u_{,ij̄αλρ̄} − g^{il̄}g^{kj̄}u_{,l̄kσ̄ρ̄}u_{,ij̄αλ}
+ g^{ip̄}g^{qr̄}g^{sj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,ij̄ασ̄} + g^{ir̄}g^{sp̄}g^{qj̄}u_{,p̄qρ̄}u_{,r̄sλ}u_{,ij̄ασ̄}
− g^{ir̄}g^{sj̄}u_{,r̄sλρ̄}u_{,ij̄ασ̄} − g^{ir̄}g^{sj̄}u_{,r̄sλ}u_{,ij̄ασ̄ρ̄}
− g^{ip̄}g^{qj̄}u_{,p̄qρ̄}u_{,ij̄ασ̄λ} + g^{ij̄}u_{,ij̄ασ̄λρ̄} − u_{,ασ̄λρ̄} − f_{,ασ̄λρ̄}.

Upon substituting this and its conjugate expression back into (5.19), we get two terms involving sixth derivatives of u which we examine below. Our goal of course is to recover Δ̃Q = g^{ij̄}(g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν})_{,ij̄} out of these two terms. In doing this, we show how to permute indices in any covariant derivative of u occurring above. The two sixth derivative terms are

g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ij̄ασ̄λρ̄}u_{,β̄δμ̄ν}, (5.22)

g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,ij̄β̄δμ̄ν.}

Consider the term u_{,ij̄ασ̄λρ̄} in (5.22). We would like to express this in terms of u_{,ασ̄λρ̄ij̄} as this is a term occurring in the expansion of Δ̃Q. We have,

(5.23) u_{,ij̄ασ̄λρ̄} = u_{,j̄iασ̄λρ̄}

= u_{,j̄αiσ̄λρ̄}

= (u_{,j̄ασ̄i} + Rā ; j̄σ̄iu_{āα} + R_{ασ̄i}^{a}u_{j̄a})_{,λρ̄}

= u_{,j̄ασ̄iλρ̄} + (Rā ; j̄σ̄iu_{āα} + R_{ασ̄i}^{a}u_{j̄a})_{,λρ̄}

= u_{,j̄ασ̄λiρ̄} + (Rā ; j̄σ̄iu_{āα} + R_{ασ̄i}^{a}u_{j̄a})_{,λρ̄}

= u_{,j̄ασ̄λρ̄i} + Rā ; j̄ρ̄iu_{āασ̄λ} + R_{αρ̄i}^{a}u_{j̄ασ̄λ}

+ Rā ; σ̄ρ̄iu_{j̄αāλ} + R_{λρ̄i}^{a}u_{j̄ασ̄a} + (Rā ; j̄σ̄iu_{āα} + R_{ασ̄i}^{a}u_{j̄a})_{,λρ̄}.

Note that so far we have expressed u_{,ij̄ασ̄λρ̄} in terms of u_{,j̄ασ̄λρ̄i}. Repeating the above process allows us to express u_{,j̄ασ̄λρ̄i} in terms of u_{,ασ̄λρ̄ij̄}. Doing the same for the term u_{,ij̄β̄δμ̄ν} in (5.22) and examining all the terms involving the curvature tensor R above, we have,

u_{,ij̄ασ̄λρ̄} = u_{,ασ̄λρ̄ij̄} + (terms), (5.24)

u_{,ij̄ασ̄λρ̄} = u_{,ασ̄λρ̄ij̄} + (terms),

where by (terms) we mean terms involving a contraction of the curvature tensor R (or up to three covariant derivatives of R) with derivatives of u such that the contraction is linear in u, and the derivatives of u are of degree at most four. This now gives

g^{ij̄}g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ij̄ασ̄λρ̄}u_{,β̄δμ̄ν} + g^{ij̄}g^{ij̄αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,ij̄β̄δμ̄ν} (5.25)

= g^{ij̄}(g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν})_{ij̄} − 2g^{ij̄}g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄i}u_{,β̄δμ̄νj̄} + (terms)

= Δ̃Q − 2|∇_{m}^{5}u|^{2} + (terms).

where now, by (terms) we mean terms involving a contraction of the curvature tensor R (or up to three covariant derivatives of R) with derivatives of u where the contraction is now quadratic in u, and the derivatives of u are of degree at most four (the notation |∇_{m}^{5}u|^{2} will be explained below). Expanding (5.19) and using the above technique to permute covariant derivatives of u, we have the following:

Remark 5.1. In (5.19), by adding terms involving contractions of the curvature tensor R (or up to three covariant derivatives of R) with either third mixed derivatives or second derivatives of u, we may assume all fifth derivatives occur in the form u_{,ij̄kl̄m} or u_{,ījk̄lm̄} and all fourth derivatives of u occur in the form u_{,ij̄kl̄}, u_{,ījk̄l}, u_{,ij̄lk} or u_{,ījl̄k̄}.

We distinguish between the different types of fourth and fifth derivatives above by establishing the following notation:

(5.26)  |∇_{m}^{3}u|^{2} = g^{αβ̄}g^{λμ̄}g^{νγ̄}u_{,αμ̄ν}u_{,β̄λγ̄}
|∇_{m}^{4}u|^{2} = g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}u_{,ασ̄λρ̄}u_{,β̄δμ̄ν}
|∇_{m′}^{4}u|^{2} = g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{ρν̄}u_{,ασ̄λρ}u_{,β̄δμ̄ν̄}
|∇_{m}^{5}u|^{2} = g^{αβ̄}g^{δσ̄}g^{λμ̄}g^{νρ̄}g^{ab̄}u_{,ασ̄λρ̄a}u_{,β̄δμ̄νb̄}.

By the remark above, and noting our previous bounds on the mixed third derivatives and all second derivatives of u we can now estimate the evolution of Q as follows:

(5.27) \frac{d|∇_{m}^{4}u|^{2}}{dt} ≤ Δ̃|∇_{m}^{4}u|^{2} − 2|∇_{m}^{5}u|^{2} + K|∇_{m}^{4}u||∇_{m}^{5}u| + K|∇_{m}^{4}u|^{3}
+ K|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u|^{2} + K|∇_{m}^{4}u||∇_{m′}^{4}u|^{2}
+ K|∇_{m′}^{4}u||∇_{m}^{4}u| + K

≤ Δ̃|∇_{m}^{4}u|^{2} − 2|∇_{m}^{5}u|^{2} + K|∇_{m}^{4}u||∇_{m}^{5}u| + K|∇_{m}^{4}u|^{3}

+ K|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u||∇_{m′}^{4}u|^{2} + K,

where in (5.27) and in all that follows, we will denote any strictly positive constants, not necessarily the same, by C and any other constants, not necessarily the same, by K. Of course, all constants C and K will be independent of T.

We now combine the evolution of |∇_{m}^{3}u|^{2} and Q. Remembering that |∇_{m}^{3}u|^{2} has already been estimated, using (5.12), we can estimate its evolution by

\frac{d|∇_{m}^{3}u|^{2}}{dt} ≤ Δ̃|∇_{m}^{3}u|^{2} − C|∇_{m}^{4}u|^{2} − C|∇_{m′}^{4}u|^{2} + K. (5.28)

Consider the quantity

S = (|∇_{m}^{3}u|^{2} + A)(|∇_{m}^{4}u|^{2} + B), (5.29)

where A and B are positive constants to be chosen later. Clearly, a bound on S implies a bound on Q. We begin by computing the evolution of S as

\frac{dS}{dt} = \frac{d|∇_{m}^{3}u|^{2}}{dt} (|∇_{m}^{4}u|^{2} + B) + (|∇_{m}^{3}u|^{2} + A)\frac{d|∇_{m}^{4}u|^{2}}{dt} (5.30)
≤ (Δ̃|∇_{m}^{3}u|^{2} − C|∇_{m}^{4}u|^{2} − C|∇_{m′}^{4}u|^{2} + K)(|∇_{m}^{4}u|^{2} + B)

+ (|∇_{m}^{3}u|^{2} + A)(Δ̃|∇_{m}^{4}u|^{2} − 2|∇_{m}^{5}u|^{2} + K|∇_{m}^{4}u||∇_{m}^{5}u| + K

+ K|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u||∇_{m′}^{4}u|^{2} + K)
≤ Δ̃|∇_{m}^{3}u|^{2}(|∇_{m}^{4}u|^{2} + B) − C|∇_{m}^{4}u|^{4} − C|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2}
− CB|∇_{m′}^{4}u|^{2} + (|∇_{m}^{4}u|^{2}m2 + A)Δ̃|∇_{m}^{4}u|^{2} − 2A|∇_{m}^{5}u|^{2}
+ AK|∇_{m}^{4}u||∇_{m}^{5}u| + AK|∇_{m}^{4}u|^{3} + AK|∇_{m′}^{4}u|^{2}
+ AK|∇_{m}^{4}u||∇_{m′}^{4}u|^{2} + K

≤ Δ̃S − 2A|∇_{m}^{5}u|^{2} − C|∇_{m}^{4}u|^{4} − C|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2} − CB|∇_{m′}^{4}u|^{2}

+ K|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| + AK|∇_{m}^{4}u||∇_{m}^{5}u| + K|∇_{m′}^{4}u||∇_{m}^{4}u||∇_{m}^{5}u|
+ AK|∇_{m}^{4}u|^{3} + AK|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|
+ AK|∇_{m}^{4}u||∇_{m′}^{4}u|^{2} + K.

The last inequality follows from the identity

(5.31)  Δ̃S = Δ̃|∇_{m}^{3}u|^{2}(|∇_{m}^{4}u|^{2} + B) + (|∇_{m}^{3}u|^{2} + A)Δ̃|∇_{m}^{4}u|^{2}

+ g^{αβ̄}|∇_{m}^{3}u|_{α}^{2}|∇_{m}^{4}u|2 ; β̄,

and by noting that the last term above is bounded by a linear combination of the terms |∇_{m}^{4}u|^{3} , |∇_{m′}^{4}u||∇_{m}^{4}u|^{2} , |∇_{m′}^{4}u||∇_{m}^{4}u||∇_{m}^{5}u| and |∇_{m}^{4}u|^{2}|∇_{m}^{5}u|.

We now state two elementary propositions which we will use to estimate the last expression in (5.30).

Proposition 5.2. Let C_{1} and C_{2} be given constants with C_{1} ≥ 0. Then there exists a constant C_{4}, depending only on C_{1} and C_{2}, such that for all x, y ≥ 0 the following inequality holds:

−C_{1}x^{2}y + C_{2}xy ≤ −(C_{1}/2)x^{2}y + C_{4}y.

Proof. Choose C_{4} such that for all x ≥ 0,

x^{2} − (C_{2}/C_{1})x ≥ (1/2)x^{2} − (C_{4}/C_{1}).  q.e.d.

Proposition 5.3. Let C_{1} and C_{2} be given constants with C_{1} ≥ 0. Then there exists a constant C_{3} ≥ 0, depending only on C_{1} and C_{2}, such that for all x, y ≥ 0 the following inequality holds:

−C_{3}x^{2} + C_{2}xy − C_{1}y^{2} ≤ −(C_{3}/2)x^{2} − (C_{1}/2)y^{2}.

Proof. Begin by writing

(\sqrt{(C_{3}/2)}x − \sqrt{(C_{1}/2)}y)^{2} = (C_{3}/2)x^{2} − \sqrt{C_{1}C_{3}}xy + (C_{1}/2)y^{2} ≥ 0.

Then from the above inequality and for a sufficiently large choice of C_{3} we have:

−(C_{3}/2)x^{2} − (C_{1}/2)y^{2} ≤ −\sqrt{C_{1}C_{3}}xy. ≤ −C_{2}xy.

This gives

−(C_{3}/2)x^{2} + C_{2}xy − (C_{1}/2)y^{2} ≤ 0,

and thus

−C_{3}x^{2} + C_{2}xy − C_{1}y^{2} ≤ −(C_{3}/2)x^{2} − (C_{1}/2)y^{2}.

q.e.d.

Using the two propositions above we now continue our estimate on the evolution of S. K_{A} will denoted any constant depending only on A.

\frac{dS}{dt} ≤ Δ̃S − 2A|∇_{m}^{5}u|^{2} − C|∇_{m}^{4}u|^{4} − C|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2} − CB|∇_{m′}^{4}u|^{2} (5.32)

+ K|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| + AK|∇_{m}^{4}u||∇_{m}^{5}u| + K|∇_{m′}^{4}u||∇_{m}^{4}u||∇_{m}^{5}u|

+ AK|∇_{m}^{4}u|^{3} + AK|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|
+ AK|∇_{m}^{4}u||∇_{m′}^{4}u|^{2} + K

≤ Δ̃S − 2A|∇_{m}^{5}u|^{2} − C|∇_{m}^{4}u|^{4} − C|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2} − CB|∇_{m′}^{4}u|^{2}

+ C|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| + K_{A}|∇_{m}^{5}u| + K|∇_{m′}^{4}u||∇_{m}^{4}u||∇_{m}^{5}u|
+ AK|∇_{m}^{4}u|^{3} + AK|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u|^{2} + K_{A}|∇_{m′}^{4}u|^{2} + K
≤ Δ̃S − A|∇_{m}^{5}u|^{2} − (C/2)|∇_{m}^{4}u|^{4} − (C/2)|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2}
− CB|∇_{m′}^{4}u|^{2} + K_{A}|∇_{m}^{5}u| + AK|∇_{m}^{4}u|^{3} + AK|∇_{m′}^{4}u|^{2}
+ K|∇_{m}^{4}u|^{2} + K_{A}|∇_{m′}^{4}u|^{2} + K.

The second inequality follows by applying Proposition 5.2 to the expressions

−(C/2)|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2} + K|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|,

and

−(C/2)|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2} + AK|∇_{m}^{4}u||∇_{m′}^{4}u|^{2},

formed by grouping terms from the right-hand side of the first inequality, and also by estimating

K|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| + AK|∇_{m}^{4}u||∇_{m}^{5}u| ≤ C|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| + K_{A}|∇_{m}^{5}u|,

for C large enough. The third inequality follows by applying the Proposition 5.3 to the terms

−A|∇_{m}^{5}u|^{2} + C|∇_{m}^{4}u|^{2}|∇_{m}^{5}u| − C|∇_{m}^{4}u|^{4},

and

−A|∇_{m}^{5}u|^{2} + K|∇_{m′}^{4}u||∇_{m}^{4}u||∇_{m}^{5}u| − C|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2},

appearing in the right-hand side of the second inequality.

By the last inequality in (5.32), we can choose the constants A and B large enough to give the following estimate:

(5.33) \frac{dS}{dt} ≤ Δ̃S − A|∇_{m}^{5}u|^{2} − (C/2)|∇_{m}^{4}u|^{4} − (C/2)|∇_{m}^{4}u|^{2}|∇_{m′}^{4}u|^{2}
− CB|∇_{m′}^{4}u|^{2} + K.

By applying the maximum principle to (5.33), we can bound sup_{(x,t)∈M×[0,T)} S(x, t), and thus also sup_{(x,t)∈M×[0,T)} Q(x, t), independent of T.

5.5. Hölder norm estimates. Recall that (M, g_{ij̄}) has bounded geometry of order ∞. In Definition 2.1, choose some large m and consider a holomorphic chart ξ_{p} : V_{p} ⊆ ℂ^{n} → U_{i} for arbitrary p ∈ M. Throughout this subsection, all local expressions will be with respect to the standard coordinates on V_{p} ∈ ℂ^{n}; C^{k+α} and C^{k+α,k/2+α/2} will denote the standard Hölder spaces on V_{p} and V_{p} × [0, T) (or subsets of these) respectively; C^{k+α}(M) and C^{k+α,k/2+α/2}(M ×[0, T)) will denote Hölder spaces relative to g_{ij̄}; all estimates derived will be independent of p and T and in general we will refer to any quantity independent of p and T as being a uniform quantity.

At any time t ∈ [0, T), in V_{p}, we have an estimate for |u|, Δu and the Hölder derivative in space of Δu (this last estimate follows from our estimate on the third mixed derivatives of u). We apply the standard elliptic Schauder interior estimates to u in V_{p} to get an estimate on the C^{2+α} norm of u in some uniformly large subset of V_{p}. Also, by looking at (1.2) we see that our third and fourth derivative estimates in space are in fact equivalent to estimates on u_{it}, u_{j̄t}, u_{ij̄t} and u_{tt} in V_{p}. Using all these we obtain an estimate on the C^{2+α,1+α/2} norm of u in U ×[0, T) for some uniformly large subset U ⊂ V_{p}. To see that we can extend this to an estimate on the C^{4+α,2+α/2} norm, we differentiate (1.2) in the natural coordinates on V_{p} giving

(5.34)  \frac{du_{k}}{dt} = g^{αβ̄}u_{,αβ̄k} − u_{k} − f_{k}
= g^{αβ̄}u_{,kαβ̄} + g^{αβ̄}R_{αkβ̄}^{a}u_{a} − u_{k} − f_{k}
= Δ̃u_{k} + g^{αβ̄}R_{αkβ̄}^{a}u_{a} − u_{k} − f_{k}
= Δ̃u_{k} − u_{k} + (g^{αβ̄}R_{αkβ̄}^{a}u_{a} − f_{k}).

We view (5.34) as a linear parabolic equation for u_{k} in V_{p} with the last term viewed as a single forcing term. It is clear that our previous estimate on the C^{2+α,1+α/2} norm of u provides an estimate on the C^{α,α/2} norm of the coefficients and forcing term in the linear equation under consideration. By standard parabolic Schauder estimates, we get an estimate on the C^{2+α,1+α/2} norm of u_{k} in Ũ × [0, T) for a uniformly large subset Ũ ⊂ U . This then gives an estimate on the C^{α,α/2} norm of the first space derivatives of the coefficients and forcing term in Ũ. We again apply the Schauder estimates to obtain an estimate on the C^{2+α,1+α/2} norm of the first space derivatives of u_{k} in U′ × [0, T) for a uniformly large subset U′ ⊂ Ũ. Noting that the index k is arbitrary, and repeating the above process for a barred index k̄ it is easily checked that this is in fact equivalent to an estimate on the C^{4+α,2+α/2} norm of u in U′ × [0, T). Noting that U′ in V_{p} can be made uniform and the choice of V_{p} was arbitrary, we see that this gives an estimate for the C^{4+α,2+α/2}(M ×[0, T)) norm of u relative to the maps ξ_{p}. In particular, we see that the curvature of the corresponding evolving metrics stays uniformly bounded. Finally, we note that the above process can be repeated successively giving an estimate for the C^{k+α,k/2+α/2}(M×[0, T)) norm of u relative to the maps ξ_{p} for every k ≤ m, and that m was chosen arbitrarily large. In particular, u(x, t) is a smooth function of space and time.

6. Long time existence and convergence

Lemma 4.1 together with our a priori estimates allow us to prove the following:

Lemma 6.1. Assume (M, g_{ij̄}) is a complete noncompact Kähler manifold with bounded geometry of infinite order and that R_{ij̄}+ g_{ij̄} = f_{ij̄} for some smooth bounded function f on M. Let C^{n+α} denote the Hölder spaces on M relative to the metric g_{ij̄}. Then (1.2) has a smooth long time solution u(x, t) which converges, as t → ∞, in every C^{n+α} norm, to a smooth limit u(x, ∞). Moreover, g̃_{ij̄}(x, t) := g_{ij̄}(x) + u_{ij̄}(x, t) provides a smooth long time solution to (1.1) and converges, as t → ∞, in every C^{n+α} norm, to a complete Kähler–Einstein metric g̃_{ij̄}(x, ∞) where g̃_{ij̄}(x, ∞) has negative scalar curvature on M, is equivalent to g_{ij̄}, and has all covariant derivatives of its curvature tensor bounded.

Proof. By Lemma 4.1 we know ∃T > 0 such that there is a smooth short time solution u(x, t) to (1.2) on M × [0, T). Moreover, we may assume that [0, T) is the maximal time interval on which we have such a solution. Our estimates from Section 5 show that the C^{2+α,1+ \frac{α}{2}} norm of u is bounded on M × [0, T) independent of T. Combining this with the arguments from Section 5.5 and the assumption of bounded geometry of order ∞ implies that for every n the C^{n+α} norm of u(x, t) is bounded independent of t. This allows us to take a limit of u(x, t), in C^{n+α} for any n, as t → T, yielding a limit metric g̃_{ij̄}(x, T) := g_{ij̄}(x) + u_{ij̄}(x, T) on M equivalent to g_{ij̄} and with bounded geometry of order ∞. Now if T < ∞, by Lemma 4.1, with g̃_{ij̄}(x, T) as initial metric, we see that we may continue our solution u(x, t) past T for some short time thus contradicting our assumption that T is maximal and thus proving long time existence of u(x, t).

It remains to be shown that for any n, u(x, t) converges in C^{n+α} as t → ∞. By subsection 5.1 u(x, t) converges in the C^{0} norm to a unique limit u(x, ∞). Moreover, we have seen that for any n the C^{n+α} norm of u(x, t) remains uniformly bounded in time. Combining these two facts, it is easily seen that u(x, ∞) ∈ C^{n+α} ∀n, and u(x, t) must converge in every C^{n+α} norm to u(x, ∞). Taking the limit of (1.2) as t → ∞, and noting the estimates of the previous section, we see that the limit metric g̃_{ij̄}(x, ∞) is a Kähler–Einstein metric as stated in the lemma. q.e.d.

Our main Theorem 1.1 now follows immediately from Lemma 4.1 and Lemma 6.1.

References

[1] T. Aubin, Some nonlinear problems in Riemannian Geometry, Springer-Verlag, Berlin, 1998, MR 1636569 (99i:58001), Zbl 0896.53003.

[2] H.-D. Cao, Deformation of Kahler metrics to Kahler Einstein metrics on compact Kahler manifolds, Invent. Math. 81(2) (1985), 359–372, MR 0799272 (87d:58051), Zbl 0574.53042.

[3] H.-D. Cao & B. Chow, Recent developements on the Ricci flow, Bull. Amer. Math. Soc. (N.S.) 36(1) (1999) 59–74, MR 1655479 (99j:53044), Zbl 0926.53016.

[4] S.Y. Cheng & S.-T. Yau, On the existence of a complete Kahler Einstein metric on noncompact complex manifolds and the regularity of Feffermans equation, Comm. Pure App. Math. 33 (1980) 507–544, MR 0575736 (82f:53074), Zbl 0506.53031.

[5] S.Y. Cheng & S.-T. Yau, Inequality between Chern numbers of singular Kahler surfaces and characterization of orbit space of discrete group of SU(2, 1), Contemporary Math. 49 (1986) 31–43, MR 0833802 (87m:53078), Zbl 0597.53052.

[6] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. of Differential Geometry 17(2) (1982) 255–306, MR 0664497 (84a:53050), Zbl 0504.53034.

[7] R.S. Hamilton, The Ricci flow on surfaces, Contemporary Mathematics 71 (1988) 237–261, MR 0954419 (89i:53029), Zbl 0663.53031.

[8] N.V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996, MR 1406091 (97i:35001), Zbl 0865.35001.

[9] W.-X. Shi, Ricci deformation of the metric on complete Riemannian manifolds, J. of Differential Geometry 30 (1989) 303–394, MR 1010165, Zbl 0686.53037.

[10] W.-X. Shi, Ricci deformation of the metric on complete noncompact Kähler manifolds, PhD Thesis, Harvard University, 1990.

[11] W.-X. Shi, Ricci Flow and the uniformization on complete noncompact Kahler manifolds, J. of Differential Geometry 45(1) (1997) 94–220, MR 1443333 (98d:53099), Zbl 0954.53043.

[12] G. Tian & S.-T. Yau, Existence of Kahler Einstein metrics on complete Kahler manifolds and their applications to algebraic geometry, in ‘Mathematical Aspects of String Theory’, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1 (1987) 574–629, MR 0915840, Zbl 0682.53064.

[13] G. Tian & S.-T. Yau, Complete Kahler manifolds with zero Ricci curvature, I, J. Amer. Math. Soc. 3(3) (1990) 579–609, MR 1040196 (91a:53096), Zbl 0719.53041.

[14] G. Tian & S.-T. Yau, Complete Kahler manifolds with zero Ricci curvature, II, Invent. Math. 106(1) (1990) 27–60, MR 1123371 (92j:32028), Zbl 0766.53053.

[15] S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge–Ampère equation I, Comm. Pure App. Math. 331 (1978) 339–411, MR 0480350 (81d:53045), Zbl 0369.53059.

[16] S.-T. Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), 197–203, MR 0486659 (58 #6370), Zbl 0424.53040.

[17] S.-T. Yau, A splitting theorem and an algebraic geometric characterization of locally Hermitian symmetric spaces, Comm. Anal. Geom. 1 (1993) 473–486, MR 1266476 (95f:32039), Zbl 0842.53035.

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E-mail address: chau@math.harvard.edu