j. differential geometry

63 (2003) 131-154

A FULLY NONLINEAR EQUATION ON FOUR-MANIFOLDS WITH POSITIVE SCALAR

CURVATURE

MATTHEW J. GURSKY & JEFF A. VIACLOVSKY

Abstract

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

1. Introduction

Let (M, g) denote a closed, 4-dimensional Riemannian manifold, and let Y [g] denote the Yamabe invariant of the conformal class [g]:

(1.1) Y [g] ≡ inf_{g̃∈[g]}Vol (g̃)^{−1/2} ∫_{M} Rg̃dvolg̃,

where Rg̃ denotes the scalar curvature. Another important conformal invariant is

F_{2}([g]) ≡ ∫_{M} (−\frac{1}{2} | Ric_{g}|^{2} + \frac{1}{6}R_{g}^{2}) dvol_{g}, (1.2)

The first author was partially supported by NSF Grants DMS-0200646 and INT-0229457. The second author was partially supported by NSF Grant DMS-0202477.

Received 01/31/2003.

where Ric_{g} is the Ricci tensor. By the Chern-Gauss-Bonnet formula ([4]),

8π^{2}χ(M) = ∫_{M} |W_{g}|^{2}dvol_{g} + F_{2}([g]). (1.3)

Thus, the conformal invariance of F_{2} follows from the well-known (pointwise) conformal invariance of the Weyl tensor W_{g} (see [13]).

Define the tensor

A_{g}^{t} = \frac{1}{2}(Ric_{g} − \frac{t}{6}R_{g}g). (1.4)

Note that for t = 1, A_{g}^{1} is the classical Schouten tensor ([13]). Let σ_{2}(g^{−1}A_{g}^{t}) denote the second elementary symmetric function of the eigenvalues of g^{−1}A_{g}^{t}, viewed as an endomorphism of the tangent bundle. Then a simple calculation gives

F_{2}([g]) = 4 ∫_{M} σ_{2}(g^{−1}A_{g}^{1})dvol_{g}. (1.5)

Our main result is the following:

Theorem 1.1. Let (M, g) be a closed 4-dimensional Riemannian manifold with positive scalar curvature. If

F_{2}([g]) + \frac{1}{6}(1 − t_{0})(2 − t_{0})(Y [g])^{2} > 0, (1.6)

for some t_{0} ≤ 1, then there exists a conformal metric g̃ = e^{−2u}g with Rg̃ > 0 and σ_{2}(At_{0} ; g̃) > 0 pointwise. This implies the pointwise inequalities

(t_{0} − 1)Rg̃g̃ < 2 Ricg̃ < (2 − t_{0})Rg̃g̃. (1.7)

As applications of Theorem 1.1, we consider two different values of t_{0}. When t_{0} = 1, we obtain a different proof of the following result in [8]:

Corollary 1.1. Let (M, g) be a closed 4-dimensional Riemannian manifold with positive scalar curvature. If F_{2}([g]) > 0, then there exists a conformal metric g̃ = e^{−2u}g with Rg̃ > 0 and σ_{2}(g^{−1}A1 ; g̃) > 0 pointwise. In particular, the Ricci curvature of g̃ satisfies

0 < 2 Ricg̃ < Rg̃g̃.

The proof in [8] involved regularization by a fourth-order equation and relied on some delicate integral estimates. By contrast, the proof of Theorem 1.1 seems more direct, and depends on general a priori estimates for fully nonlinear equations developed in [30], [20], [27], and [23].

Our second application is to the spectral properties of a conformally invariant differential operator known as the Paneitz operator. Let δ denote the L^{2}-adjoint of the exterior derivative d; then the Paneitz operator is defined by

P_{g}ϕ = Δ^{2}ϕ + δ(\frac{2}{3}R_{g}g − 2 Ric_{g})dϕ. (1.8)

The Paneitz operator is conformally invariant, in the sense that if g̃ = e^{−2u}g, then

Pg̃ = e^{4u}P_{g}. (1.9)

Since the volume form of the conformal metric g̃ is dvolg̃ = e^{−4u}dvol_{g}, an immediate consequence of (1.9) is the conformal invariance of the Dirichlet energy

⟨Pg̃ϕ, ϕ⟩_{L^{2}(M,g̃)} = ⟨P_{g}ϕ, ϕ⟩_{L^{2}(M,g)}.

In particular, positivity of the Paneitz operator is a conformally invariant property, and clearly the kernel is invariant as well.

To appreciate the geometric significance of the Paneitz operator we need to define the associated Q-curvature, introduced by Branson:

Q_{g} = −\frac{1}{12}ΔR_{g} + 2σ_{2}(g^{−1}A_{g}^{1}). (1.10)

Under a conformal change of metric g̃ = e^{−2u}g, the Q-curvature transforms according to the equation

−P u + 2Q_{g} = 2Qg̃e^{−4u}, (1.11)

see, for example, [5]. Note that

∫_{M} Q_{g}dvol_{g} = \frac{1}{2}F_{2}([g]), (1.12)

so the integral of the Q-curvature is conformally invariant.

The Q-curvature and Paneitz operator have become important objects of study in the geometry of four-manifolds, and play a role in such diverse topics as Moser-Trudinger inequalities ([3], [6]), compactification of complete conformally flat manifolds ([9]), twistor theory ([11]), gauge choices for Maxwell’s equations ([12]), and most recently in the study of conformally compact AHE manifolds ([15], [18]).

Our interest here is in the spectral properties of the Paneitz operator and the related question of the existence of metrics with constant Q-curvature. The most general work on this subject was done by Chang and Yang [10], who studied the problem of constructing conformal metrics with constant Q-curvature by variational methods. They considered the functional

F [ϕ] = ⟨P_{g}ϕ, ϕ⟩ − 4 ∫_{M} Qϕdvol − (∫_{M} Qdvol) log ∫_{M} e^{−4ϕ}dvol, (1.13)

and analyzed the behavior of a minimizing sequence. Of course, it is not clear a priori that F is even bounded from below. Indeed, if the Paneitz operator has a negative eigenvalue and the conformal invariant (1.12) is positive, then Chang and Yang showed that inf F = −∞ (see [10], p. 177). For example, take a compact surface Σ of curvature −1 with first eigenvalue λ_{1}(−Δ) << 1. Then the product manifold M = Σ × Σ will have λ_{1}(P) < 0 and ∫ Qdvol > 0.

Chang and Yang also pointed out the connection between the conformal invariant (1.12) and the best constant in the inequality of Adams [1], another key point for establishing the W^{2,2} compactness of a minimizing sequence. Combining these observations, they proved:

Theorem 1.2 ([10]). Let (M, g) be a compact 4-manifold. Assume:

(i) The Paneitz operator P_{g} is nonnegative with Ker P = {constants}.

(ii) The conformal invariant (1.12) is strictly less than the value attained by the round sphere.

Then there exists a minimizer of F, which satisfies (1.11) with Qg̃ = constant.

Subsequently, the first author proved that any four-manifold of positive scalar curvature which is not conformally equivalent to the sphere already satisfies the second assumption of Chang-Yang. In addition:

Theorem 1.3 ([21]). Let (M, g) be a compact 4-manifold. If the scalar curvature of g is nonnegative and ∫ Qdvol ≥ 0, then the Paneitz operator is positive and Ker P = {constants}.

Because of the example of Chang-Yang, it is clear that one cannot relax the condition on the scalar curvature in the above theorem. On the other hand, the positivity of the conformal invariant (1.12) is a rather strong assumption. For example, if the scalar curvature is strictly positive, then the positivity of (1.12) implies the vanishing of the first Betti number of M (see [22]). Thus, for example, the manifold N#(S^{1}× S^{3}) can not admit a metric of positive scalar curvature with ∫ Qdvol > 0.

It is interesting to note that the positivity of the Paneitz operator was studied by Eastwood and Singer in [11] for reasons motivated by twistor theory. They constructed metrics on k(S^{3}×S^{1}) for all k > 0 with P ≥ 0 and Ker P = {constants}. Since these manifolds have ∫ Qdvol < 0, the Eastwood-Singer construction is in some respects complementary to the result of [21].

By combining Theorem 1.1 with t_{0} = 0, and an integration by parts argument, we obtain a new criterion for the positivity of P:

Theorem 1.4. Let (M, g) be a closed 4-dimensional Riemannian manifold with positive scalar curvature. If

∫ Q_{g}dvol_{g} + \frac{1}{6}(Y [g])^{2} > 0, (1.14)

then the Paneitz operator is nonnegative, and Ker P = {constants}. Therefore, by the results in [10], there exists a conformal metric g̃ = e^{−2u}g with Qg̃ = constant.

Since Theorem 1.4 allows the integral of the Q-curvature to be negative, we are able to use surgery techniques to construct many new examples of manifolds which admit metrics with constant Q. For example, we will show that

N = (S^{2} × S^{2})#k(S^{1} × S^{3}), k ≤ 5,
N = ℂℙ^{2}#k(S^{1} × S^{3}), k ≤ 5,
N = ℂℙ^{2}#k(ℝℙ^{4}), k ≤ 8,
N = k(S^{1} × S^{3})#l(ℝℙ^{4}), 2k + l ≤ 9,

all admit metrics with constant Q. See Section 7 for additional examples.

For the proof of Theorem 1.1, we will be concerned with the following equation for a conformal metric g̃ = e^{−2u}g:

σ_{2}^{1/2}(g^{−1}At ; g̃) = f(x)e^{2u}, (1.15)

where f(x) > 0. We have the following formula for the transformation of A^{t} under a conformal change of metric g̃ = e^{−2u}g:

At ; g̃ = A_{g}^{t} + ∇^{2}u + \frac{1 − t}{2}(Δu)g + du ⊗ du − \frac{2 − t}{2}|∇u|^{2}g. (1.16)

Since A^{t} = A^{1} + \frac{1−t}{2}tr(A^{1})g, this formula follows easily from the standard formula for the transformation of the Schouten tensor (see [30]):

A1 ; g̃ = A_{g}^{1} + ∇^{2}u + du ⊗ du − \frac{1}{2}|∇u|^{2}g. (1.17)

Using (1.16), we may write (1.15) with respect to the background metric g

(1.18) σ_{2}^{1/2}(g^{−1}(∇^{2}u + \frac{1 − t}{2}(Δu)g

− \frac{2 − t}{2}|∇u|^{2}g + du ⊗ du + A_{g}^{t})) = f(x)e^{2u}.

The choice of the right-hand side in (1.18) is quite flexible; the key requirement is simply that the exponent is a positive multiple of u. For negative exponents we lose the invertibility of the linearized equation and some key a priori estimates; see the proofs of Propositions 2.2 and 3.1.

Equation (1.18) was considered in our earlier work ([23]) in the context of negative curvature. Li and Li ([27]) used a similar path to prove existence of solutions of the conformally invariant equation involving more general symmetric functions of the eigenvalues, assuming the manifold is locally conformally flat. After completing this paper, we also received the preprint of Guan, Lin and Wang ([19]), where they used a similar deformation technique to obtain various results in the locally conformally flat setting.

We will use the continuity method: the assumption of positive scalar curvature will allow us to start at some t = δ very negative. We will then use the conformally invariant assumption (1.6) in Section 3, together with the Harnack inequality of [20] and [27] in Section 4, to prove compactness of the space of solutions. Existence of a solution at time t_{0} and verification of the inequalities (1.7) will be proved in Section 5, thus completing the proof of Theorem 1.1. Theorem 1.4 will be proved in Section 6, and in Section 7 we give many new examples of manifolds admitting metrics with constant Q-curvature.

2. Ellipticity

In this section we will discuss the ellipticity properties of Equation (1.18).

Definition 1. Let (λ_{1}, …, λ_{4}) ∈ R^{4}. We view the second elementary symmetric function as a function on R^{4}:

σ_{2}(λ_{1}, …, λ_{4}) = ∑_{i<j} λ_{i}λ_{j}, (2.1)

and we define

Γ_{2}^{+} = {σ_{2} > 0} ∩ {σ_{1} > 0}, (2.2)

where σ_{1} = λ_{1} + ⋯ + λ_{4} denotes the trace.

For a symmetric linear transformation A : V → V, where V is an n-dimensional inner product space, the notation A ∈ Γ_{2}^{+} will mean that the eigenvalues of A lie in the corresponding set. We note that this notation also makes sense for a symmetric tensor on a Riemannian manifold. If A ∈ Γ_{2}^{+}, let σ_{2}^{1/2}(A) = {σ_{2}(A)}^{1/2}.

Definition 2. Let A : V → V be a symmetric linear transformation, where V is an n-dimensional inner product space. The first Newton transformation associated with A is

T_{1}(A) = σ_{1}(A) · I − A. (2.3)

Also, for t ∈ R we define the linear transformation

L^{t}(A) = T_{1}(A) + \frac{1 − t}{2}σ_{1}(T_{1}(A)) · I. (2.4)

We note that if A_{s} : R → Hom(V, V), then

\frac{d}{ds}σ_{2}(A_{s}) = ∑_{i,j} T_{1}(A_{s})_{ij} \frac{d}{ds}(A_{s})_{ij}, (2.5)

that is, the first Newton transformation is what arises from differentiation of σ_{2}.

Proposition 2.1.

(i) The set Γ_{2}^{+} is an open convex cone with vertex at the origin.

(ii) If the eigenvalues of A are in Γ_{2}^{+}, then T_{1}(A) is positive definite. Consequently, for t ≤ 1, L^{t}(A) is also positive definite.

(iii) For symmetric linear transformations A ∈ Γ_{2}^{+}, B ∈ Γ_{2}^{+}, and s ∈ [0, 1], we have the following inequality

{σ_{2}((1 − s)A + sB)}^{1/2} ≥ (1 − s){σ_{2}(A)}^{1/2} + s{σ_{2}(B)}^{1/2}. (2.6)

Proof. The proof of this proposition is standard, and may be found in [7] and [16]. q.e.d.

For u ∈ C^{2}(M), we define

(2.7)  A_{u}^{t} = A_{g}^{t} + ∇^{2}u + \frac{1 − t}{2}(Δu)g + du ⊗ du − \frac{2 − t}{2}|∇u|^{2}g.

Proposition 2.2. Let u ∈ C^{2}(M) be a solution of

σ_{2}^{1/2}(g^{−1}A_{u}^{t}) = f(x)e^{2u}, (2.8)

for some t ≤ 1 with A_{u}^{t} ∈ Γ_{2}^{+}. Then the linearized operator at u, L^{t} : C^{2,α}(M) → C^{α}(M), is invertible (0 < α < 1).

Proof. We define

F_{t}[x, u, ∇u, ∇^{2}u] = σ_{2}(g^{−1}A_{u}^{t}) − f(x)^{2}e^{4u},

so that solutions of (2.8) are zeroes of F_{t}. We then suppose that u ∈ C^{2}(M) satisfies F_{t}[x, u, ∇u, ∇^{2}u] = 0, with A_{u}^{t} ∈ Γ_{2}^{+}. Define u_{s} = u + sφ, then

(2.9)  L^{t}(φ) = \frac{d}{ds}F_{t}[x, u_{s}, ∇u_{s}, ∇^{2}u_{s}]|_{s=0}

= \frac{d}{ds}(σ_{2}(g^{−1}A_{u_{s}}^{t}))|_{s=0} − \frac{d}{ds} (f^{2}e^{4u_{s}}) |_{s=0}.

From (2.5), we have (using the summation convention)

\frac{d}{ds}(σ_{2}(g^{−1}A_{u}^{t}))|_{s=0} = T_{1}(g^{−1}A_{u}^{t})_{ij} \frac{d}{ds}((g^{−1}A_{u_{s}}^{t})_{ij})|_{s=0}.

We compute

\frac{d}{ds}((g^{−1}A_{u_{s}}^{t}))|_{s=0}

= g^{−1}(∇^{2}φ + \frac{1 − t}{2}(Δφ)g − (2 − t)⟨du, dφ⟩g + 2du ⊗ dφ).

Therefore,

\frac{d}{ds}(σ_{2}(g^{−1}A_{u_{s}}^{t}))|_{s=0} = T_{1}(g^{−1}A_{u}^{t})_{ij}\{g^{−1}(∇^{2}φ + (1 − t)(Δφ)(g/2) (2.10)
− (2 − t)⟨du, dφ⟩g + 2du ⊗ dφ)}_{ij}.

For the second term on the right-hand side of (2.9) we have

\frac{d}{ds} (f^{2}e^{4u_{s}}) |_{s=0} = 4f^{2}e^{4u}φ. (2.11)

Combining (2.10) and (2.11), we conclude

L^{t}(φ) = T_{1}(g^{−1}A_{u}^{t})_{ij}{g^{−1}(∇^{2}φ + (1 − t)(Δφ)(g/2))}_{ij} − 4f^{2}e^{4u}φ + ⋯ (2.12)

where + ⋯ denotes additional terms which are linear in ∇φ. Using the definition of L^{t} in (2.4), we can rewrite the leading term of (2.12) and obtain

L^{t}(φ) = L^{t}(g^{−1}A_{u}^{t})_{ij}(g^{−1}∇^{2}φ)_{ij} − 4f^{2}e^{4u}φ + ⋯ (2.13)

For t ≤ 1, Proposition 2.1 implies that L^{t}(g^{−1}A_{u}^{t}) is positive definite, so L^{t} is elliptic. Since the coefficient of φ in the zeroth-order term of (2.13) is strictly negative, the lineariztion is furthermore invertible on the stated Hölder spaces (see [17]). q.e.d.

3. C^{0} estimate

Throughout the sequel, (M, g) will be a closed 4-dimensional Riemannian manifold with positive scalar curvature. Since R_{g} > 0, there exists δ > −∞ so that A_{g}^{δ} is positive definite. For t ∈ [δ, 1], consider the path of equations

σ_{2}^{1/2}(g^{−1}A_{u_{t}}^{t}) = f(x)e^{2u_{t}}, (3.1)

where f(x) = σ_{2}^{1/2}(g^{−1}A_{g}^{δ}) > 0. Note that u ≡ 0 is a solution of (3.1) for t = δ.

Proposition 3.1. Let u_{t} ∈ C^{2}(M) be a solution of (3.1) for some δ ≤ t ≤ 1. Then u_{t} ≤ \overline{δ}, where \overline{δ} depends only upon g.

Proof. From Newton’s inequality \frac{4}{\sqrt{6}}σ_{2}^{1/2} ≤ σ_{1}, so

\frac{4}{\sqrt{6}}f(x)e^{2u_{t}} ≤ σ_{1}(g^{−1}A_{u_{t}}^{t}). (3.2)

Let p be a maximum of u_{t}, then the gradient terms vanish at p, and Δu_{t} ≤ 0, so by (1.16)

\frac{4}{\sqrt{6}}f(p)e^{2u_{t}(p)} ≤ σ_{1}(g^{−1}A_{u_{t}}^{t})(p)
= σ_{1}(g^{−1}A_{g}^{t}) + (3 − 2t)Δu_{t}
≤ σ_{1}(g^{−1}A_{g}^{t}).

Since t ≥ δ, this implies u_{t} ≤ \overline{δ}. q.e.d.

Proposition 3.2. Assume that for some δ ≤ t ≤ 1,

F_{2}([g]) + \frac{1}{6}(1 − t)(2 − t)(Y [g])^{2} = λ_{t} > 0. (3.3)

If u_{t} ∈ C^{2}(M) is a solution of (3.1) satisfying ∥∇u_{t}∥_{L^{∞}} < C_{1}, then u_{t} > \underline{δ}, where \underline{δ} depends only upon g, C_{1}, and log λ_{t}.

Proof. Using Lemma 24 in [30], we have

σ_{2}(A^{t}) = σ_{2}(A^{1} + \frac{1 − t}{2}σ_{1}(A^{1})g)

= σ_{2}(A^{1}) + 3\frac{1 − t}{2}σ_{1}(A^{1})^{2} + 6(\frac{1 − t}{2}σ_{1}(A^{1}))^{2}

= σ_{2}(A^{1}) + \frac{3}{2}(1 − t)(2 − t)σ_{1}(A^{1})^{2}.

Letting g̃ = e^{−2u_{t}}g,

e^{4u_{t}}f^{2} = σ_{2}(g^{−1}A_{u_{t}}^{t}) = σ_{2}(g^{−1}A_{u_{t}}^{1}) + \frac{3}{2}(1 − t)(2 − t)(σ_{1}(g^{−1}A_{u_{t}}^{1}))^{2}

= e^{−4u_{t}}(σ_{2}(g^{−1}A_{u_{t}}^{1}) + \frac{1}{24}(1 − t)(2 − t)R2 ; g̃).

a fully nonlinear equation 141

Integrating this, we obtain C′ ∫_{M} e^{4u_{t}}dvol_{g} ≥ ∫_{M} f^{2}e^{4u_{t}}dvol_{g}

= ∫_{M} σ_{2}(g^{−1}A_{u_{t}}^{1})e^{−4u_{t}}dvol_{g} + \frac{1}{24}(1 − t)(2 − t) ∫_{M} R2 ; g̃e^{−4u_{t}}dvol_{g}

= ∫_{M} σ_{2}(g^{−1}A1 ; g̃)dvolg̃ + \frac{1}{24}(1 − t)(2 − t) ∫_{M} R2 ; g̃dvolg̃,

where C′ > 0 is chosen so that f^{2} ≤ C′.

Lemma 3.1. For any metric g′ ∈ [g], we have ∫_{M} R_{g′}^{2}dvol_{g′} ≥ (Y [g])^{2}. (3.4)

Proof. From Hölder’s inequality, (3.5) ∫_{M} R_{g′}dvol_{g′} ≤ {∫_{M} R_{g′}^{2}dvol_{g′}}^{1/2} · {Vol (g′)}^{1/2} . Since g has positive scalar curvature, Y [g] > 0, so the left-hand side of (3.5) must be positive. We then obtain

(Y [g])^{2} ≤ (Vol (g′)^{−1/2} ∫_{M} R_{g′}dvol_{g′})^{2} ≤ ∫_{M} R_{g′}^{2}dvol_{g′}.

q.e.d.

Using the lemma, and the conformal invariance of F_{2}, we obtain

C′ ∫_{M} e^{4u_{t}}dvol_{g} ≥ \frac{1}{4}F_{2}([g]) + \frac{1}{24}(1 − t)(2 − t)(Y [g])^{2} = \frac{1}{4}λ_{t} > 0. (3.6)

This implies

max u_{t} ≥ \frac{1}{4} log λ_{t} − C(g). (3.7)

The assumption |∇u_{t}| < C_{1} implies the Harnack inequality max u_{t} ≤ min u_{t} + C(C_{1}, g), (3.8) by simply integrating along a geodesic connecting points at which u_{t} attains its maximum and minimum. Combining (3.7) and (3.8) we obtain

min u_{t} ≥ \frac{1}{4} log λ_{t} − C.

q.e.d.

4. Harnack inequality

We next have the following C^{1} estimate for solutions of Equation (1.18).

Proposition 4.1. Let u_{t} be a C^{3} solution of (3.1) for some δ ≤ t ≤ 1, satisfying u_{t} < \overline{δ}. Then ∥∇u_{t}∥_{L^{∞}} < C_{1}, where C_{1} depends only upon \overline{δ} and g.

Remark. A Harnack inequality was proved for the conformally invariant equation for t = 1 in [20], and then extended to t < 1 in [27]. More specifically, in [27] was considered the equation

σ_{k}^{1/k}(sA^{1} + (1 − s)σ_{1}(A^{1})g) = f(x)e^{−2u}. (4.1)

The left-hand side is just a reparametrization of A^{t}, but (3.1) has a different right-hand side, so the Harnack inequality now depends on the sup. The differences are minor, but for convenience, we present an outline of the proof here, and also provide a simple direct proof which works for t < 1.

Proof. Consider the function h = |∇u|^{2} (we will omit the subscript on u_{t}). Since M is compact, and h is continuous, we suppose the maximum of h occurs and a point p ∈ N. Take a normal coordinate system (x^{1}, …, x^{n}) at p, then g_{ij}(p) = δ_{ij}, and Γ_{jk}^{i}(p) = 0, where g = g_{ij}dx^{i}dx^{j}, and Γ_{jk}^{i} is the Christoffel symbol (see [4]).

Locally, we may write h as

h = g^{lm}u_{l}u_{m}. (4.2)

In a neighborhood of p, differentiating h in the x^{i} direction we have

∂_{i}h = h_{i} = ∂_{i}(g^{lm}u_{l}u_{m}) = ∂_{i}(g^{lm})u_{l}u_{m} + 2g^{lm}∂_{i}(u_{l})u_{m}. (4.3)

Since in a normal coordinate system, the first derivatives of the metric vanish at p, and since p is a maximum for h, evaluating (4.3) at p, we have

u_{li}u_{l} = 0. (4.4)

Next we differentiate (4.3) in the x^{j} direction. Since p is a maximum, ∂_{j}∂_{i}h = h_{ij} is negative semidefinite, and we get (at p)

(4.5)  0 ≫ h_{ij} = \frac{1}{2}∂_{j}∂_{i}g^{lm}u_{l}u_{m} + u_{lij}u_{l} + u_{li}u_{lj}.

We recall from Section 2 that

L_{ij}^{t} = T_{ij} + \frac{1 − t}{2} ∑_{l} T_{ll}δ_{ij}, (4.6)

is positive definite, where T_{ij} means (T_{1}(g^{−1}A_{u}^{t}))_{ij}. We sum with (4.5) with L_{ij}^{t} to obtain the inequality

0 ≥ \frac{1}{2}L_{ij}^{t}∂_{i}∂_{j}g^{lm}u_{l}u_{m} + L_{ij}^{t}u_{lij}u_{l} + L_{ij}^{t}u_{li}u_{lj}. (4.7)

We next differentiate Equation (3.1) in order to replace the u_{lij} term with lower order terms. With respect to our local coordinate system, from (2.7) we have

(4.8) (A_{u}^{t})_{ij} = (A_{g}^{t})_{ij} + u_{ij} − u_{r}Γ_{ij}^{r} + \frac{1 − t}{2} ∑_{k}(u_{kk} − u_{r}Γ_{kk}^{r})g_{ij} + u_{i}u_{j}

− \frac{2 − t}{2}(g^{r_{1}r_{2}}u_{r_{1}}u_{r_{2}})g_{ij}.

At the point p, this simplifies to

(A_{u}^{t})_{ij} = (A_{g}^{t})_{ij} + u_{ij} + \frac{1 − t}{2} ∑_{k}(u_{kk})g_{ij} + u_{i}u_{j} − \frac{2 − t}{2}(|∇u|^{2})δ_{ij}. (4.9)

Next we take m with 1 ≤ m ≤ n, and differentiate (3.1) with respect to x^{m} in our local coordinate system:

∂_{m} {σ_{2}(g^{lj}(A_{u}^{t})_{ij})} = ∂_{m}(f(x)^{2}e^{4u}). (4.10)

144 m.j. gursky & j.a. viaclovsky

Differentiating and evaluating at p, we obtain T_{ij}(∂_{m}(A_{g}^{t})_{ij} + u_{ijm} − u_{r}∂_{m}Γ_{ij}^{r} (4.11)

+ \frac{1 − t}{2} ∑_{k}(u_{kkm} − u_{r}∂_{m}Γ_{kk}^{r})δ_{ij} + 2u_{im}u_{j})

= (∂_{m}f^{2})e^{4u} + 4f^{2}e^{4u}u_{m}.

Note that the third order terms in the above expression are

T_{ij}(u_{ijm} + \frac{1 − t}{2} ∑_{k} u_{kkm}δ_{ij}) = L_{ij}^{t}u_{ijm}.

Next we sum (4.11) with u_{m}, using (4.4) we have the following formula L_{ij}^{t}u_{m}u_{ijm} + T_{ij}(u_{m}∂_{m}(A_{g}^{t})_{ij} (4.12)

− u_{m}u_{r}∂_{m}Γ_{ij}^{r} − \frac{1 − t}{2} ∑_{k}(u_{r}u_{m}∂_{m}Γ_{kk}^{r})δ_{ij})

= u_{m}(∂_{m}f^{2})e^{4u} + 4f^{2}e^{4u}|∇u|^{2}.

Substituting (4.12) into (4.7), we arrive at the inequality

0 ≥ \frac{1}{2}L_{ij}^{t}∂_{i}∂_{j}g^{lm}u_{l}u_{m} + T_{ij}( − u_{m}∂_{m}(A_{g}^{t})_{ij}

+ u_{m}u_{r}∂_{m}Γ_{ij}^{r} + \frac{1 − t}{2} ∑_{k}(u_{r}u_{m}∂_{m}Γ_{kk}^{r})δ_{ij})

+ u_{m}(∂_{m}f^{2})e^{4u} + L_{ij}^{t}u_{li}u_{lj}.

Using (4.6) and Lemma 2 in [31], we obtain

(4.13) 0 ≥ T_{ij}(\frac{1 − t}{2} ∑_{k} R_{klkm}u_{l}u_{m}δ_{ij} + R_{iljm}u_{l}u_{m} − u_{m}∂_{m}(A_{g}^{t})_{ij})

+ u_{m}(∂_{m}f^{2})e^{4u} + T_{ij}u_{li}u_{lj} + \frac{1 − t}{2} ∑_{l} T_{ll}u_{ij}u_{ij},

where R_{iljm} are the components of the Riemann curvature tensor of g.

Lemma 4.1. There exists a constant β > 0 such that for t ∈ [δ, 1],

T_{ij}u_{li}u_{lj} + \frac{1 − t}{2} ∑_{l} T_{ll}u_{ij}u_{ij} ≥ β ∑_{l} T_{ll}|∇u|^{4}. (4.14)

Remark. This was proved in [27], using the result in [20].

Using the lemma, we have

(4.15) 0 ≥ T_{ij} (\frac{1 − t}{2} ∑_{k} R_{klkm}u_{l}u_{m}δ_{ij} + R_{iljm}u_{l}u_{m} − u_{m}∂_{m}(A_{g}^{t})_{ij})

+ u_{m}(∂_{m}f^{2})e^{4u} + β ∑_{l} T_{ll}|∇u|^{4}.

Since we are assuming u is bounded above, the |∇u|^{4} term dominates, and the proof proceeds as in [20] or [27]. q.e.d.

Remark. For convenience, we would like to present here a simplified proof of Lemma 4.1 which works for t < 1. Although the argument breaks down as t → 1, it covers the case t_{0} = 0, and therefore suffices for proving Theorem 1.4.

To begin, we claim that if β′_{t} > 0 is sufficiently small, then for at least one i_{0}, u_{i_{0}i_{0}} ≥ β′_{t}|∇u|^{2}. If not, then u_{ii} < β′_{t}|∇u|^{2} for i = 1 … 4. Since Γ_{2} ⊂ {σ_{1} > 0},

Δu + 2(1 − t)(Δu) − 2(2 − t)|∇u|^{2} + |∇u|^{2} + σ_{1}(A_{g}^{t}) > 0.

Without loss of generality, we may assume that σ_{1}(A_{g}^{t}) ≤ ϵ|∇u|^{2}, we then have

(1 + 2(1 − t))Δu + (1 − 2(2 − t))|∇u|^{2} + ϵ|∇u|^{2} > 0.

From the assumption, Δu = ∑ u_{ii} < 4β′_{t}|∇u|^{2}, so

4β′_{t}(1 + 2(1 − t))|∇u|^{2} + (1 − 2(2 − t))|∇u|^{2} + ϵ|∇u|^{2} > 0,

which is a contradiction for ϵ and β′_{t} sufficiently small. We then have

(4.16)  T_{ij}u_{li}u_{lj} + \frac{1 − t}{2} ∑_{l} T_{ll}u_{ij}u_{ij} ≥ \frac{1 − t}{2} ∑_{l} T_{ll}u_{i_{0}i_{0}}^{2}

≥ \frac{1 − t}{2}(β′_{t})^{2} ∑_{l} T_{ll}|∇u|^{4},

so choose β_{t} = \frac{1−t}{2}(β′_{t})^{2}. This completes the proof.

5. Proof of Theorem 1.1

Proposition 5.1. Let u_{t} be a C^{4} solution of (3.1) for some δ ≤ t ≤ 1 satisfying \underline{δ} < u_{t} < \overline{δ}, and ∥∇u_{t}∥_{L^{∞}} < C_{1}. Then for 0 < α < 1, ∥u_{t}∥_{C^{2,α}} ≤ C_{2}, where C_{2} depends only upon \underline{δ}, \overline{δ}, C_{1}, and g.

Proof. The C^{2} estimate follows from the global estimates in [23], or the local estimates [20] and [27]. We remark that the main fact used in deriving these estimates is that σ_{2}^{1/2}(A^{t}) is a concave function of the second derivative variables, which follows easily from the inequality (2.6). Since f(x) > 0, the C^{2} estimate implies uniform ellipticity, and the C^{2,α} estimate then follows from the work of [25] and [14] on concave, uniformly elliptic equations. q.e.d.

To finish the proof of Theorem 1.1, we use the continuity method. Recall that we are considering the 1-parameter family of equations, for t ∈ [δ, t_{0}],

σ_{2}^{1/2}(g^{−1}A_{u_{t}}^{t}) = f(x)e^{2u_{t}}, (5.1)

with f(x) = σ_{2}^{1/2}(g^{−1}A_{g}^{δ}) > 0, and δ was chosen so that A_{g}^{δ} is positive definite. We define

S = {t ∈ [δ, t_{0}] | ∃ a solution u_{t} ∈ C^{2,α}(M) of (5.1) with A_{u_{t}}^{t} ∈ Γ_{2}^{+}}.

The function f(x) was chosen so that u ≡ 0 is a solution at t = δ. Since A_{g}^{δ} is positive definite, and the positive cone is clearly contained in Γ_{2}^{+}, S is nonempty. Let t ∈ S, and u_{t} be any solution. From Proposition 2.2, the linearized operator at u_{t}, L^{t} : C^{2,α}(M) → C^{α}(M), is invertible. The implicit function theorem (see [17]) implies that S is open. Note that since f ∈ C^{∞}(M), it follows from classical elliptic regularity theory that u_{t} ∈ C^{∞}(M). Proposition 3.1 implies a uniform upper bound on solutions u_{t} (independent of t). We may then apply Proposition 4.1 to obtain a uniform gradient bound, and Lemma 3.2 then implies a uniform lower bound on u_{t}. Proposition 5.1 then implies that S is closed, therefore S = [δ, t_{0}]. The metric g̃ = e^{−2u_{t_{0}}} g then satisfies

σ_{2}(At_{0} ; g̃) > 0 and Rg̃ > 0.

We next verify the inequalities (1.7). We decompose A^{t} into its trace-free and pure-trace components,

(5.2)  A^{t} = A^{t} − \frac{1}{n}σ_{1}(A^{t})g + \frac{1}{n}σ_{1}(A^{t})g

≡ A^{t}^{∘} + \frac{1}{n}σ_{1}(A^{t})g.

We now associate to A^{t} the symmetric transformation Â^{t}, defined by

Â^{t} ≡ −A^{t}^{∘} + \frac{1}{n}σ_{1}(A^{t})g. (5.3)

That is, Â^{t} is the (unique) symmetric transformation which has the same pure-trace component as A, but the opposite trace-free component.

Lemma 5.1. The tensors Â^{t} and A^{t} satisfy the equalities

σ_{1}(Â^{t}) = σ_{1}(A^{t}), (5.4)

σ_{2}(Â^{t}) = σ_{2}(A^{t}). (5.5)

Proof. The proof of (5.4) is immediate from the definition of Â^{t}. To prove (5.5), we use the identity

σ_{2}(A^{t}) = −\frac{1}{2}|A^{t}|^{2} + \frac{1}{2}σ_{1}(A^{t})^{2}

= −\frac{1}{2}|A^{t}^{∘} + \frac{1}{n}σ_{1}(A^{t})g|^{2} + \frac{1}{2}σ_{1}(A^{t})^{2}.

Since the decomposition (5.2) is orthogonal with respect to the norm | · |^{2}, we conclude

σ_{2}(A^{t}) = −\frac{1}{2}| − Â^{t}|^{2} + \frac{1}{2}σ_{1}(A^{t})^{2} = σ_{2}(Â^{t}).

q.e.d.

Combining Lemma 5.1 and Proposition 2.1 we have:

Proposition 5.2. If the eigenvalues of A^{t} are in Γ_{2}^{+}, then

(5.6)  −A^{t} + σ_{1}(A^{t})g > 0, and

A^{t} + \frac{n − 2}{n}σ_{1}(A^{t})g > 0. (5.7)

Proof. The tensor in (5.6) is simply the first Newton transformation of A^{t}, which is positive definite by Proposition 2.1. By Lemma 5.1, σ_{2}(Â^{t}) > 0 and σ_{1}(Â^{t}) > 0. Thus, the eigenvalues of Â^{t} are also in Γ_{2}^{+}. By Proposition 2.1, the first Newton tranform T_{1}(Â^{t}) is positive definite. By definition,

T_{1}(Â^{t}) = −Â^{t} + σ_{1}(Â^{t})g

= − (−A^{t}^{∘} + \frac{1}{n}σ_{1}(A^{t})g) + σ_{1}(A^{t})g

= A^{t} + \frac{(n − 2)}{n}σ_{1}(A^{t})g. q.e.d.

When expressed in terms of Ric and n = 4, (5.6) and (5.7) are exactly (1.7).

6. Proof of Theorem 1.4

The assumption (1.14) corresponds to t_{0} = 0 in Theorem 1.1. From (1.7) we find a conformal metric (which for simplicity, we again denote by g) with Ric_{g} < R_{g}g. Theorem 1.4 then follows from:

Proposition 6.1. If Ric_{g} ≤ R_{g}g, then P ≥ 0, and Ker P = {constants}.

Proof. We again recall that the Paneitz operator is defined by

P ϕ = Δ^{2}ϕ + δ(\frac{2}{3}R_{g}g − 2Ric_{g})dϕ. (6.1)

Integrating by parts,

⟨P ϕ, ϕ⟩_{L^{2}} = ∫_{M} ((Δϕ)^{2} + \frac{2}{3}R_{g}|∇ϕ|^{2} − 2Ric_{g}(∇ϕ, ∇ϕ)) dvol_{g}. (6.2)

From the Bochner formula,

0 = ∫ (|∇^{2}ϕ|^{2} + Ric_{g}(∇ϕ, ∇ϕ) − (Δϕ)^{2}) dvol_{g}. (6.3)

Substituting (6.3) into (6.2), we have

⟨P ϕ, ϕ⟩_{L^{2}}

= ∫_{M} (−\frac{1}{3}(Δϕ)^{2} + \frac{4}{3}(Δϕ)^{2} + \frac{2}{3}R_{g}|∇ϕ|^{2} − 2Ric_{g}(∇ϕ, ∇ϕ)) dvol_{g}

= ∫_{M} (−\frac{1}{3}(Δϕ)^{2} + \frac{4}{3}|∇^{2}ϕ|^{2} + \frac{2}{3}R_{g}|∇ϕ|^{2} − \frac{2}{3}Ric_{g}(∇ϕ, ∇ϕ)) dvol_{g}

= ∫_{M} (\frac{4}{3}|∇^{2}^{∘}ϕ|^{2} + \frac{2}{3}(R_{g}g − Ric_{g})(∇ϕ, ∇ϕ)) dvol_{g}

≥ ∫_{M} \frac{4}{3}|∇^{2}^{∘}ϕ|^{2}dvol_{g},

where ∇^{2}^{∘}ϕ = ∇^{2}ϕ − (1/4)(Δϕ)g. Consequently, P ≥ 0. Assume by contradiction that P ϕ = 0, and ϕ is not constant. From the above, we conclude that ∇^{2}^{∘}ϕ ≡ 0. By [28, Theorem A], g is homothetic to S^{4}. We then have

0 = ⟨P ϕ, ϕ⟩_{L^{2}} = ∫_{M} \frac{4}{3}|∇^{2}^{∘}ϕ|^{2}dvol_{g} + \frac{1}{2}R_{g} ∫_{M} |∇ϕ|^{2}dvol_{g},

and therefore ϕ = constant. q.e.d.

Remark. In [11], the nonnegativity of the Paneitz operator was shown assuming Rg − λRic ≥ 0 for λ ∈ (1, 3]. The above proposition extends this to the endpoint λ = 1.

Corollary 6.1. If Ric_{g} ≥ 0, then P ≥ 0, and Ker P = {constants}.

Proof. Clearly, Ric_{g} ≥ 0 implies that Ric_{g} ≤ R_{g}g, so this follows directly from Proposition 6.1. q.e.d.

Remark. The construction in [29] yields metrics with positive Ricci curvature on the connect sums k(S^{2} × S^{2}), kℂℙ^{2}#\overline{ℂℙ}^{2}), and (k + l)ℂℙ^{2}#k\overline{ℂℙ}^{2}. Consequently, from Corollary 6.1, and the results in [10], these manifolds admit metrics with Q = constant.

7. Examples

The following theorem will allow us to give many examples of metrics satisfying the conditions of Theorem 1.4.

Theorem 7.1. Let (M, g) satisfy ∫ Q_{g}dvol_{g} ≥ 0. If Y [g] > 4\sqrt{3k}π, k < 8, then the manifold N = M#k(S^{1} ×S^{3}) admits a metric g̃ satisfying (1.14). If Y [g] > 8\sqrt{3}π, then the manifold N = M#l(ℝℙ^{4}) admits a metric g̃ satisfying (1.14) for l < 9. Consequently, these manifolds N admit metrics with Q = constant.

Proof. From the assumption on ∫ Qdvol and the Chern-Gauss-Bonnet formula, we have

∫_{M} |W_{g}|^{2}dvol_{g} ≤ 8π^{2}χ(M). (7.1)

From [2, Proposition 4.1], given a point p ∈ M, and ϵ > 0, M admits a metric g′ so that g′ is locally conformally flat in a neighborhood of p, and

|Y [g′] − Y [g]| < ϵ, (7.2)

| ∫_{M} |W_{g}|^{2}dvol_{g} − ∫_{M} |W_{g′}|^{2}dvol_{g′} | < ϵ.

We next put a metric on the connect sum using the technique in [24]. Since g′ is locally conformally flat near p, there is a conformal factor on M − {p} which makes the metric look cylindrical near p.

For the first case, since S^{1} × S^{3} is locally conformally flat, for any p′ ∈ S^{1} × S^{3} there is a conformal factor on S^{1} × S^{3} − {p′} which makes the metric look cylindrical near p′. Therefore one can put a metric on N by identifying the cylindrical regions together along their boundaries. From the construction in [24], there are sequences of locally conformally flat metrics on k(S^{1} × S^{3}) whose Yamabe invariants approach σ(k(S^{1} × S^{3})) = σ(S^{4}) = 8\sqrt{6}π > Y [g′], where σ denotes the diffeomorphism Yamabe invariant, so we choose a locally conformally flat metric g_{1} on k(S^{1} × S^{3}) satisfying Y [g_{1}] ≥ 8\sqrt{6}π − ϵ. We have min\{Y [g_{1}], Y [g′]\} = Y [g′], so following the proof [24, Theorem 2], by changing the length of the cylindrical region, one can put a metric g̃ on the connect sum N = M#k(S^{1} × S^{3}) with Y [g̃] > Y [g′] − ϵ. Clearly we also have

| ∫_{M} |W_{g}|^{2}dvol_{g} − ∫_{N} |Wg̃|^{2}dvolg̃| < ϵ,

which along with (7.1) implies

∫_{N} |Wg̃|^{2}dvolg̃ ≤ 8π^{2}χ(M) + ϵ. (7.3)

We next verify that, for appropriate ϵ, the metric g̃ satisfies the condition (1.14). To see this, write (Y [g])^{2} = 48kπ^{2} + 3δ, with δ > 0, and noting that χ(N) = χ(M) − 2k we have

2 ∫_{N} Qg̃dvolg̃ + \frac{1}{3}(Y [g̃])^{2} = 8π^{2}χ(N) − ∫_{N} |Wg̃|^{2}dvolg̃ + \frac{1}{3}(Y [g̃])^{2}

≥ 8π^{2}χ(N) − 8π^{2}χ(M) + \frac{1}{3}(Y [g])^{2} − Cϵ

= −16kπ^{2} + \frac{1}{3}(Y [g])^{2} − Cϵ

= δ − Cϵ > 0,

for ϵ sufficiently small.

For the second case, since ℝℙ^{4} is locally conformally flat, we do exactly the same gluing as before. Again we use [2, Proposition 4.1] to find a metric g′ on M satisfying (7.2). We fix the standard metric g_{0} on ℝℙ^{4}, which is locally conformally flat. Since Y ([g′]) > 8\sqrt{3}π = Y [g_{0}] we have min{Y [g_{0}], Y [g′]} = Y [g_{0}], so from the construction in [24], we can put a metric g̃ on the connect sum N = M#l(ℝℙ^{4}) with Y [g̃] > 8\sqrt{3}π − ϵ, and which also satisfies (7.3).

Write (Y [g])^{2} = 3(64π^{2} + δ), with δ > 0, and noting that χ(N) = χ(M) − l,

2 ∫_{N} Qg̃dvolg̃ + \frac{1}{3}(Y [g̃])^{2} = 8π^{2}χ(N) − ∫_{N} |Wg̃|^{2}dvolg̃ + \frac{1}{3}(Y [g̃])^{2}

≥ 8π^{2}χ(N) − 8π^{2}χ(M) + \frac{1}{3}(Y [g])^{2} − Cϵ

= −8lπ^{2} + 64π^{2} + δ − Cϵ > 0,

for ϵ sufficiently small and l < 9. q.e.d.

We next write down some specific examples of (M, g) satisfying the assumptions of Theorem 7.1. We will use the fact that if (M, g) is a positive Einstein manifold, then ∫ Q_{g}dvol_{g} > 0 and the Yamabe invariant is attained by g.

(1) M = S^{2} × S^{2} with the product metric, Y [g] = 16π > 4\sqrt{3k}π for k < 6, so we have

N = (S^{2} × S^{2})#k(S^{1} × S^{3}), k ≤ 5. (7.4)

(2) M = ℂℙ^{2} with the Fubini-Study metric, Y [g] = 12\sqrt{2}π (see [26]). Since 12\sqrt{2}π > 4\sqrt{3k}π for k < 6, this yields the examples

N = ℂℙ^{2}#k(S^{1} × S^{3}), k ≤ 5. (7.5)

(3) Again, we take M = ℂℙ^{2}. We have 12\sqrt{2}π > 8\sqrt{3}π, so from the second statement in Theorem 7.1 we have

N = ℂℙ^{2}#k(ℝℙ^{4}), k ≤ 8. (7.6)

(4) M = ℂℙ^{2}#l\overline{ℂℙ}^{2}, 3 ≤ l ≤ 8, M admits Kähler-Einstein metrics satisfying Y [g] = 4π\sqrt{2(9 − l)} (see [22], [26]). Since 4π\sqrt{2(9 − l)} > 4\sqrt{3}π for l < 8, we have the examples

N = ℂℙ^{2}#l\overline{ℂℙ}^{2}#(S^{1} × S^{3}), 3 ≤ l ≤ 7. (7.7)

(5) N = k(S^{1} × S^{3})#l(ℝℙ^{4}), 2k + l ≤ 9. We do not need Theorem 7.1 for this example, we argue directly. By the construction in [24], these manifolds admit locally conformally flat metrics g̃ with Y [g̃] ≈ Y [ℝℙ^{4}, g_{0}] = 8\sqrt{3}π, We have χ(N) = −2k − l + 2, so the assumption (1.14) is that

0 < 8π^{2}(−2k − l + 2) + \frac{1}{3}(Y [g])^{2} ≈ 8π^{2}(−2k − l + 2) + 64π^{2},

which is satisfied for 2k + l < 10.

The above examples are all summing with locally conformally flat manifolds, but this is not necessary in our construction. We end with a corollary, whose proof is similar to the proof of Theorem 7.1.

Corollary 7.1. Let (M_{1}, g_{1}) and (M_{2}, g_{2}) satisfy ∫_{M_{i}} Q_{g_{i}}dvol_{g_{i}} ≥ 0, and Y [g_{i}] > 4\sqrt{3}π. Then the manifold N = M_{1}#M_{2} admits a metric g̃ satisfying (1.14). Consequently, N admits a metric with Q = constant.

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Department of Mathematics University of Notre Dame Notre Dame, IN 46556

Department of Mathematics

Massachusetts Institute of Technology Cambridge, MA 02139