j. differential geometry

57 (2001) 339-388

POISSON EQUATION, POINCARÉ-LELONG EQUATION AND CURVATURE DECAY ON

COMPLETE KÄHLER MANIFOLDS

LEI NI, YUGUANG SHI & LUEN-FAI TAM

Abstract

In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kähler manifolds. In particular, the Poincaré-Lelong equation on complete noncompact Kähler manifolds with nonnegative holomorphic bisectional curvature is studied. Several applications are then derived, which include the Steinness of the complete Kähler manifolds with nonnegative curvature and the flatness of a class of complete Kähler manifolds satisfying a curvature pinching condition. Liouville type results for plurisubharmonic functions are also obtained.

0. Introduction

In this paper, we will discuss the Poisson equation on complete noncompact manifolds and derive some applications on Kähler manifolds.

Let M^{m} be a complete noncompact Kähler manifold, where m ≥ 2 is the complex dimension. Assume M has nonnegative holomorphic bisectional curvature and has maximal volume growth such that the scalar curvature decays like r^{−2} where r is the distance from a fixed point. Then it was proved in [20] by Mok-Siu-Yau that one can solve the following Poincaré-Lelong equation

(0.1)  \sqrt{−1}∂∂u = ρ

Received November 25, 2000, and, in revised form, June 28, 2001. The first author was partially supported by NSF grant DMS9970284, USA, the second author was partially supported by NSF of China, project 10001001, and the third author was partially supported by Earmarked Grant of Hong Kong #CUHK4217/99P.

by first solving the Poisson equation 1/2Δu = trace(ρ), where ρ is the Ricci form of M. They then applied the results to study the analytic and geometric properties of M. On the other hand, in [27] Yau discussed certain differential inequalities. Again, applications on Riemannian and Kähler manifolds were given. For example, some vanishing results for L^{p} holomorphic sections of holomorphic vector bundles over Kähler manifolds were obtained; see also [10], [18], [22]. In some cases, if one can solve the Poisson equation then it is rather easy to apply the methods in [27]. These motivate our study on the Poisson equation on complete noncompact manifolds.

We are mainly concerned with manifolds with nonnegative Ricci curvature. Let M^{n} be such a manifold and consider the Poisson equation:

(0.2)  Δu = f.

The first question is to find sufficient conditions for the existence of solutions of (0.2). If a solution u exists, it is also important for applications to estimate u together with its gradient and Hessian.

Our main result is that if f decays faster than r^{−1} in a certain sense, then (0.2) has a solution. More precisely, assume f ≥ 0 and let k(x, t) = k_{f}(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} f be the average of f over the geodesic ball B_{x}(t) with center at x and radius t, where V_{x}(t) is the volume of B_{x}(t). Let o ∈ M be a fixed point and let k(t) = k(o, t). We prove that if ∫_{0}^{∞} k(t)dt < ∞ and if there exist a constant 1 > δ > 0 and a nonnegative function h(t) ≥ 0, 0 ≤ t < ∞ with h(t) = o(t) as t → ∞ such that

∫_{0}^{t} sk(x, s)ds ≤ h(t)

for all x and for all t ≥ δr(x), then (0.2) has a solution u. Moreover, lower and upper estimates of u are obtained. In case that M is nonparabolic (that is, M supports a positive Green’s function) such that the volume of geodesic balls satisfy certain assumptions then the condition ∫_{0}^{∞} k(t)dt < ∞ alone will be sufficient. This is the case if M has maximal volume growth with n ≥ 3. In any case, pointwise and integral estimates for the gradient of the solution u and an integral estimate of the Hessian of u are obtained.

The above conditions on the average of f over geodesic balls for the existence of solution of (0.2) are reasonable, because in some cases they are also necessary. For example, we prove the following results. Let f ≥ 0 be a function on a complete noncompact manifold with nonnegative Ricci curvature. Then:

(i) Δu = f has a bounded solution if and only if there is a constant

C > 0 such that

∫_{0}^{∞} tk(x, t)dt ≤ C

for all x.

(ii) Δu = f has a solution with sup_{B_{o}(r)} |u| ≤ C log(2 + r) for some constants C for all r if any only if

∫_{0}^{t} sk(x, s)ds ≤ C′ log(2 + t)

for some constant C′ for all r = r(x) and for all t ≥ \frac{1}{5}r.

(iii) Δu = f has a solution with sup_{B_{o}(r)} |u| ≤ C(1 + r)^{1−δ} for some constant C and 1 > δ > 0 for all r if any only if

∫_{0}^{t} sk(x, s)ds ≤ C′(1 + t)^{1−δ}

for some constant C′ for all r = r(x) and for all t ≥ \frac{1}{5}r.

In [12], Li proved that if u is a bounded subharmonic function on a complete noncompact manifold M with nonnegative Ricci curvature, then the average of u over a geodesic ball of radius r with a fixed center converges to sup_{M} u as r → ∞. Using the result (i) in the above, we give another proof of Li’s result. Furthermore, one can estimate the difference between sup_{M} u and the average of u over a geodesic ball of radius r in terms of Δu, r and the dimension of M. Using the pointwise estimate for the gradient of the solution of (0.2), we prove that if in addition that f = Δu decays like r^{−2} then u will actually be asymptotically constant.

The rest of this work is to apply these results to Kähler manifolds. One of the applications is to study plurisubharmonic functions. It is easy to see that if n ≥ 3, then there are nonconstant bounded subharmonic functions on ℝ^{n} which are asymptotically constant. On the other hand, it was proved by Ni in [22] that if u is a plurisubharmonic function on a complete noncompact Kähler manifold M^{m} with nonnegative Ricci curvature and if u satisfies

(0.3)  lim sup_{x→∞} \frac{u(x)}{log r(x)} = 0

then (∂∂u)^{m} ≡ 0. We prove that u is actually constant under some assumptions on M and Δu by using a minimum principle in [2], [1] or a method in [20].

The above mentioned result of Ni can be generalized to nonparabolic manifolds with scalar curvature R satisfying ∫_{M} R_{−} < ∞, where R_{−} is the negative part of R. This generalization is a consequence of one of the vanishing results we obtain in this work. Consider a complete noncompact Kähler manifold M^{m} with nonnegative Ricci curvature and a Hermitian holomorphic line bundle L over M. We prove that given any τ > 0 and 0 < ϵ < 1 there is a constant a depending only on τ, ϵ > 0 and m such that if the average of the trace of positive part of the curvature of L over B(r) is less than ar^{−2}, then any holomorphic (p, 0) form ϕ with value in L is trivial if

\frac{1}{V_{o}(r)} ∫_{B(r)} |ϕ|^{τ} = O(r^{−ϵ})

as r → ∞, where o ∈ M is a fixed point and V_{o}(r) is the volume of the ball of radius r centered at o. The proof is a combination of our results on the Poisson equation and the mean value inequality in [13]. If M is nonparabolic then a similar result is true. In this case, we assume that the negative part of the scalar curvature of M and the positive part of the trace of the curvature of L are both integrable. The vanishing theorems are similar to some results in [27], [10], [22].

Using the vanishing results and the L^{2} estimate in [11], [7] one can prove the following: Let M be a complete noncompact Käher manifold with nonnegative Ricci curvature. Suppose the scalar curvature R satisfies

(0.4)  lim sup_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} R = 0,

where o ∈ M is a fixed point, then the Ricci form ρ of M must satisfy ρ^{m} ≡ 0. Observe that if M is nonparabolic and R is integrable, then (0.4) is true. Hence this generalizes a result in [22, Theorem 3.6].

From the arguments in [23], under the additional assumption that M has nonnegative holomorphic bisectional curvature which is also bounded, one can conclude that if (0.4) is true for all base point o so that the convergence is uniform then M is flat as observed in [4]. In our case, we only assume that the Ricci curvature is nonnegative and we do not assume the scalar curvature being bounded. The result is weaker and it is interesting to see whether M is actually Ricci flat in this case. In fact, for Riemannian case, it is proved by Chen and Zhu [3] that if (0.4) is true uniformly and if the Riemannian manifold is locally conformally flat then the manifold is flat.

Finally we solve the Poincaré-Lelong equation (0.1). Let M^{m} be a complete Kähler manifold with nonnegative bisectional curvature and let ρ be a real closed (1, 1) form with trace f. We prove that if f ≥ 0 and ρ satisfies the following conditions:

(0.5)  ∫_{0}^{∞} \frac{1}{V_{o}(t)} ∫_{B_{o}(t)} ||ρ||dt < ∞,

and that

(0.6)  lim inf_{r→∞} \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ||^{2} = 0,

then (0.1) has a solution u. It is easy to see that if M^{m} has maximal volume growth with m ≥ 2 and ||ρ|| decays like r^{−2}, then the above conditions will be satisfied. In fact in this case we have

(0.7)  \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ|| ≤ Cr^{−2}

for some C for all r. Hence our result is a generalization of a related result in [20], see also [19]. Note that we do not assume ||ρ|| to be bounded.

Using solutions of (0.1), we can discuss properties of Kähler manifolds with nonnegative holomorphic bisectional curvature. For example, we prove that if in addition M has positive Ricci curvature which satisfies (0.5) and (0.6), then M is Stein, provided the sectional curvature is nonnegative. This is related to the works of [8] and [19], [20]. In [8], it was proved that M is Stein under the assumption that M has positive biholomorphic bisectional curvature and nonnegative sectional curvature. In [19], it was proved that M is Stein under the assumptions that M has positive Ricci curvature and nonnegative holomorphic bisectional curvature, has maximal volume growth and the scalar curvature decays likes r^{−2}. Recently, it is proved by Chen and Zhu [4] that a complete noncompact Kähler manifold M^{m} with nonnegative holomorphic bisectional curvature and with maximal volume growth is Stein if the scalar curvature decays like r^{−1−ϵ} for some ϵ > 0.

In [24], it was proved that if M^{m} is a complete noncompact Kähler manifold of complex dimension m ≥ 3 with nonnegative holomorphic bisectional curvature and with a pinching condition, then (0.7) is satisfied with a constant independent of the point o. Using solution of (0.1), we prove that if in addition the scalar curvature has a pointwise decay like r^{−2} or the volume of geodesic ball of radius r is no greater than r^{m}, then M is actually flat. Here m is the complex dimension of M.

Using solutions of (0.1), we can also study relations between the decay of the scalar curvature and volume growth of a complete Kähler manifold with nonnegative biholomorphic bisectional curvature. For example, we prove that if the Ricci form ρ is positive at some point, ρ satisfies (0.7) and ||ρ|| decays like r^{−2}, then M must have maximal volume growth. In this case, the scalar curvature cannot decay too fast in the sense that we have a reverse inequality of (0.7):

\frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ|| ≥ Cr^{−2}

for some positive constant C for all r. If we only assume that M is not Ricci flat in the above, then one can prove that V_{o}(r) ≥ Cr^{2} for some positive constant C.

The arrangement of the paper is as follows. In Section 1 and Section 2 we study the Poisson equation. Section 3 contains some vanishing theorems. Section 4 is a discussion of Liouville property of plurisubharmonic functions. Section 5 gives a solution to the Poincaré-Lelong equation with applications on manifolds with nonnegative holomorphic bisectional curvature.

The authors would like to thank Xi-Ping Zhu for many useful discussions. The authors would also like to thank the referee for helpful comments, which were of great help in improving the exposition and readability of this paper.

1. The Poisson equation (I)

Let M^{n} be a complete noncompact manifold. Given any function f ≥ 0 on M, define

k_{f}(x, t) = \frac{1}{V_{x}(t)} ∫_{B_{x}(t)} f.

In the following C(a, b, …) will denote a constant depending only on a, b, . . .. We also denote r(x, y) to be the distance between x and y, and r(x) = r(x, o) where o ∈ M is a fixed point. In this section, we will discuss the conditions on f so that Δu = f has a solution u and we will also discuss the properties of u.

Theorem 1.1. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature. Assume M is nonparabolic and there is a constant σ > 0 such that the minimal positive Green’s function G(x, y) satisfies

(1.1)  σ^{−1} \frac{r^{2}(x, y)}{V_{x}(r(x, y))} ≤ G(x, y) ≤ σ \frac{r^{2}(x, y)}{V_{x}(r(x, y))}

for all x ≠ y in M. Let f ≥ 0 be a locally Hölder continuous function and let k(x, t) = k_{f}(x, t) and k(t) = k(o, t), where o ∈ M is a fixed point. Suppose that ∫_{0}^{∞} k(t)dt < ∞. Then the Poisson equation Δu = f has a solution u such that for all 1 > ϵ > 0

α_{1}r ∫_{2r}^{∞} k(t)dt + β_{1} ∫_{0}^{2r} tk(t)dt ≥ u(x)
≥ − α_{2}r ∫_{2r}^{∞} k(t)dt

− β_{2} ∫_{0}^{ϵr} tk(x, t)dt + β_{3} ∫_{0}^{2r} tk(t)dt

for some positive constants α_{1}(n, σ), α_{2}(n, σ, ϵ) and β_{i}(n), 1 ≤ i ≤ 3, where r = r(x). Moreover u(o) = 0.

Proof. By the estimate of Green’s function in [17, Theorem 5.2], (1.1) implies

(1.2)  C^{−1} \frac{r^{2}(x, y)}{V_{x}(r(x, y))} ≤ ∫_{r(x,y)}^{∞} \frac{t}{V_{x}(t)} dt ≤ C \frac{r^{2}(x, y)}{V_{x}(r(x, y))}

for some constant C = C(n, σ) > 0. For all R > 0, let G_{R} be the positive Green’s function on B_{o}(R) with zero boundary value and let

u_{R}(x) = ∫_{B_{o}(R)} (G_{R}(o, y) − G_{R}(x, y)) f(y)dy.

Then Δu_{R} = f in B_{o}(R) and u_{R}(o) = 0. For any x with r(x) = r, suppose R ≫ r, then

(1.3) u_{R}(x) = {∫_{B_{o}(R)\B_{o}(2r)} + ∫_{B_{o}(2r)}} (G_{R}(o, y) − G_{R}(x, y)) f(y)dy
= I + II.

To estimate I, let y be any point in B_{o}(R) \ B_{o}(2r), then r_{1} = r(y) ≥ 2r = 2r(x) and so r(z, y) ≥ \frac{1}{2}r_{1} if z ∈ B_{o}(r). Also B_{z}(\frac{1}{4}r) ⊂ B_{o}(2r). Hence by the gradient estimate [5, Theorem 6],

|G_{R}(o, y) − G_{R}(x, y)| ≤ r sup_{z∈B_{o}(r)} |∇_{z}G_{R}(z, y)|

≤ C_{1} \frac{r}{r_{1}} sup_{z∈B_{o}(r)} G_{R}(z, y)

≤ C_{2} \frac{r}{r_{1}} G(o, y)

≤ C_{3} \frac{r}{r_{1}} ∫_{r_{1}}^{∞} \frac{t}{V_{o}(t)} dt,

where C_{1} − C_{3} are constants depending only on n by [5], [17]. Here we have used the Harnack inequality for G_{R}(·, y), the fact that G_{R}(o, y) ≤ G(o, y) [17, Theorem 5.2].

(1.4)  |I| ≤ C_{3}r ∫_{B_{o}(R)\B_{o}(2r)} r^{−1}(y) (∫_{r(y)}^{∞} \frac{t dt}{V_{o}(t)}) f(y)dy

= C_{3}r ∫_{2r}^{R} t^{−1} (∫_{t}^{∞} \frac{s}{V_{o}(s)} ds) (∫_{∂B_{o}(t)} f) dt

≤ C_{3}r[R^{−1} (∫_{R}^{∞} \frac{s}{V_{o}(s)} ds) (∫_{B_{o}(R)} f)

+ ∫_{2r}^{R} (\frac{1}{t^{2}} ∫_{t}^{∞} \frac{s}{V_{o}(s)}ds + \frac{1}{V (t)}) (∫_{B_{o}(t)} f) dt]

≤ C_{4}r (Rk(R) + ∫_{2r}^{R} k(t)dt)

for some constant C_{4}(n, σ), where we have used (1.2).

(1.5) ∫_{B_{o}(2r)} G_{R}(o, y)f(y)dy ≤ ∫_{B_{o}(2r)} G(o, y)f(y)dy

≤ C_{5}∫_{0}^{2r} (∫_{t}^{∞} \frac{s}{V_{o}(s)} ds)(∫_{∂B_{o}(t)} f) dt

= C_{5} [(∫_{2r}^{∞} \frac{t}{V_{o}(t)} dt) ( ∫_{B_{o}(2r)} f)

+ ∫_{0}^{2r} \frac{t}{V_{o}(t)}( ∫_{B_{o}(t)} f) dt]

≤ C_{5} [C_{6}r^{2}k(2r) + ∫_{0}^{2r} tk(t)dt]

for some constants C_{5}(n) and C_{6}(n, σ). Combining (1.3), (1.4) and (1.5), we have

(1.6) u_{R}(x) ≤ C_{7} (rRk(R) + r^{2}k(2r) + r ∫_{2r}^{R} k(t)dt))+β_{1} ∫_{0}^{2r} tk(t)dt

for some constants C_{7}(n, σ) and β_{1}(n).

As in the proof of (1.5), using the lower bound of the Green’s function, we have

(1.7)  ∫_{B_{o}(2r)} G(o, y)f(y)dy ≥ C_{8} [C_{9}r^{2}k(2r) + ∫_{0}^{2r} tk(t)dt]

for some constants C_{8}(n) > 0 and C_{9}(n, σ) > 0. For 1 > ϵ > 0,

(1.8) ∫_{B_{x}(ϵr)} G_{R}(x, y)f(y)dy ≤ ∫_{B_{x}(ϵr)} G(x, y)f(y)dy

≤ β_{2} [(∫_{ϵr}^{∞} \frac{t}{V_{x}(t)} dt) (∫_{B_{x}(ϵr)} f)

+ ∫_{0}^{ϵr} \frac{t}{V_{x}(t)} (∫_{B_{x}(t)} f) dt]

≤ C_{10}(ϵr)^{2}k(x, ϵr) + β_{2} ∫_{0}^{ϵr} tk(x, t)dt

≤ C_{11}r^{2}k ((1 + ϵ)r)) + β_{2} ∫_{0}^{ϵr} tk(x, t)dt for some constants C_{10}(n, σ), C_{11}(n, σ, ϵ) and β_{2}(n). Here we have used volume comparison and the fact that B_{x}(ϵr) ⊂ B_{o} ((1 + ϵ)r)).

(1.9) ∫_{B_{o}(2r)\B_{x}(ϵr)} G_{R}(x, y)f(y)dy ≤ ∫_{B_{o}(2r)\B_{x}(ϵr)} G(x, y)f(y)dy

≤ σ · \frac{16r^{2}}{V_{x}(ϵr)} ∫_{B_{o}(2r)} f(y)dy

≤ C_{12}r^{2}k(2r)

for some constant C_{12}(n, σ, ϵ). By (1.3), (1.4), (1.8) and (1.9), if R ≥ 4r, we have

(1.10)  u_{R}(x) ≥ − C_{13}r (Rk(R) + ∫_{2r}^{R} k(t)dt)
− β_{2} ∫_{0}^{ϵr} tk(x, t)dt
+ ∫_{B_{o}(2r)} G_{R}(o, y)f(y)dy

where β_{3}(n) and C_{13}(n, σ, ϵ) are positive constants. Here we have used the fact that for any α > 1, k(αR) ≥ Ck(R) for some positive constant C(n, α) for all R. Since ∫_{0}^{∞} k(t)dt < ∞, lim_{R→∞} Rk(R) = 0. Hence from (1.6) and (1.10), u_{R} is bounded on compact sets and there exists R_{i} → ∞ such that u_{R_{i}} converges uniformly on compact sets to a function u which satisfies Δu = f. By (1.6), (1.7) and (1.10), let R_{i} → ∞ we can conclude that u satisfies the estimates in the theorem. q.e.d.

Note that if n ≥ 3 and M has maximal volume growth, then M satisfies the assumptions in Theorem 1.1. In general, if M is a complete noncompact manifold with nonnegative Ricci curvature, then M × ℝ^{4} with the standard metric on ℝ^{4} satisfies the condition of the theorem as observed in [23]. Using this, in some cases one can remove the assumptions that M is nonparabolic and that its Green’s function satisfies (1.1).

Lemma 1.1. Let M = M_{1} × M_{2}, where M_{1} and M_{2} are complete noncompact manifolds with nonnegative Ricci curvature. Let f ≥ 0 be a function on M_{1}, and be considered also as a function on M, which is independent of the second variable. Let x = (x_{1}, x_{2}) ∈ M and r > 0. Then

\frac{C^{−1}}{V_{x_{1}}^{(1)}(\frac{1}{\sqrt{2}}r)} ∫_{B_{x_{1}}^{(1)}(\frac{1}{\sqrt{2}} r)} f ≤ \frac{1}{V_{x}(r)} ∫_{B_{x}(r)} f ≤ \frac{C}{V_{x_{1}}^{(1)}(r)} ∫_{B_{x_{1}}^{(1)}(r)} f

for some constant C > 0 depending only on the dimensions of M_{1} and M_{2}. Here B_{x}(t), B_{x_{1}}^{(1)}(t) are geodesic balls with radius t in M, M_{1}, with centers at x, x_{1} respectively, and V_{x}(t), V_{x_{1}}^{(1)}(t) are the respective volumes.

Proof. Denote the geodesic ball with center x_{2} and radius t in M_{2} by B_{x_{2}}^{(2)}(t) and its volume by V_{x_{2}}^{(2)}(t). Then

B_{x_{1}}^{(1)} (\frac{1}{\sqrt{2}}r) × B_{x_{2}}^{(2)} (\frac{1}{\sqrt{2}}r) ⊂ B_{x}(r) ⊂ B_{x_{1}}^{(1)}(r) × B_{x_{2}}^{(2)}(r).

∫_{B_{x}(r)} f ≤ ∫_{B_{x_{1}}^{(1)}(r)×B_{x_{2}}^{(2)}(r)} f
= V_{x_{2}}^{(2)}(r) ∫_{B_{x_{1}}^{(1)}(r)} f

since f is independent of x_{2}. On the other hand,

V_{x}(r) ≥ V_{x_{1}}^{(1)} (\frac{1}{\sqrt{2}}r) V_{x_{2}}^{(2)} (\frac{1}{\sqrt{2}}r)
≥ CV_{x_{1}}^{(1)}(r)V_{x_{2}}^{(2)}(r)

for some constant C > 0 depending only on the dimensions of M_{1} and M_{2} by volume comparison. Hence

\frac{1}{V_{x}(r)} ∫_{B_{x}(r)} f ≤ \frac{C}{V_{x_{1}}^{(1)}(r)} ∫_{B_{x_{1}}^{(1)}(r)} f.

The other inequality can be proved similarly. q.e.d.

Theorem 1.2. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature. Let f ≥ 0 be a locally Hölder continuous function and let k(x, t) = k_{f}(x, t) and k(t) = k(o, t), where o ∈ M is a fixed point. Suppose that ∫_{0}^{∞} k(t)dt < ∞ and suppose that there exist 1 > δ > 0, h(t) ≥ 0, 0 ≤ t < ∞ with h(t) = o(t) as t → ∞ such that

∫_{0}^{t} sk(x, s)ds ≤ h(t)

for all x and for all t ≥ δr(x). Then the Poisson equation Δu = f has a solution u such that for all 1 > ϵ > 0

α_{1}r ∫_{2r}^{∞} k(t)dt + β_{1} ∫_{0}^{2r} tk(t)dt ≥ u(x)

≥ −α_{2}r ∫_{2r}^{∞} k(t)dt − β_{2} ∫_{0}^{ϵr} tk(x, t)dt

+ β_{3} ∫_{0}^{2r} tk(t)dt

for some positive constants α_{1}(n), α_{2}(n, ϵ) and β_{i}(n), 1 ≤ i ≤ 3. In particular, |u(x)| = o (r(x)) as x → ∞.

Proof. Let M̃ = M × ℝ^{4} with the flat metric on ℝ^{4}. Then M̃ is nonparabolic by [17, Theorem 5.2]. By the volume comparison, it is easy to see that (1.2) is satisfied by M̃ and hence (1.1) is also satisfied by M̃ with σ depending only on n. Denote a point on M̃ by x̃ = (x, x′), and let õ = (o, 0). Let r̃(x̃) and r(x) be the distance functions on M̃ and M from the õ and o respectively. Let f̃(x̃) = f(x) for x̃ = (x, x′) and let k̃(x̃, t) be the average of f̃ over the geodesic ball of radius t with center at x̃. By Lemma 1.1, for any t > 0 and x̃ = (x, x′) in M̃,

C_{1}^{−1}k(x, 1/\sqrt{2}t) ≤ k̃(x̃, t) ≤ C_{1}k(x, t)

for some constant C_{1}(n). Since ∫_{0}^{∞} k(t)dt < ∞ and

∫_{0}^{t} sk(x, s)ds ≤ h(t)

for all t ≥ δr(x) with h(t) = o(t) as t → ∞, we have

∫_{0}^{∞} k̃(t)dt < ∞ and ∫_{0}^{δr̃} tk̃(x̃, t)dt = o(r̃) (1.11)

as r̃ = r̃(x) → ∞

where we have used the fact that r̃ ≥ r(x). Let Δ̃ be the Laplacian of M̃. By Theorem 1.1, there is a solution ũ of Δ̃ũ = f̃ on M̃ such that for all 1 > ϵ > 0

(1.12)  α_{1}r̃∫_{2r̃}^{∞} k̃(t)dt + β_{1} ∫_{0}^{2r̃} tk̃(t)dt ≥ ũ(x)
≥ −α_{2}r̃∫_{2r̃}^{∞} k̃(t)dt
− β_{2} ∫_{0}^{ϵr̃} tk̃(x̃, t)dt
+ β_{3} ∫_{0}^{2r̃} tk̃(t)dt

for some positive constants α_{1}(n), α_{2}(n, ϵ), and β_{i}(n), 1 ≤ i ≤ 3. Moreover, ũ(õ) = 0. By (1.11) and (1.12), it is easy to see that

|ũ(x̃)| = o (r̃(x))

as x̃ → ∞. Let x′_{0} ∈ ℝ^{4} be fixed, then ṽ(x, x′) = ũ(x, x′ + x′_{0}) is also a solution of Δ̃ṽ = f̃. Hence ṽ− ũ is harmonic and is of sublinear growth. By the result of [5], ũ − ṽ must be a constant. Hence

ũ(x, x′ + x′_{0}) − ũ(x, x′) = ũ(o, x′_{0}).

Let x = o, we conclude that

ũ(o, x′ + x′_{0}) = ũ(o, x′_{0}) + ũ(o, x′)

for all x′, x′_{0} ∈ ℝ^{4}. Since ũ(o, x′) is continuous, it must be a linear function. Using the fact that u is of sublinear growth, we conclude that ũ(o, x′) is a constant which is zero because ũ(õ) = 0. Hence ũ is independent of x′ and if we let u(x) = ũ(x, 0), then Δu = f in M and u satisfies the estimates in the theorem by (1.12), the fact that C_{1}^{−1}k(x, 1/\sqrt{2}t) ≤ k̃(x̃, t) ≤ C_{1}k(x, t) and the fact that if x̃ = (x, 0) then r̃(x̃) = r(x).

The last assertion of the theorem follows easily from the assumptions that ∫_{0}^{∞} k(t)dt < ∞ and ∫_{0}^{t} sk(x, s)ds = o(t) as t → ∞ uniformly on x.

q.e.d.

Observe that if ∫_{0}^{∞} k(o, t)dt < ∞ then ∫_{0}^{∞} k(x, t)dt < ∞ for all x. If u is the solution obtained in Theorem 1.1 or 1.2, then for any x_{0} ∈ M

u(x) − u(x_{0}) = ∫_{M} (G(x_{0}, y) − G(x, y)) f(y)dy.

From the proof of the theorem, it is easy to see that

(1.13) α_{1}r ∫_{2r}^{∞} k(x_{0}, t)dt + β_{1} ∫_{0}^{2r} tk(x_{0}, t)dt ≥ u(x) − u(x_{0})
≥ − α_{2}r ∫_{2r}^{∞} k(x_{0}, t)dt
− β_{2} ∫_{0}^{ϵr} tk(x, t)dt

+ β_{3} ∫_{0}^{2r} tk(x_{0}, t)dt

where α_{i} and β_{j} are the constants in Theorem 1.1 or 1.2 and r = r(x, x_{0}).

In the following proposition, we will give a criteria for f to satisfy the assumptions in Theorem 1.2.

Proposition 1.1. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature and let f ≥ 0 be a function on M. Define k(x, t) = k_{f}(x, t) and k(t) = k(o, t) as before. Suppose ∫_{0}^{∞} k(t)dt < ∞ and sup_{∂B_{o}(r)} f = o(r^{−1}) as r → ∞. Then

∫_{0}^{t} sk(x, s)ds = o(t)

as t → ∞ uniformly on x. In particular f satisfies the assumptions in Theorem 1.2.

Proof. Given ϵ > 0, there exists r_{0} > 0 such that if r ≥ r_{0}, then

sup_{∂B_{o}(r)} f ≤ ϵr^{−1},

and there exists t_{0} > 0 such that

∫_{0}^{t} sk(s)ds < ϵt

for t ≥ t_{0} because ∫_{0}^{∞} k(t)dt < ∞. Let t_{1} = max\{r_{0}, \frac{1}{3}t_{0}\}. Let x ∈ M be such that r(x) = r ≥ 2r_{0}. If \frac{r}{2} ≥ t ≥ t_{1}, then r ≥ 2r_{0} and

∫_{0}^{t} sk(x, s)ds ≤ C_{1}ϵr^{−1}t^{2} ≤ \frac{1}{2}C_{1}ϵt

for some absolute constant C_{1}. If t ≥ \frac{r}{2}, then

∫_{0}^{t} sk(x, s)ds = ∫0^{\frac{r}{2}} sk(x, s)ds + ∫ t_{\frac{r}{2}} sk(x, s)ds

≤ \frac{1}{2}C_{1}ϵt + C_{2} ∫_{0}^{3t} sk(s)ds

≤ C_{3}ϵt

for some constants C_{2}(n), C_{3}(n). This completes the proof of the first part of the proposition. As for the second part, we just take h(t) = sup_{x∈M} ∫_{0}^{t} sk(x, s)ds which is well-defined because f is bounded.

q.e.d.

Suppose f(x) ≤ Cr^{−2}(x) and k(t) = 1/V_{o}(t) ∫_{B_{o}(t)} f(y) dy ≤ Ct^{−2}, then the solution u obtained in Theorem 1.2 is bounded if and only if ∫_{0}^{∞} tk(t)dt < ∞. We will discuss bounded subharmonic functions in the next section. Here we consider the case when ∫_{0}^{∞} tk(t)dt = ∞.

Corollary 1.1. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature and let f ≥ 0 be a locally Hölder continuous function on M. Assume that f(x) ≤ Cr^{−2}(x) and that k(t) = 1/V_{o}(t) ∫_{B_{o}(t)} f(y) dy ≤ Ct^{−2} for some constant C for all x ∈ M and t > 0. Let u be the solution of the Poisson equation Δu = f which is obtained in Theorem 1.2. Suppose ∫_{0}^{∞} tk(t)dt = ∞. We have:

(i)

β_{1} ≥ lim sup_{r→∞} \frac{sup_{∂B_{o}(r)} u}{∫_{0}^{r} tk(t)dt} ≥ lim inf_{r→∞} \frac{inf_{∂B_{o}(r)} u}{∫_{0}^{r} tk(t)dt} ≥ β_{3}

where β_{1} and β_{3} are the positive constants in Theorem 1.2.

(ii) If M^{n} has maximal volume growth with n ≥ 3 then

lim_{x→∞} \frac{u(x)}{∫_{0}^{r(x)} tk(t)dt} = \frac{1}{n}.

Proof. Since f satisfies the conditions in Proposition 1.1, one can solve Δu = f by Theorem 1.2. The solution satisfies the estimates in the theorem. By the assumptions on f, we can prove that

C_{1} + β_{1} ∫_{0}^{2r} tk(t)dt ≥ u(x) ≥ −C_{2} + β_{3} ∫_{0}^{2r} tk(t)dt

for some positive constants C_{1} and C_{2} independent of x, where r = r(x). From this, (i) follows easily.

If n ≥ 3, M has maximal volume growth and u is unbounded, then by (i) and the proof of Theorem 1.1, it is easy to see that

lim_{x→∞} \frac{u(x)}{∫_{B_{o}(2r)} G(o, y)f(y)dy} = 1.

(ii) follows from the sharp bound of the Green’s function in [6], see also [16]. q.e.d.

Next we will estimate the gradient and the Hessian of the solution u obtained in Theorem 1.1 or 1.2.

Theorem 1.3. With the same assumptions and notations as in Theorem 1.1 or 1.2 and let u be the solution of Δu = f obtained in Theorem 1.1 or 1.2. We have the following:

(i)

|∇u(x)| ≤ C(n, σ) ∫_{0}^{∞} k(x, t)dt.

(ii) For any p ≥ 1 and α ≥ 2, if u is the solution obtained in Theorem 1.1, then

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)} |∇u|^{p} ≤ C′ (∫_{αR}^{∞} k(t)dt)^{p} + \frac{C′′R^{p}}{V_{o}(R)} ∫_{B_{o}(αR)} f^{p}

for some constants C′(n, σ, p) and C′′(n, σ, p, α), and if u is the solution obtained in Theorem 1.2, then

\frac{1}{V_{o}(\frac{1}{\sqrt{2}}R)} ∫_{B_{o}(\frac{1}{\sqrt{2}} R)} |∇u|^{p} ≤ C′ (∫_{αR}^{∞} k(t)dt)^{p} + \frac{C′′R^{p}}{V_{o}(R)} ∫_{B_{o}(αR)} f^{p}

for some constants C′(n, p) and C′′(n, p, α).

(iii) If u is the solution obtained in Theorem 1.1, then

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)} |∇^{2}u|^{2} ≤ C [R^{−2} (∫_{4R}^{∞} k(t)dt)^{2} + \frac{1}{V_{o}(4R)} ∫_{B_{o}(4R)} f^{2}]

for some constant C(n), and if u is the solution obtained in Theorem 1.2, then

\frac{1}{V_{o}(\frac{1}{\sqrt{2}}R)} ∫_{B_{o}(\frac{1}{\sqrt{2}} R)} |∇^{2}u|^{2} ≤ C [R^{−2} (∫_{4R}^{∞} k(t)dt)^{2} + \frac{1}{V_{o}(4R)} ∫_{B_{o}(4R)} f^{2}]

for some constant C(n, σ).

Proof. Let us first consider the solution u obtained in Theorem 1.1. Divide (1.13) by r, choose ϵ = \frac{1}{4} and let r → 0, it is easy to see (i) is true. To prove (ii), by Theorem 1.1 for any x ∈ B_{o}(R), we have

(1.15)  |∇u(x)| ≤ ∫_{M} |∇_{x}G(x, y)|f(y)dy

≤ C_{1} ∫_{M} r^{−1}(x, y)G(x, y)f(y)dy

≤ C_{2} ∫_{M\B_{o}(αR)} \frac{r(x, y)}{V_{x}(r(x, y))}f(y)dy

+ C_{1} ∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)f(y)dy

for some constants C_{1}(n), C_{2}(n, σ). Here we have used (1.1) and (1.2) and the gradient estimate in [5]. Now

(1.16) _{∫M\B_{o}(αR)} \frac{r(x, y)}{V_{x}(r(x, y))}f(y)dy

≤ C_{3} ∫_{M\B_{o}(αR)} \frac{r(y)}{V_{o}(r(y))}f(y)dy

= C_{3} ∫_{αR}^{∞} \frac{t}{V_{o}(t)} (∫_{∂B_{o}(t)} f) dt

= C_{3} [\frac{t}{V_{o}(t)} ∫_{B_{o}(t)} f|_{αR}^{∞}

− ∫_{αR}^{∞} (\frac{1}{V_{o}(t)} − \frac{tA_{o}(t)}{V_{o}^{2}(t)}) (∫_{B_{o}(t)} f) dt]

≤ C_{4} ∫_{αR}^{∞} \frac{1}{V_{o}(t)} (∫_{B_{o}(t)} f) dt

= C_{4} ∫_{αR}^{∞} k(t)dt for some constants C_{3}(n), C_{4}(n), where we have used the fact that α ≥ 2, volume comparison, tA_{o}(t) ≤ nV_{o}(t) and the fact that tk(t) → 0 as t → ∞. Note that for any z ∈ M and for any ρ > 0

(1.17)  ∫_{B_{z}(ρ)} \frac{r(z, y)}{V_{z}(r(z, y))}dy = ∫_{0}^{ρ} \frac{tA_{z}(t)}{V_{z}(t)}dt ≤ nρ.

Let us first assume p > 1 and let q = p/(p − 1). By (1.15) and (1.16), we have

∫_{B_{o}(R)} |∇u|^{p} ≤ C_{5}V_{o}(R) (∫_{αR}^{∞} k(t)dt)^{p}

+ C_{6} ∫_{B_{o}(R)} (∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)f(y)dy)^{p} dx (1.18)

for some constants C_{5}(n, σ, p) and C_{6}(n, p).

\begin{aligned}_{∫B_{o}(R)} & (∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)f(y)dy)^{p} dx \\ & ≤ ∫_{B_{o}(R)} (∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)dy)^{\frac{p}{q}} \\ & · (∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)f^{p}(y)dy) dx \\ & ≤ C_{7}(n, σ, p)R^{\frac{p}{q}} ∫_{B_{o}(R)} (∫_{B_{o}(αR)} r^{−1}(x, y)G(x, y)f^{p}(y)dy) dx \\ & = C_{7}R^{\frac{p}{q}} ∫_{B_{o}(αR)} (∫_{B_{o}(R)} r^{−1}(x, y)G(x, y)dx) f^{p}(y)dy \\ & ≤ C_{8}R^{p} ∫_{B_{o}(αR)} f^{p}(y)dy \end{aligned} (1.19)

for some constants C_{7}(n, σ, p, α) and C_{8}(n, σ, p, α), where we have used (1.1) and (1.17). Combine this with (1.18), (ii) follows if p > 1. The case that p = 1 can be proved similarly.

To prove (iii), in terms of a local orthonormal frame,

\frac{1}{2}Δ|∇u|^{2} = ∑_{k,l} u_{kl}^{2} + ∑_{k} u_{k}(Δu)_{k} + ∑_{k,l} R_{kl}u_{k}u_{l}
≥ |∇^{2}u|^{2} + ⟨∇u, ∇f⟩

where R_{kl} is the Ricci curvature tensor of M which is positive semidefinite. Let φ ≥ 0 be a smooth function with compact support in B_{o}(2R). Multiplying the above inequality by φ^{2} and integrating by parts, we have

∫B_{o}(2R)φ^{2}|∇^{2}u|^{2}

≤ ∫_{B_{o}(2R)} φ^{2}f^{2} + ∫_{B_{o}(2R)} φ|∇φ| |∇u| |f|

+ 2 ∫_{B_{o}(2R)} φ|∇φ| |∇(|∇u|^{2})|

≤ C_{9} [∫_{B_{o}(2R)} φ^{2}f^{2}

+ ∫_{B_{o}(2R)} |∇φ|^{2}|∇u|^{2} + ∫_{B_{o}(2R)} φ|∇φ| |∇u||∇^{2}u|]

≤ C_{9} [∫_{B_{o}(2R)} φ^{2}f^{2} + (1 + \frac{1}{ϵ}) ∫_{B_{o}(2R)} |∇φ|^{2}|∇u|^{2}

+ϵ ∫_{B_{o}(2R)} φ^{2}|∇^{2}u|^{2}]

for any ϵ > 0, for some absolute constant C_{9}. Hence choose ϵ = (2C_{9})^{−1}, we have

∫_{B_{o}(2R)} φ^{2}|∇^{2}u|^{2} ≤ 2C_{9} (∫_{B_{o}(2R)} φ^{2}f^{2} + ∫_{B_{o}(2R)} |∇φ|^{2}|∇u|^{2})

for all φ ≥ 0 with compact support in B_{o}(R). Choose a suitable φ we obtain

∫_{B_{o}(R)} |∇^{2}u|^{2} ≤ C_{10} (∫_{B_{o}(2R)} f^{2} + R^{−2} ∫_{B_{o}(2R)} |∇u|^{2})

for some absolute constant C_{10}. Combining this with (ii) the results follows.

Suppose the assumptions of Theorem 1.2 are satisfied. Using the same notations as in the proof of Theorem 1.2. Then the gradient and the Hessian of ũ can be estimated as before. Since ũ is independent of x′ ∈ ℝ^{4}, the results then follow easily from Lemma 1.1. q.e.d.

Remark 1.1. The assumption that f ≥ 0 in Theorem 1.1, 1.2 and 1.3 can be relaxed. For general f, let k(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} |f| instead. Under similar assumptions as in Theorem 1.1 or 1.2, we can solve Δu_{1} = max{f, 0} and Δu_{2} = max{−f, 0} using these theorems. Then u = u_{1} − u_{2} satisfies Δu = f. Even though the estimates for u in Theorem 1.1 or 1.2 will no longer be true, however the estimates for |∇u| and |∇^{2}u| in Theorem 1.3 still hold.

Corollary 1.2. With the same assumptions on M and f and with the same notations as in Theorem 1.1 or 1.2. Let u be the solution of Δu = f obtained in Theorem 1.1 or 1.2. We have the following:

(i) Suppose there is a constant C > 0 such that ∫_{0}^{∞} k(x, t)dt ≤ C for all x. Then sup_{M} |∇u| < ∞.

(ii) Suppose f(x) ≤ Cr^{−2}(x) and k(t) ≤ Ct^{−2} for some constant C for all x ∈ M and t > 0. Then

|∇u(x)| ≤ C′r^{−1}(x)

for some constant C′ for all x.

(iii) Suppose there is a constant C > 0 such that f(x) ≤ Cr^{−2}(x) and k(t) ≤ t^{−2}h(t) with lim_{t→∞} h(t) = 0. Then

|∇u(x)| = o (r^{−1}(x))

as x → ∞.

Proof. (i) follows easily from Theorem 1.3(i). To prove (iii), by Theorem 1.3(i), it is sufficient to estimate ∫_{0}^{∞} k(x, t)dt. For any \frac{1}{2} > ϵ > 0, let x ∈ M and let r = r(x), then

(1.20)  ∫_{0}^{ϵr} k(x, t)dt ≤ C_{1}ϵr^{−1}

for some constant C_{1} independent of x and ϵ. For t ≥ ϵr,

(1.21)  k(x, t) = \frac{1}{V_{x}(t)} ∫_{B_{x}(t)} f

= \frac{V_{x}(t + r)}{V_{x}(t)} · \frac{1}{V_{x}(t + r)} ∫_{B_{o}(t+r)} f

≤ C_{2}(n) (1 + ϵ^{−1})^{n} \frac{1}{V_{o}(t)} ∫_{B_{o}((1+ϵ^{−1}))t} f

≤ C_{1} (1 + ϵ^{−1})^{2n} k ((1 + ϵ^{−1})t) for some constant C_{2}(n). By (1.20) and (1.21), we have

∫_{0}^{∞} k(x, t)dt ≤ C_{1}ϵr^{−1} + ∫_{ϵr}^{∞} (1 + ϵ^{−1})^{2n} k ((1 + ϵ^{−1})t)

≤ βϵr^{−1} + C_{2} ( sup_{t≥(1+ϵ)r} h(t)) · r^{−1}

for some constant C_{2}(n, ϵ). Since h(t) → 0 as t → ∞ (iii) follows. The proof of (ii) is similar. q.e.d.

2. The Poisson equation (II)

In the previous section, we have obtained some conditions on f so that the Poisson equation Δu = f has a solution. In this section, we will study the problem from another perspective. Namely, suppose a solution u of the Poisson equation Δu = f exists, we want to discuss the properties of f. We have the following general result.

Theorem 2.1. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature. Suppose u is a solution of Δu = f on M, where f ≥ 0 is a nonnegative function. Suppose that there exist nondecreasing functions h and g such that

−g(r) ≤ inf_{B_{o}(r)} u ≤ sup_{B_{o}(r)} u ≤ h(r)

for all r, where o ∈ M is a fixed point. Then for any R > 0 and x ∈ M

C(n) [R^{2}k(x, R) + ∫_{0}^{R} tk(x, t)dt] ≤ −u(x) + h(5R + r)
≤ g(r) + h(5R + r)

for some positive constant C(n), where r = r(x),

k(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} f.

In particular, if we let k(t) = k(o, t), then

C(n) [R^{2}k(R) + ∫_{0}^{R} tk(t)dt] ≤ −u(o) + h(5R)

for all R.

Proof. Let x ∈ M and let r = r(x). For any R > 0, let G_{R} be the positive Green’s function on B_{x}(R) with Dirichlet boundary data. Then

∫_{B_{x}(R)} G_{R}(x, y)f(y)dy = ∫_{B_{x}(R)} G_{R}(x, y)Δu(y)dy

= −u(x) − ∫_{∂B_{x}(R)} u\frac{∂G_{R}}{∂ν}

≤ −u(x) + h(R + r)

where we have used the fact that \frac{∂G_{R}}{∂ν} < 0 on ∂B_{x}(R) and that ∫_{∂B_{x}(R)} \frac{∂G_{R}}{∂ν} = −1. By Lemma 1.1 in [25]

G_{R}(x, y) ≥ C_{1} ∫_{r(x,y)}^{R} \frac{t}{V_{x}(t)}dt

for all y ∈ B_{x}(\frac{1}{5}R) for some constant C_{1}(n) > 0. Hence

g(r) + h(R + r) ≥ −u(x) + h(R + r)

≥ C_{1} ∫0^{\frac{R}{5}} (∫_{t}^{R} \frac{s}{V_{x}(s)} ds) (∫_{∂B_{x}(t)} f) dt

≥ C_{1} [(∫ R_{\frac{R}{5}} \frac{t}{V_{o}(t)}dt) (∫_{B_{x}( \frac{R}{5} )} f)

+ ∫0^{\frac{R}{5}} (\frac{t}{V_{x}(t)} ∫_{B_{x}(t)} f) dt]

≥ C_{2} [R^{2}k(x, \frac{R}{5}) + ∫0^{\frac{R}{5}} tk(x, t)dt]

for some positive constant C_{2}(n), where we have used volume comparison. From this the theorem follows. q.e.d.

Note that similar estimate has been obtained in [15, Lemma 2.1], where no curvature assumption was made and hence the result was weaker.

Using Theorem 2.1 and the results in Section 1, we can obtain necessary and sufficient conditions for a function f so that Δu = f has a solution with certain growth rate.

Theorem 2.2. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature and let m(t), 0 ≤ t < ∞, be a nonnegative nondecreasing function such that for any A > 1 there exists C > 0 with

(2.1)  m(At) ≤ Cm(t)

for all t, and

(2.2)  ∫_{1}^{∞} t^{−2}m(t)dt < ∞.

Let f ≥ 0 be a locally Hölder continuous function on M and let k(x, t) as in Theorem 2.1. Then the Poisson equation Δu = f has a solution on M with sup_{B_{o}(r)} |u| ≤ Cm(r) for some constant C for all r if and only if

(2.3)  ∫_{0}^{t} sk(x, s)ds ≤ C′m(t)

for some constant C′ for all x and for all t ≥ \frac{1}{5}r(x).

Proof. By (2.2) and the fact that m is nondecreasing, m(t) = o(t) as t → ∞. Suppose f satisfies (2.3). Then it is easy to see that f satisfies the assumptions in Theorem 2.1. Hence Δu = f has a solution u such that

α_{1}r ∫_{2r}^{∞} k(t)dt + β_{1} ∫_{0}^{2r} tk(t)dt ≥ u(x)
≥ −α_{2}r ∫_{2r}^{∞} k(t)dt
− β_{2} ∫_{0}^{\frac{1}{5} r} tk(x, t)dt
+ β_{3} ∫_{0}^{2r} tk(t)dt

for some positive constants α_{1}(n), α_{2}(n) and β_{i}(n), 1 ≤ i ≤ 3, where r = r(x), k(t) = k(o, t). Combine this with (2.1)–(2.3), we have

(2.4)  sup_{B_{o}(r)} |u| ≤ C_{1}m(r)

for some constant C_{1} for all r.

Conversely, suppose Δu = f has a solution satisfying (2.4). Then by Theorem 2.1 and (2.1), it is easy to see that (2.3) is true. q.e.d.

Remark 2.1. (i) We assume condition (2.1) so that we can state the theorem more simply. Otherwise, we may replace some of the m(t) in the theorem by m(At) for some constant A. (ii) From the proof it is easy to see that if Δu = f has a solution satisfying sup_{B_{o}(r)} |u| ≤ Cm(r), then u is the solution obtained in Theorem 1.2. Because in this case, both u and the solution in Theorem 1.2 are of sub-linear growth.

If we take m(t) =constant, m(t) = log(2 + t) or m(t) = (1 + t)^{1−δ} for some 1 > δ > 0, then we have the following.

Corollary 2.1. Let M^{n} be a complete noncompact Riemannian manifold with nonnegative Ricci curvature and let f ≥ 0 be a locally Hölder continuous function on M. Let k(x, t) and k(t) be as in Theorem 2.2. Then:

(i) Δu = f has a bounded solution if and only if there is a constant

C > 0 such that

∫_{0}^{∞} tk(x, t)dt ≤ C

for all x.

(ii) Δu = f has a solution with sup_{B_{o}(r)} |u| ≤ C log(2 + r) for some constant C for all r if any only if

∫_{0}^{t} sk(x, s)ds ≤ C′ log(2 + t)

for some constant C′ for all r = r(x) and for all t ≥ \frac{1}{5}r.

(iii) Δu = f has a solution with sup_{B_{o}(r)} |u| ≤ C(1 + r)^{1−δ} for some constants C and 0 < δ < 1 for all r if any only if

∫_{0}^{t} sk(x, s)ds ≤ C′(1 + t)^{1−δ}

for some constant C′ for all r = r(x) and for all t ≥ \frac{1}{5}r.

Suppose ∫_{0}^{∞} tk(t)dt < ∞ and if f is not identically zero, then ∫_{1}^{∞} t/V_{o}(t)dt < ∞ and M must be nonparabolic. In this case, it is easy to see that u(x) = − ∫_{M} G(x, y)f(y)dy + C for some constant C. As an application of this remark and Corollary 2.1, we will give another proof of a result of Li [12, Theorem 4] on bounded subharmonic functions. The method is not simpler, but we obtain some estimates that may be useful.

Theorem 2.3. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature. Let u be a bounded subharmonic function, and let α = sup_{M} u, then:

(i)

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)}(α − u)

≤ C (\frac{R^{2}}{V_{o}(2R)} ∫_{B_{o}(2R)} f + ∫_{2R}^{∞} (\frac{t}{V_{o}(t)} ∫_{B_{o}(t)} f) dt)

for some constant C(n) for all R > 0, where f = Δu.

(ii) (Li [12])

lim_{R→∞} \frac{1}{V_{o}(R)} ∫_{B_{o}(R)} u = sup_{M} u.

Proof. First we assume that M is nonparabolic and satisfies (1.1) and (1.2). Then f ≥ 0 because u is subharmonic. By Corollary 2.1, we have

(2.5)  ∫_{0}^{∞} (\frac{t}{V_{x}(t)} ∫_{B_{x}(t)} f) dt ≤ C_{1}

for some constant C_{1} for all x ∈ M. Moreover, by adding a constant to u, we may assume that

u(x) = − ∫_{M} G(x, y)f(y)dy.

Note that u is nonpositive. Using similar methods as in the proof of Theorem 1.3, let R > 0

(2.6) ∫_{B_{o}(R)}(−u) = ∫_{x∈B_{o}(R)} (∫_{y∈M} G(x, y)f(y)dy) dx

= ∫_{x∈B_{o}(R)} (∫_{y∈B_{o}(2R)} G(x, y)f(y)dy) dx

+ ∫_{x∈B_{o}(R)} (∫_{y∈M\B_{o}(2R)} G(x, y)f(y)dy) dx.

For any y ∈ M,

∫_{x∈B_{y}(3R)} G(x, y)dx ≤ C_{2} ∫_{0}^{3R} A_{y}(t)\frac{t^{2}}{V_{y}(t)}dt
= C_{3} ∫_{0}^{3R} tdt
≤ C_{4}R^{2}

for some constants C_{2}, C_{3} and C_{4} depending only on n and σ, where we have used (1.1) and (1.2). Hence

∫_{x∈B_{o}(R)} (∫_{y∈B_{o}(2R)} G(x, y)f(y)dy) dx (2.7)

= ∫_{y∈B_{o}(2R)} f(y) (∫_{x∈B_{o}(R)} G(x, y)dx) dy

≤ ∫_{y∈B_{o}(2R)} f(y) (∫_{x∈B_{y}(3R)} G(x, y)dx) dy
≤ C_{4}R^{2} ∫_{B_{o}(2R)} f.

For any x ∈ B_{o}(R), using (2.5)

\begin{aligned}∫_{y∈M\B_{o}(2R)} G(x, y)f(y)dy & ≤ C_{5} ∫_{y∈M\B_{o}(2R)} G(o, y)f(y)dy \\ & ≤ C_{6} ∫_{2R}^{∞} (∫_{t}^{∞} \frac{s}{V_{o}(s)}ds) (∫_{∂B_{o}(t)} f) dt \\ & ≤ C_{6} ∫_{2R}^{∞} (\frac{t}{V_{o}(t)} ∫_{B_{o}(t)} f) dt \end{aligned} (2.8)

for some constants C_{5}, C_{6} depending only on n and σ. By (2.6)–(2.8), we have

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)}(−u)

≤ (C_{4} + C_{6}) (\frac{R^{2}}{V_{o}(R)} ∫_{B_{o}(2R)} f + ∫_{2R}^{∞} (\frac{t}{V_{o}(t)} ∫_{B_{o}(t)} f) dt) .

This implies (i) because α = sup_{M} u ≤ 0. By (2.5), the right side of the above inequality will tend to 0 as R → ∞. This implies (ii) by noting that

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)}(−u) ≥ inf_{M}(−u) ≥ 0.

For general cases, we just take M × ℝ^{4} with flat metric on ℝ^{4} and consider u as a subharmonic function on M × ℝ^{4} and use Lemma 1.1. Note that in this case, we can choose σ to depend only on n. q.e.d.

Consider the following example: let u be a nonconstant bounded subharmonic function on ℝ^{3} and consider u as a bounded subharmonic function on ℝ^{4}. Then the average of u over B_{o}(r) will tends to sup_{M} u as r → ∞. However, it is obvious that u will not be asymptotically constant at infinity. In this respect, we have:

Theorem 2.4. Let M^{n} be a complete noncompact manifold with nonnegative Ricci curvature and let u be a smooth bounded subharmonic function on M. Suppose f = Δu is such that f(x) ≤ Cr^{−2}(x). Then

lim_{x→∞} u(x) = sup_{M} u

where x → ∞ means that r(x) → ∞.

Proof. Since u is bounded and subharmonic, by Corollary 2.1 we can conclude that

t^{−2}k(t) → 0

as t → ∞, where k(t) = 1/V_{o}(t) ∫_{B_{o}(t)} f. By Corollary 1.2(iii) and the assumption that f(x) ≤ Cr^{−2}(x), we have

(2.9)  |∇u(x)| = o(r^{−1}(x))

as r(x) → ∞. We may assume that sup_{M} u = 0. By Li’s result Theorem 2.3(ii)

lim_{r→∞} \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} u = 0.

Since u ≤ 0, for any ϵ > 0, let D_{r} = {x ∈ B_{o}(r)| u(x) ≥ −ϵ}, then it is easy to see that

V (D_{r}) ≥ (1 − ϵ)V (r)

if r is large enough. Hence if x is such that r(x) = R and if R is large enough, V_{x}(\frac{1}{2}R) ∩ D_{2R} ≠ ∅. By (2.9) we conclude that u(x) → 0 as x → ∞. q.e.d.

3. Some vanishing results

In this section, we will apply the results in Section 1 and Section 2 to obtain some vanishing theorems on holomorphic line bundles over complete noncompact Kähler manifolds. The results are related to those in [22] and [10]. We need the following Kodaira-Bochner formula [21, Chapter 3, §6]:

Lemma 3.1. Let M be a Kähler manifold, let L be a Hermitian holomorphic line bundle over M and let ϕ be a holomorphic (p, 0) form with value in L. Denote |ϕ| to be the norm of ϕ with respect to the Kähler metric on M and the Hermitian metric h on L. Then

|ϕ|^{2}Δ|ϕ|^{2} − |∇|ϕ|^{2}|^{2}

≥ 4 (−Ω + min_{1≤i_{1}<i_{2}<···<i_{p}≤m}(γ_{i_{1}} + γ_{i_{2}} + ⋯ + γ_{i_{p}})) |ϕ|^{4}.

where γ_{i} are the eigenvalues of the Ricci form R_{ij} of M, Ω is the trace of the curvature form Ω_{ij} of h.

Theorem 3.1. Let M^{m} be a complete noncompact Kähler manifold of complex dimension m with nonnegative Ricci curvature. Let L be a Hermitian holomorphic line bundle over M and let Ω be the trace of the curvature form of L. For any 0 < ϵ < 1 and τ > 0, there exists a constant a(m, τ, ϵ) such that if

(3.1)  lim sup_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} Ω_{+} ≤ a,

where Ω_{+} is the positive part of Ω, then any holomorphic (p, 0) form ϕ (valued in L) is trivial if the norm of ϕ satisfies

(3.2)  \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} |ϕ|^{2τ} = O(r^{−ϵ})

as r → ∞.

Proof. Suppose Ω_{+} satisfies the condition (3.1) with a to be determined later. Let M̃ = M × ℝ^{4} and denote x̃ = (x, x′). Consider Ω_{+} as a function on M̃. By Theorem 1.1 and Lemma 1.1, we can find a solution u(x̃) of Δ̃u = 4τΩ_{+} such that

u(x̃) ≤ α_{1}r̃∫ ∞ ; r̃ k(t)dt + β_{1}τ ∫_{0}^{2r̃} tk(t)dt

where α_{1} and β_{1} are the constants depending only on m, r̃ = r̃(x̃) is the distance from x̃ to õ = (o, 0) and k(t) = 1/V_{o}(t) ∫_{B_{o}(t)} Ω_{+}. Here Δ̃ is the

Laplacian on M̃. Choose a > 0 such that aβ_{1}τ < \frac{1}{2}ϵ, then a depends only on m, τ, ϵ. By (3.1), we have

lim sup_{r(x̃)→∞} \frac{u(x̃)}{aβ_{1}τ log r̃(x)} ≤ 1

and hence

(3.3)  e^{u(x̃)} ≤ C_{1}(1 + r̃)^{\frac{1}{2} ϵ}(x̃)

for some constant C_{1} for all x̃. Let ϕ be a holomorphic (p, 0) form such that

(3.4)  \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} |ϕ|^{τ} = O(r^{−ϵ}),

and let f = |ϕ|^{2}. By Lemma 3.1, we have

fΔf − |∇f|^{2} ≥ −4Ω_{+}f^{2}

and if we consider f as a function on M̃, then

(3.5)  fΔ̃f − | ˜∇f|^{2} ≥ −4Ω_{+}f^{2}

where ˜∇ is the gradient on M̃. For any δ > 0, let g = (f + δ)^{τ}. At a point x̃ where f(x̃) > 0, we have

gΔ̃g − | ˜∇g|^{2} = g^{2}Δ̃ log g

= τg^{2} (\frac{Δ̃f}{f + δ} − \frac{| ˜∇f|^{2}}{(f + δ)^{2}} )

≥ τg^{2} (−\frac{4Ω_{+}f}{f + δ} + \frac{| ˜∇f|^{2}}{f(f + δ)} − \frac{| ˜∇f|^{2}}{(f + δ)^{2}} )

≥ −4τΩ_{+}g^{2}.

On the other hand, suppose f(x̃) = 0, then g attains minimum at x̃. Hence we still have

gΔ̃g − | ˜∇g|^{2} ≥ −4τΩ_{+}g^{2}.

Let v = e^{u}g, then

vΔ̃v = e^{u}g (e^{u}Δ̃g + 2e^{u} < ˜∇u, ˜∇g > +e^{u}gΔ̃u + e^{u}g| ˜∇u|^{2})

≥ e^{2u} (−4τΩ_{+}g^{2} + | ˜∇g|^{2} − 2g| ˜∇u| | ˜∇g| + 4τΩ_{+}g^{2} + g^{2}| ˜∇u|^{2})

≥ 0.

By the mean value inequality of Li-Schoen ([13, Theorem 2.1]), for any r̃ > 0,

sup_{Bõ(r̃)} v ≤ \frac{C_{2}}{Võ(2r̃)} ∫_{Bõ(2r̃)} v

for some constant C_{2}(m). Let δ → 0, we have

(3.6)  sup_{B_{õ(r̃)}} e^{u}f^{τ} ≤ \frac{C_{2}}{Võ(2r̃)} ∫_{Bõ(2r̃)} e^{u}f^{τ}

≤ C_{3}(1 + r̃)^{\frac{1}{2} ϵ}V −1 ; õ(2r̃) ∫_{Bõ(2r̃)} f^{τ}

≤ C_{4}(1 + r̃)^{\frac{1}{2} ϵ}V_{o}^{−1}(2r̃) ∫_{B_{o}(2r̃)} f^{τ}

for some constants C_{3} and C_{4} independent of r̃. Here we have used (3.3) and Lemma 1.1. For any x ∈ M with r = r(x), if we take r̃ = R > 2r in (3.6), we have

e^{u(x,0)}|ϕ(x)|^{2τ} = e^{u(x,0)}f^{τ} (x)

≤ \frac{C_{5}(1 + R)^{\frac{1}{2} ϵ}}{V_{o}(2R)} ∫_{B_{o}(2R)} f^{τ}

= \frac{C_{5}(1 + R)^{\frac{1}{2} ϵ}}{V_{o}(2R)} ∫_{B_{o}(2R)} |ϕ|^{2τ}

= O (R^{− \frac{1}{2} ϵ})

for some constant C_{5} independent of r and R. Let R → ∞, we conclude that ϕ ≡ 0. q.e.d.

Remark 3.1. If ∫_{M} |ϕ|^{2τ} < ∞, then obviously ϕ satisfies (3.2) because the volume growth of M is at least linear by [27].

As an application, we have:

Corollary 3.1. Let M^{m} be a complete noncompact Kähler manifold with nonnegative Ricci curvature and let L be a holomorphic line bundle over M with Hermitian metric h. Let ρ = \sqrt{−1}Ω_{ij}dz^{i} ∧ dz^{j} be the curvature form of L, h and let Ω be the trace of ρ. Suppose ρ ≥ 0 and

lim sup_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} Ω = 0.

Then ρ^{m} ≡ 0.

Proof. Suppose ρ^{m} ≠ 0 at some point, then there exists a positive integer ℓ and a nontrivial holomorphic section ϕ of L^{ℓ} such that |ϕ| ∈ L^{2}(M) by Corollary 3.3 in [22]. Note that the trace of the curvature form of L^{ℓ} is ℓΩ. By Theorem 3.1 and the assumption on Ω, we have a contradiction. q.e.d.

Later in Section 5, we will discuss conditions so that L is actually flat, see Proposition 5.2.

If we take L to be the anti-canonical bundle of M, then we have the following generalization of the first part of Corollary 3.5 in [22].

Corollary 3.2. Let M^{m} be a complete noncompact Kähler manifold with nonnegative Ricci curvature and let R be the scalar curvature of M. Suppose

(3.7)  lim sup_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} R = 0.

Then the Ricci form ρ of m satisfies ρ^{m} = 0.

Remark 3.2. (i) If M is nonparabolic and if R is integrable, then (3.7) is true. Hence Corollary 3.2 is a generalization of Corollary 3.5 and Theorem 3.6 in [22] for the case that M has nonnegative Ricci curvature. (ii) As observed in [4], from the arguments in [23], if (3.7) is true for all base point o so that the convergence is uniform and if the holomorphic bisectional curvature of M is bounded and nonnegative, then M is flat. In the above corollary, we only assume that the Ricci curvature is nonnegative and we do not assume that the scalar curvature is bounded. The result is weaker and it is interesting to see whether M is actually Ricci flat in this case. In fact, for Riemannian case, it is proved by Chen and Zhu [3] that if (3.7) is true uniformly and if the Riemannian manifold is locally conformally flat then the manifold is flat.

In the next result, we will relax the assumption that M has nonnegative Ricci curvature.

Theorem 3.2. Let M^{m} be a complete noncompact nonparabolic Kähler manifold with complex dimension m. Let L be a Hermitian holomorphic line bundle and let Ω be the trace of the curvature form of L. For any 1 ≤ p ≤ m, let

S(x) = min_{1≤i_{1}<i_{2}<···<i_{p}≤m}(γ_{i_{1}}(x) + γ_{i_{2}}(x) + ⋯ + γ_{i_{p}}(x)),

where γ_{j} are the eigenvalues of the Ricci form of M, and let S ≡ 0 if p = 0. Suppose

∫_{M} (−Ω + S)_{−} < ∞,

in particular, suppose

∫_{M} (Ω_{+} + S_{−}) < ∞

where Ω_{+} is the positive part of Ω, S_{−} is the negative part of S and (−Ω + S)_{−} is the negative part of −Ω+S. Then any holomorphic (p, 0) form ϕ with value in L must be trivial if ϕ satisfies

∫_{B_{o}(r)} |ϕ|^{2τ} = o(r^{2})

as r → ∞ for some τ > 0.

Proof. The result follows immediately from Lemma 3.1 and the result in [18, Corollary 2.2]. For the sake of completeness, we prove the result in the same spirit as the proof of Theorem 3.1.

First, we claim that there is a solution u of Δu = (−Ω + S)_{−} with u ≤ 0. To prove the claim, let R > 0 be fixed, and let G be the minimal positive Green’s function of M. G exists because M is nonparabolic. Let σ = 4 (−Ω + S)_{−}, then for x ∈ B_{o}(R),

∫_{M} G(x, y)σ(y)dy = ∫_{B_{o}(2R)} G(x, y)σ(y)dy + ∫_{M\B_{o}(2R)} G(x, y)σ(y)dy

≤ C_{1} (1 + ∫_{M\B_{o}(2R)} G(o, y)σ(y)dy)
≤ C_{2}

for some constants C_{1}, C_{2} independent of x with x ∈ B_{o}(R). Here we have used Harnack inequality, the fact that sup_{M\B_{o}(2R)} G(o, y) < ∞ (cf. [14, p.1138]), the fact that for σ ∈ L^{1}(M), and that there exists a constant C_{3} such that

∫_{B_{o}(2R)} G(x, y)σ(y)dy ≤ C_{3}

for all x ∈ B_{o}(R). Hence ∫_{M} G(x, y)σ(y)dy is locally bounded and u(x) = − ∫_{M} G(x, y)σ(y) is well defined with Δu = σ. Obviously u ≤ 0.

To complete the proof of the theorem, let ϕ be a holomorphic (p, 0) form such that ∫_{B_{o}(r)} |ϕ|^{2τ} = o(r^{2}). As in the proof of Theorem 3.1, for any ϵ > 0, let f = |ϕ|^{2} and let g = (f + ϵ)^{τ}. If v = e^{τu}g, then

vΔv ≥ 0.

We can then apply the method in [27], [18]. Namely, multiplying the above inequality by a suitable cut off function, we have

∫_{B_{o}(r)∩M_{a}} |∇v|^{2} ≤ ∫_{B_{o}(r)} |∇v|^{2} ≤ \frac{C_{4}}{r^{2}} ∫_{B_{o}(2r)} v^{2}

where M_{a} = {x ∈ M| f(x) > a} and C_{4} is a constant independent of r and a. Let ϵ → 0, we have

(3.8)  ∫_{B_{o}(r)∩M_{a}} |∇w|^{2} ≤ \frac{C_{4}}{r^{2}} ∫_{B_{o}(2r)} w^{2}

where w = e^{τu}f^{τ}. By the assumption on f = |ϕ|^{2} and the fact that τ > 0, u ≤ 0, we conclude that

(3.9)  ∫_{B_{o}(2r)} w^{2} = ∫_{B_{o}(2r)} (e^{τu}f^{τ})^{2} = o(r^{2})

as r → ∞. Combine this with (3.8) and let r → ∞, |∇w| ≡ 0 on M_{a}. Since a is arbitrary, w must be a constant. Since M is nonparabolic,

lim sup_{r→∞} \frac{V_{o}(r)}{r^{2}} > 0.

and hence (3.9) implies that w must be identically zero. q.e.d.

Theorem 3.2 generalizes Theorem 2.3 in [22] which deals with holomorphic section of line bundles. By taking L to be the trivial bundle with flat metric, it is easy to see that the theorem also generalizes part of Theorem 2 in [10] for the case of holomorphic p-forms.

4. Liouville property of plurisubharmonic functions

In this section, we will apply the results in Section 1 and Section 2 to study plurisubharmonic functions on a complete noncompact Kähler manifold M^{m} with nonnegative Ricci curvature, where m is the complex dimension of M. In [22, Proposition 4.1], it was proved that if u is a plurisubharmonic function such that

lim sup_{r(x)→∞} \frac{u(x)}{log r(x)} = 0

then

(4.1)  (∂∂u)^{m} = 0

on M. Let us first prove a more general result as an application of Theorem 3.2.

Proposition 4.1. Let M^{m} be a complete noncompact nonparabolic Kähler manifold such that the scalar curvature R satisfies

∫_{M} R_{−} < ∞.

where R_{−} is the negative part of R. Suppose u is plurisubharmonic function on M such that

lim sup_{x→∞} \frac{u(x)}{log r(x)} = 0.

Then (∂∂u)^{m} = 0.

Proof. Suppose \sqrt{−1}∂∂u(x_{0}) > 0 at some point x_{0}. We can find a coordinate neighborhood U with holomorphic coordinates z(x) where z = (z_{1}, …, z_{m}) so that x_{0} corresponds to the origin and U corresponds to |z| < 4, and that \sqrt{−1}∂∂u > 0 in U. Let λ ≥ 0 be a smooth function on U such that λ(z(x)) = 1 in |z(x)| < 1 and λ = 0 outside |z(x)| = 2. Let ϕ be the function on M such that ϕ(x) = 2(m + 1)λ(z(x)) log |z(x)| on U and zero outside U. Then ϕ is smooth on M \ {x_{0}} with compact support. Since ∂∂ϕ ≥ 0, in the weak sense, within {|z(x)| ≤ 1} and \sqrt{−1}∂∂u > 0 in U. Hence there is a positive constant A such that if ψ = Au + ϕ then \sqrt{−1}∂∂ψ ≥ ϵω for some nonnegative continuous function ϵ which is positive on |z(x)| ≤ 1. Here ω is the Kähler form of M. Let ρ ≥ 0 be a smooth cutoff function such that ρ(z(x)) = 1 if |z(x)| ≤ 1/2 and ρ = 0 outside |z(x)| = 1. Let η = ρdz_{1} ∧ ⋯ ∧ dz_{m}. It is easy to see that

∫_{M} \frac{||∂η||^{2}}{ϵ} e^{−ψ} < ∞.

By Theorem 5.1 in [7], there is an (m, 0) form τ such that ∂τ = ∂η and

∫_{M} ||τ||^{2}e^{−ψ} ≤ C ∫_{M} \frac{||∂η||^{2}}{ϵ} e^{−ψ} < ∞.

Note, here we do not need assumptions on the curvature of M because we are dealing with (m, 0) forms. By the definition of ψ, we conclude that τ(x_{0}) = 0. Hence η̃ = τ − η is holomorphic (m, 0) form which is nontrivial. Moreover, by the above inequality and the growth assumption on u, we have

∫_{B_{o}(r)} |η̃|^{2} = O(r).

as r → ∞. This contradicts Theorem 3.2 with L being the trivial line bundle with flat metric. q.e.d.

Because of Ni’s result [22], it is interesting to see whether u is actually constant if u satisfies (4.1).

Let u be a plurisubharmonic function and let f = Δu. As before, let k(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} f and k(t) = k(o, t). First, we assume that M supports a strictly plurisubharmonic function.

Theorem 4.1. Let M^{m} be a complete noncompact Kähler manifold with nonnegative Ricci curvature. Let u be a plurisubharmnonic function satisfying (4.1) such that f(x) = Δu(x) ≤ Cr^{−2}(x) for some constant C > 0 for all x. Suppose M supports a strictly plurisubharmonic function. Then u must be constant if one of the following is true:

(a) u is bounded.

(b) sup_{B_{o}(r)} u = o(r) as r → ∞ and there exist r_{i} → ∞ and a constant C such that r_{i} ∫_{r_{i}}^{∞} k(t)dt ≤ C.

(c) u(x) ≤ a log r(x) for some constant a for all x with r(x) ≥ 2 and M is nonparabolic with Green’s function satisfying (1.1).

Proof. (a) Since f(x) ≤ Cr^{−2}(x), by Theorem 2.3, we have

lim_{x→∞} u(x) = sup_{M} u.

By the minimum principle of [2], [1], u must be a constant.

Suppose (b) is true. Then ∫_{0}^{∞} k(t)dt < ∞. Since f(x) ≤ Cr^{−2}(x), we can find a solution v of Δv = f such that |v(x)| = o (r(x)) as x → ∞ by Proposition 1.1 and Theorem 1.2. Hence u−v is a harmonic function and sup_{B_{o}(r)}(u − v) = o(r) because sup_{B_{o}(r)} u = o(r). By the gradient estimate [5, Theorem 6], u − v must be a constant. Without loss of generality, we may assume that u = v which is the solution obtained in Theorem 1.2. In particular,

u(x) ≥ −α_{2}r ∫_{2r}^{∞} k(t)dt − β_{2} ∫_{0}^{\frac{1}{2} r} tk(x, t)dt + β_{3} ∫_{0}^{2r} tk(t)dt

where α_{2} and β_{2} are positive constants depending only on m and r = r(x). Since f(x) ≤ Cr^{−2}(x),

(4.2)  u(x) ≥ −C_{1} − α_{2}r ∫_{2r}^{∞} k(t)dt + β_{3} ∫_{0}^{2r} tk(t)dt

for some constant C_{1} independent of x. By the assumption, there exist r_{i} → ∞ and a constant C_{2} such that

(4.3)  r_{i} ∫_{r_{i}}^{∞} k(t)dt ≤ C_{2}.

Hence

inf_{∂B_{o}( \frac{1}{2} r_{i})} u ≥ −C_{3} + β_{3} ∫_{0}^{r_{i}} tk(t)dt

for some constant C_{3} for all i. Suppose ∫_{0}^{∞} tk(t)dt = ∞, then u(0) = ∞ by the minimum principle of [2], [1]. This is impossible. Hence ∫_{0}^{∞} tk(t)dt < ∞. By the minimum principle again, we conclude that u is bounded from below. By Theorem 1.2, u also has an upper bound

u(x) ≤ α_{1}r ∫_{2r}^{∞} k(t)dt + β_{1} ∫_{0}^{2r} tk(t)dt

for some constants α_{1} and β_{1} depending only on m, where r = r(x). By (4.3), the fact that ∫_{0}^{∞} tk(t)dt < ∞ and the maximum principle for subharmonic function, we conclude that u must be also bounded from above. Hence u is constant by (a).

Suppose (c) is true. By the assumption on the upper bound of u(x) and Theorem 2.1, there exists a constant C_{4} such that

(4.4)  ∫_{0}^{R} tk(t)dt ≤ C_{4} log R

for all R large enough. It is sufficient to show that there exist r_{i} → ∞ such that (4.3) is true. First note that

(4.5)  I = ∫_{r}^{∞} t^{−1} (∫_{t}^{∞} \frac{s}{V_{o}(s)} ds) (∫_{∂B_{o}(t)} f) dt

≤ C_{5} ∫_{r}^{∞} \frac{A(t)}{tV (t)}dt

≤ C_{6}r^{−1}

for some constants C_{5} and C_{6} independent of r. Here we have used the fact that f(x) ≤ Cr^{−2}(x) and the fact that tA(t) ≤ 2mV (t). On the other hand,

(4.6)  I = t^{−1} (∫_{t}^{∞} \frac{s}{V_{o}(s)} ds) (∫_{B_{o}(t)} f) |_{r}^{∞}

+ ∫_{r}^{∞} t^{−2} (∫_{t}^{∞} \frac{s}{V_{o}(s)} ds) (∫_{B_{o}(t)} f) dt

+ ∫_{r}^{∞} \frac{1}{V_{o}(t)} (∫_{B_{o}(t)} f) dt

≥ −\frac{C_{7}r}{V_{o}(r)} (∫_{B_{o}(r)} f) + ∫_{r}^{∞} k(t)dt

for some positive constant. Here we have used the fact that the positive Green’s function satisfies (1.1). By (4.4), (4.5), (4.6) it is easy to see that (4.3) is true for some r_{i} → ∞. q.e.d.

By the method of [20, pp.195-199], we can obtain the following Liouville result for bounded plurisubharmonic functions on Kähler manifold with nonnegative sectional curvature with maximal volume growth.

Proposition 4.2. Let M^{m} be a complete noncompact Kähler manifold with nonnegative sectional curvature and with maximal volume growth. Let u be a bounded plurisubharmnonic function satisfying (4.1) such that f = Δu satisfies f(x) ≤ Cr^{−2}(x) for some constant C > 0 for all x. Then u must be constant.

Proof. Since u is bounded and f(x) ≤ Cr^{−2}(x), u is asymptotically constant by Theorem 2.4. Using the method in [20], we conclude that u is constant. q.e.d.

Suppose M has positive holomorphic bisectional curvature, then M supports a strictly plurisubharmonic function by [9]. Hence we have the following.

Corollary 4.1. Let M^{m} be a complete noncompact Kähler manifold with positive biholomorphic sectional curvature. Let u be a plurisubharmonic function satisfying (4.1) such that f(x) ≤ Cr^{−2}(x). Suppose one of the conditions (a), (b) or (c) in Theorem 4.1 is true, then u must be constant.

5. The Poincaré-Lelong equation

Let M^{m} be a complete Kähler manifold with nonnegative bisectional curvature. In [20, Theorem 1.1], it was proved that if M has maximal volume growth and if ρ is a closed (1, 1) form on M such that the norm ||ρ|| of ρ satisfies ||ρ(x)|| ≤ Cr^{−2}(x) for some constant for all x, then one can solve the Poincaré-Lelong equation by solving 1/2Δu = trace(ρ). In this section, we will apply the results in Section 1 and Section 2 to show that given a closed (1, 1) form ρ with trace f, one can solve the following Poincaré-Lelong equation under rather general assumptions on ρ:

(5.1)  \sqrt{−1}∂∂u = ρ.

We will also give some applications of the result.

In this section, m always denotes the complex dimension of M^{m}.

Theorem 5.1. Let M^{m} be a complete Kähler manifold with nonnegative holomorphic bisectional curvature. Let ρ be a real closed (1, 1) form with trace f. Suppose f ≥ 0 and ρ satisfies the following conditions:

(5.2)  ∫_{0}^{∞} \frac{1}{V_{o}(t)} ∫_{B_{o}(t)} ||ρ||dt < ∞,

and

(5.3)  lim inf_{r→∞} \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ||^{2} = 0.

Then there is a solution u of the Poincaré-Lelong Equation (5.1). Moreover, for any 0 < ϵ < 1, u satisfies

α_{1}r ∫_{2r}^{∞} k(t)dt + β_{1} ∫_{0}^{2r} tk(t)dt ≥ u(x)
≥ −α_{2}r ∫_{2r}^{∞} k(t)dt
− β_{2} ∫_{0}^{ϵr} tk(x, t)dt
+ β_{3} ∫_{0}^{2r} tk(t)dt

for some positive constants α_{1}(m), α_{2}(m, ϵ) and β_{i}(m), 1 ≤ i ≤ 3, where r = r(x). Here as before, k(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} f and k(t) = k(o, t), where o ∈ M is a fixed point. Moreover, the gradient of u satisfies the estimates in Theorem 1.3.

Proof. Let us first consider the case that M is nonparabolic and its Green’s function satisfies (1.1). By (5.2), since f is the trace of ρ we have

∫_{0}^{∞} k(t)dt < ∞.

By Theorem 1.1 we can find a solution u of \frac{1}{2}Δu = f. Moreover, u satisfies the estimates in Theorems 1.1 and 1.3. We claim that u satisfies (5.1). By (5.3), we can find R_{j} → ∞ such that

lim_{j→∞} \frac{1}{V_{o}(R_{j})} ∫_{B_{o}(R_{j})} ||ρ||^{2} = 0.

It is known that ||\sqrt{−1}∂∂u − ρ||^{2} is subharmonic, see [20, p.187] for example. For any x ∈ M, if j is large enough so that R_{j} ≥ 8r(x), then by the mean value inequality for subharmonic function in [13, Theorem 2.1], using Theorem 1.3(iii) we have

||\sqrt{−1}∂∂u − ρ||^{2}(x) ≤ \frac{C_{1}}{V_{x}(\frac{R_{j}}{8} )} ∫_{B_{x}( \frac{R_{j}}{8} )} ||\sqrt{−1}∂∂u − ρ||^{2}

≤ \frac{C_{2}}{V_{o}(\frac{R_{j}}{4} )} ∫_{B_{o}( \frac{R_{j}}{4} )} (|∇^{2}u|^{2} + ||ρ||^{2})

≤ C_{3} [R^{−2} (∫_{R_{j}}^{∞} k(t)dt)^{2} + \frac{1}{V_{o}(R_{j})} ∫_{B_{o}(R_{j})} ||ρ||^{2}]

→ 0 as j → ∞, where C_{1} − C_{3} are constants independent of j. Hence \sqrt{−1}∂∂u ≡ ρ and the proof is completed in this case.

In general, let M̃ = M ×ℂ^{2}. Then M̃ is nonparabolic and its Green’s function satisfies (1.1). We may consider ρ as a closed (1, 1) form on M̃. Moreover, the trace of ρ is still f which is independent of the variable in ℂ^{2}. It is easy to see that ρ still satisfies (5.2) and (5.3) by Lemma 1.1. Hence we can find ũ such that \sqrt{−1}∂∂ũ = ρ. It is easy to see that for any fixed x_{0} ∈ M, ũ(x_{0}, ·) is pluriharmonic on ℂ^{2}. Moreover, since ũ satisfies the estimates in the theorem, we have

lim sup_{y∈ℂ^{2},y→∞} \frac{ũ(x_{0}, y)}{|y|} = 0.

Hence ũ(x_{0}, ·) is constant on ℂ^{2} by Harnack inequality, and ũ(x, y) = u(x) which satisfies (5.1) on M. Moreover, u satisfies the estimates in the theorem. The estimates of the gradient of u follows from the construction and Theorem 1.3. q.e.d.

Remark 5.1. (i) If ρ satisfies (5.2), and if

lim inf_{r→∞} r^{−1} sup_{∂B_{o}(r)} ||ρ|| < ∞,

then (5.3) will also be satisfied. In particular, ||ρ|| may be unbounded. (ii) By Remark 1.1, it is easy to see that the theorem is still true without the assumptions that ρ is real and f is nonnegative. What we need is (5.2) and (5.3).

In the following we give some applications of the theorem.

(I) Steinness of Kähler manifolds.

Theorem 5.2. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature. M is Stein if one of the following is true:

(i) There exists a closed real (1, 1) form ρ which is positive everywhere such that ||ρ||(x) ≤ Cr^{−2}(x) and k(t) ≤ Ct^{−2} for some constant C for all x and t. Here f is the trace of ρ and k(t) is as in Theorem 5.1.

(ii) M has nonnegative sectional curvature and there exists a real closed (1, 1) form ρ which is positive everywhere and satisfies (5.2) and (5.3).

Proof. (i) By Theorem 5.1, we can solve the Poincaré-Lelong equation \sqrt{−1}∂∂u = ρ. Since ρ is positive everywhere, u is a strictly plurisubharmonic function. Since u satisfies the estimate in the theorem, by Corollary 1.1 and Corollary 3.1, we see that u is an exhaustion function. Hence M is Stein.

To prove (ii), by the assumption, we can obtain a strictly plurisubharmonic function as before. Since the sectional curvature is nonnegative, one can apply the method in [8] to show that the manifold is Stein.

q.e.d.

Corollary 5.1. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature. Suppose that M has positive Ricci curvature. Then M is Stein if M satisfies one of the following:

(i) The Ricci form ρ satisfies ||ρ||(x) ≤ Cr^{−2}(x) and 1/V_{o}(t) ∫_{B_{o}(t)} ||ρ|| ≤ Ct^{−2} for some constant C for all x ∈ M and t > 0.

(ii) M has nonnegative sectional curvature and ρ satisfies (5.2) and (5.3).

Corollary 5.1(i) was basically proved in [19] (see also [20]) under the assumption is that ||ρ||(x) ≤ Cr^{−2}(x) and M has maximal volume growth, which will imply that 1/V_{o}(t) ∫_{B_{o}(t)} ||ρ|| ≤ Ct^{−2} provided m ≥ 2. Corollary 5.1(i) seems to be more general at first sight, but we will see later that if m ≥ 2, the assumptions in Corollary 5.1(i) will imply that M has maximal volume growth. Also, it was proved in [8] that if M has nonnegative sectional curvature and has positive holomorphic bisectional curvature, then M is Stein. In (ii) of the corollary, we still have the same assumption on sectional curvature, but we replace the assumption on the positivity of holomorphic bisectional curvature by the assumption that the Ricci curvature is positive and whose norm decays faster than linearly in a certain sense. We would like to mention that in [4], it is proved that if M has nonnegative holomorphic bisectional curvature, has maximal volume growth such that its scalar curvature R satisfies R(x) ≤ Cr^{−1−ϵ}(x) for some constants C and ϵ > 0 for all x, then M is Stein.

(II) Plurisubharmonic functions revisited

Proposition 5.1. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature and positive Ricci curvature with Ricci form ρ satisfies (5.2) and (5.3). Let u be a plurisubharmonic function satisfying (4.1) such that f(x) ≤ Cr^{−2}(x). Suppose one of (a), (b) or (c) in Theorem 4.1 is true, then u must be constant.

Proof. By Theorem 5.1, (5.1) has a solution with ρ being the Ricci form of M. Since ρ > 0 everywhere, M supports a strictly plurisubharmonic function. The result follows from Theorem 4.1. q.e.d.

Proposition 5.2. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature and let L be a holomorphic line bundle over M with Hermitian metric h with nonnegative curvature. Let Ω be the trace of the curvature of L with respect to h. Suppose

(i) M supports a strictly plurisubharmonic function;

(ii)

lim sup_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} Ω = 0;

and

(iii) Ω(x) ≤ Cr^{−2}(x).

Then L is flat.

Proof. By Theorem 5.1, we can find a solution u of (5.1) satisfying the estimate in the theorem with f = Ω and ρ be the curvature form of L. Since ρ^{m} = 0 by Theorem 3.1, u must be constant by Theorem 4.1. Hence ρ ≡ 0. q.e.d.

(III) Volume and curvature estimates

Lemma 5.1. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature. Let ρ be a real closed (1, 1) form with trace f. Suppose ρ ≥ 0 and ρ > 0 at some point o and satisfies (5.2) and (5.3). Then for any α > 2 and p ≥ 1, there exists constants C_{1} > 0, C_{2} > 0 independent of R and C^{∗} independent of R and α such that

\frac{C_{1}R}{V_{o}(R)} ≤ [C^{∗} ∫_{αR}^{∞} k(t)dt + C_{2}R (\frac{1}{V_{o}(R)} ∫_{B_{o}(R)} f^{p})^{\frac{1}{p}} ]

· [\frac{1}{V_{o}(R)} ∫_{B_{o}(2R)\B_{o}( \frac{R}{2} )} f^{q(m−1)}]^{\frac{1}{q}}

where q = p/(p − 1) if p > 1, and if p = 1, then the last integral is interpreted as sup_{B_{o}(2R)\B_{o}( \frac{R}{2} )} f^{m−1}.

Proof. By Theorem 5.1 we can solve the Poincaré-Lelong equation \sqrt{−1}∂∂u = ρ. Let us first assume that p > 1. Since ρ ≥ 0 and is strictly positive at o, there is a constant C_{1} > 0 such that for all R ≥ 1

C_{1} ≤ ∫_{B_{o}(R)} ρ^{m}
= ∫_{B_{o}(R)} (\sqrt{−1}∂∂u ∧ ρ^{m−1})
= ∫_{∂B_{o}(R)} \sqrt{−1}∂u ∧ ρ^{m−1}

≤ (∫_{∂B_{o}(R)} |∇u|^{p})^{\frac{1}{p}} (∫_{∂B_{o}(R)} f^{q(m−1)})^{\frac{1}{q}}

for some constant C_{4} independent of R and q = p/(p − 1). For R ≥ 2, integrating from \frac{1}{2}R to R, there is a constant C_{5} > 0 independent of R and α such that

(5.4) \frac{C_{1}}{2} R ≤ (∫_{B_{o}(R)\B_{o}( \frac{R}{2} )} |∇u|^{p})^{\frac{1}{p}} (∫_{B_{o}(R)\B_{o}( \frac{R}{2} )} f^{q(m−1)})^{\frac{1}{q}} .

By Theorem 5.1, the gradient of u satisfies

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)} |∇u|^{p} ≤ C^{∗} (∫_{αR}^{∞} k(t)dt)^{p} + \frac{C_{2}R^{p}}{V_{o}(R)} ∫_{B_{o}(αR)} f^{p}

where α > 2 is a constant, C^{∗} is a constant independent of R and α and C_{2} is a constant independent of R. Combine this with (5.4), the theorem is true if p > 1. The case that p = 1 is similar. q.e.d.

Theorem 5.3. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature. Let ρ be a closed real (1, 1) form with trace f. Suppose ρ ≥ 0 and ρ > 0 at some point o and suppose that

(5.5)  (\frac{1}{V_{o}(r)} ∫_{B_{o}(r)} f^{p})^{\frac{1}{p}} ≤ Cr^{−1−ϵ}

and

(5.6)  (\frac{1}{V_{o}(r)} ∫_{B_{o}(r)\B_{o}( \frac{r}{2} )} f^{q(m−1)})^{\frac{1}{q(m−1)}} ≤ Cr^{−1−ϵ}

for some 1 ≤ p ≤ m and for some constant C for all r, where 0 < ϵ ≤ 1, q = p/(p − 1). If p = 1, (5.6) means that

sup_{B_{o}(r)\B_{o}( \frac{r}{2} )} f ≤ Cr^{−1−ϵ}.

Then

V_{o}(R) ≥ CR^{m(1+ϵ)}

for some constant C > 0 for all R ≥ 2. If, in addition,

V_{o}(R) ≤ C′R^{m(1+ϵ)}

for some constant C′ for all R, then

(\frac{1}{V_{o}(r)} ∫_{B_{o}(r)} f^{p})^{\frac{1}{p}} ≥ C′′r^{−1−ϵ}

for some constant C′′ > 0 for all r large enough. In particular, if ϵ = 1, then M has maximal volume growth and

(\frac{1}{V_{o}(r)} ∫_{B_{o}(r)} f^{p})^{\frac{1}{p}} ≥ C′′r^{−2}.

Proof. Let us consider the case that p = 1. Then ρ satisfies the conditions in Lemma 5.1. Hence we have for R ≥ 2,

\frac{C_{1}R}{V_{o}(R)} ≤ (\frac{C^{∗}}{αR} + \frac{C_{2}R}{V_{o}(R)} ∫_{B_{o}(R)} f) · R^{−(m−1)(1+ϵ)}
≤ C_{3}R^{−m(1+ϵ)+1}

where C_{1} and C^{∗} are positive constants independent of α and R, and C_{2} and C_{3} are constants independent of R. From this it is easy to see that

V_{o}(R) ≥ CR^{m(1+ϵ)}

for some constant C > 0 for all R ≥ 2. If in addition,

V_{o}(R) ≤ C′R^{m(1+ϵ)}

for some constant C′ for all R. Then for R large enough, we have

\frac{C_{4}}{R^{ϵ}} ≤ \frac{C^{∗}}{αR} + \frac{C_{2}R}{V_{o}(R)} ∫_{B_{o}(R)} f

where C_{4} > 0 is a constant independent of R and α. If we take α large enough, we can conclude that

\frac{1}{V_{o}(R)} ∫_{B_{o}(R)} f ≥ C_{5}R^{−1−ϵ}

for some constant C_{5} > 0 independent of R. Suppose p > 1. By (5.5) we have

\frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ|| ≤ Cr^{−1−ϵ}.

Let q^{∗} = q(m − 1). Since p ≤ m, we have q∗ ≥ m ≥ 2. Let k ≥ 1 be an integer. By (5.6), for ℓ ≤ k, we have

∫_{B_{o}(2^{ℓ})\B_{o}(2^{ℓ−1})} ||ρ||^{q^{∗}} ≤ C2^{−ℓq^{∗}(1+ϵ)} × V_{o}(2^{ℓ})

where C is a constant independent of ℓ. Hence for any k_{0} ≤ k, we have

\frac{1}{V_{o}(2^{k})} ∫_{B_{o}(2^{k})\B_{o}(1)} ||ρ||^{q^{∗}} ≤ C \frac{1}{V_{o}(2^{k})} ∑_{ℓ=1}^{k_{0}} 2^{−ℓq^{∗}(1+ϵ)} × V_{o}(2^{ℓ})

+ ∑_{ℓ=k_{0}+1}^{∞} 2^{−ℓq^{∗}(1+ϵ)}.

Fix k_{0}, let k → ∞ and then let k_{0} → ∞, we have

lim_{r→∞} \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ||^{q^{∗}} = 0.

Since q^{∗} ≥ 2, we have

lim_{r→∞} \frac{1}{V_{o}(r)} ∫_{B_{o}(r)} ||ρ||^{2} = 0.

Hence the conditions of Theorem 5.1 are satisfied and we can proceed as in the case that p = 1 and complete the proof of the theorem. q.e.d.

In particular, we have:

Corollary 5.2. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature. Suppose the Ricci curvature is positive at some point and scalar curvature R satisfies R(x) ≤ Cr^{−2}(x) and 1/V_{o}(t) ∫_{B_{o}(t)} R ≤ Ct^{−2}. Then M has maximal volume growth, and

lim inf_{r→∞} \frac{r^{2}}{V_{o}(r)} ∫_{B_{o}(r)} R > 0.

Proof. Set ϵ = 1 in Theorem 5.3, the result follows easily by noting that V_{o}(r) ≤ Cr^{2m}, by the volume comparison. q.e.d.

This result says that under the assumptions of the corollary, even though the scalar curvature decays, but it actually cannot decay too fast. One should compare this with Corollary 3.1.

Remark 5.2. If we assume that ρ has rank ℓ ≥ 1 at o rather than ρ is positive at o and if we assume that (5.5) and (5.6) are true with ϵ = 1 and with m replaced by ℓ, then we can modify the proof of Lemma 5.1 and conclude that V_{o}(r) ≥ Cr^{2ℓ} for some positive constant C for all r. In particular, if M is not Ricci flat, then V (r) ≥ Cr^{2} for some constant C > 0 for all r ≥ 1.

(IV) Positive (1,1) forms satisfying a pinching condition

Theorem 5.4. Let M^{m} be a complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature, with m ≥ 2. Let ρ ≥ 0 be a closed real (1, 1) form on M such that inf_{1≤j≤m} λ_{j}(x) ≥ ϵf(x) for some positive constant ϵ, for all x, where λ_{j} are the eigenvalues of ρ and f is the trace of ρ. Define k(x, t) = 1/V_{x}(t) ∫_{B_{x}(t)} f and k(t) = k(o, t) as before. Then ρ ≡ 0 if one of the following is satisfied:

(i) k(t) ≤ Ct^{−2} and f(x) ≤ Cr^{−2}(x) for some constant C for all t > 0 and x.

(ii) V_{o}(r) ≤ Cr^{m} for some constant C for all r, ρ satisfies (5.2) and (5.3) and there exists a constant C such that ∫_{0}^{∞} k(x, t)dt ≤ C for all x.

Proof. If (i) is true and if f > 0 at some point, then M has maximal volume growth by Theorem 5.3. Combining this with the assumptions in (i) and the fact that m ≥ 2, it is not hard to prove that k(x, t) ≤ Ct^{−2} for some constant C for all t and x. By Theorem 5.1, (5.1) has a solution u. Moreover, by the gradient estimate in Corollary 1.2, there exists a constant C_{1} such that

(5.7)  |∇u(x)| ≤ C_{1}r^{−1}(x).

As in the proof of Theorem 5.3, using the pinching condition that λ_{j} ≥ ϵf, we have

(5.8)  ∫_{B_{o}(r)} f^{m} ≤ C_{2} ∫_{B_{o}(r)} ρ^{m}
= C_{2} ∫_{B_{o}(r)} (\sqrt{−1}∂∂u ∧ ρ^{m−1})
= C_{2} ∫_{∂B_{o}(r)} \sqrt{−1}∂u ∧ ρ^{m−1}

≤ C_{3} (∫_{∂B_{o}(r)} |∇u|^{m})^{\frac{1}{m}} (∫_{∂B_{o}(r)} f^{m})^{\frac{m−1}{m}}

≤ C_{4} (r ∫_{∂B_{o}(r)} f^{m})^{\frac{m−1}{m}}

for some constants C_{2} − −C_{4} independent of r, where we have used (5.7) and the fact that A_{o}(r) ≤ Cr^{2m−1} for some constant depending only on m. Since f > 0 at some point, there exists r_{0} > 0 such that F (r) = ∫_{B_{o}(r)} f^{m} > 0 for all r ≥ r_{0}. By (5.8), we have

\frac{F′}{F^{\frac{m}{m−1}}} ≥ C_{5}r^{−1}

for some positive constant C_{5} for all r ≥ r_{0}. Integrating from r_{0} to r with r > r_{0}, we have

F^{− \frac{1}{m−1}} (r_{0}) − F^{− \frac{1}{m−1}} (r) ≥ \frac{C_{5}}{m − 1} log \frac{r}{r_{0}} .

Let r → ∞, we have a contradiction. Hence f ≡ 0 and ρ ≡ 0.

Under the assumptions of (ii), we conclude that |∇u(x)| ≤ C_{6} for some constant C_{6} for all x by Corollary 1.2. As in (5.8), using the same notations as before, we have

F (r) ≤ C_{7}Ao^{\frac{1}{m}} (r) (F′(r))^{\frac{m−1}{m}}
≤ C_{8} (rF′(r))^{\frac{m−1}{m}}

for some constants C_{7}, C_{8} independent of r, where we have used the assumption that V_{o}(r) ≤ Cr^{m} and the fact that rA_{o}(r) ≤ 2mV_{o}(r). We can proceed as before to show that f and hence ρ must be identically zero. q.e.d.

Remark 5.3. If we let g(r) = ∫_{∂B_{o}(r)} |∇u|^{m}, then it is easy to see that f ≡ 0 provided that

∫_{r_{0}}^{∞} g^{− \frac{1}{m−1}} (t)dt = +∞.

Hence the conditions of Theorem 5.4 may be relaxed a little bit further.

In [24], Shi and Yau proved the following: Suppose M^{m} is a complete noncompact Kähler manifold with m ≥ 3 and with bounded nonnegative holomorphic bisectional curvature and suppose that R_{ααββ} ≥ ϵR for some positive constant ϵ, where R_{ααββ} is the holomorphic bisectional

curvature and R is the scalar curvature. Then 1/V_{x}(t) ∫_{B_{x}(t)} R ≤ Ct^{−2} for some constant for all x. Using Theorem 5.1, we have:

Corollary 5.3. Let M^{m} a complete noncompact Kähler manifold with complex dimension m ≥ 3. Suppose that R_{ααββ} ≥ ϵR for some positive constant ϵ and suppose that the scalar curvature satisfies (i) R(x) ≤ Cr^{−2}(x) for all x or (ii) V_{o}(r) ≤ Cr^{m} for some constant C for all r. Then M is flat.

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