ERRATUM: CONFORMALLY FLAT 3-MANIFOLDS

WITH CONSTANT SCALAR CURVATURE

Qing-Ming Cheng*, Susumu Ishikawa* & Katsuhiro Shiohama*

Abstract. In this note, we corrected the coefficients of formula (2.12) in Proposition 1 and the third formula in Proposition 2 in our paper [1]. We should remark that our theorems in [1] are not harmed.

In this note, we must mention that the coefficients of formula (2.12) in Proposition 1 and the third formula in Proposition 2 in our paper [1] should be corrected as the following:

(2.12)  \frac{1}{2}Δ ∑_{i,j,k} R_{ij,k}^{2} = ∑_{i,j,k,l} R_{ij,kl}^{2} + \frac{3(n + 2)}{n − 2} ∑_{i,j,k} λ_{i}R_{ij,k}^{2}

− \frac{(n + 4)r}{(n − 1)(n − 2)} ∑_{i,j,k} R_{ij,k}^{2}.

(2.21)  ∑_{i,j,k,l} R_{ij,kl}^{2} = −24 ∑_{i,j,k} μ_{i}R_{ij,k}^{2} − \frac{3}{4}r ∑_{i,j,k} R_{ij,k}^{2} − \frac{9}{4}B(B − \frac{1}{6}r^{2}).

We should remark that their proofs do not need to be corrected except the statements of results of formulas.

For the proofs of Theorem 1 and Theorem 2 in the section 3 of our paper [1], except the coefficient of formula (3.7), we do not need to correct any thing. Hence, the assertions of Theorem 1 and Theorem 2 in [1] are true. Thus, Main Theorem 1 in [1] is also true. Although the proof of Theorem 3 in [1] needs to be corrected as the following (in fact, we use the same idea as before), we know that the assertion is true. Hence, the assertion in Main Theorem 2 in [1] holds.

Typeset by AMS-TEX

Key words and phrases. space form, conformally flat manifold, constant scalar curvature and Ricci curvature tensor.

* Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Curvature, Japan.

2000 Mathematics Subject Classification 53C20

Proof of Theorem 3. For the proof of Theorem 3 in [1], by using the corrected formula (2.21), the formula (3.39) becomes

(3.39)  x = −\frac{1}{6}(B + r \frac{B_{3}^{∗}}{B} )±\frac{1}{6}\sqrt{r^{2}(\frac{B_{3}^{∗}}{B} )^{2}+\frac{1}{2}rB_{3}^{∗} + 10r^{2}B − \frac{121}{2}B^{2}}.

Since formula (3.42) and Lemma 3 in [1] hold, we obtain

0 ≥ (−\frac{3}{B} x(μ_{l}^{∗}^{2} − \frac{2}{3}B) + 3μ_{l}^{∗}^{2} + \frac{1}{2}rμ_{l}^{∗} − B)(μ_{l}^{∗}^{2} − \frac{1}{6}B), (3.42)

and

(3.43)  x = −\frac{1}{6}(B + r \frac{B_{3}^{∗}}{B} )−\frac{1}{6}\sqrt{r^{2}(\frac{B_{3}^{∗}}{B} )^{2}+\frac{1}{2}rB_{3}^{∗} + 10r^{2}B − \frac{121}{2}B^{2}}.

Next we do necessary correction of Lemma 4 as follows:

Lemma 4. If B_{3}^{∗} ≠ B^{3/2}/\sqrt{6}, then

B < \frac{32}{121} \frac{r^{2}}{3} .

Proof. By taking l = 2 in (3.42) and using the inequality μ_{2}^{∗}^{2} − B/6 < 0 in (3.41), we have

(3.44)  0 ≤ −\frac{3}{B} x(μ_{2}^{∗}^{2} − \frac{2}{3}B)+3μ_{2}^{∗}^{2} + \frac{1}{2}rμ_{2}^{∗} − B.

First of all, we assert B < 4r^{2}/27. In fact, since B and r are constant by the assumption and B < r^{2}/6 as noted in Lemma 2, there exists a constant a > 0 such that (1 + a)B < r^{2}/6. If a ≥ 1/8, then B < 4r^{2}/27. In this case our assertion holds. If a < 1/8, then we have

(3.45)  x = −\frac{1}{6}(B + r \frac{B_{3}^{∗}}{B} )−\frac{1}{6}\sqrt{((1 − 6a)B + r \frac{B_{3}^{∗}}{B} )^{2}+G},

where G is defined by

G = −(1 − 6a)^{2}B^{2} − 2(1 − 6a)rB_{3}^{∗} + \frac{1}{2}rB_{3}^{∗} + 10Br^{2} − \frac{121}{2}B^{2}.

According to rB_{3}^{∗} ≤ −B^{2} in Lemma 2 and (1 + a)B < r^{2}/6 we have G > 0 Therefore, we obtain x ≤ −aB. By using this inequality, μ_{2}^{∗}^{2} − B/6 < 0 in (3.41), μ_{2}^{∗} < 0 and (3.44), we can prove that it is impossible. Hence, we have

(1 + 6a)B < \frac{r^{2}}{6} .

If 6a ≥ 1/8, then our assertion is true. If 6a < 1/8, then by applying the above procedure k times as in [1] so that 6^{k}a ≥ 1/8, then we have

(1 + 6^{k}a)B < \frac{r^{2}}{6} .

Thus we obtain B < 4r^{2}/27, that is, our assertion holds.

Next, by taking account of B < 4r^{2}/27, we will show B < \frac{32}{121} \frac{r^{2}}{3} . In fact, from (3.43), rB_{3}^{∗} ≤ −B^{2} of (4) in Lemma 2 and B < 4r^{2}/27, we have x < −\frac{B}{8} . Hence, by substituting the above inequality into (3.44), from 0 < μ_{2}^{∗}^{2} < B/6 and μ_{2}^{∗} < 0, we infer

B < \frac{32}{121} \frac{r^{2}}{3} .

Thus the proof of Lemma 4 is completed.

In view of the inequality rB_{3}^{∗} ≤ −B^{2} and r < 0, we have B_{3}^{∗} > 0. Thus, from Lemma 1 in [1], we have 0 < B_{3}^{∗} = sup B_{3} < B^{3/2}/\sqrt{6}. From Lemma 4, we have r^{2} > 11B. Hence, we obtain, from B_{3}^{∗} < B^{3/2}/\sqrt{6} and r^{2} > 11B,

10r^{2}B − \frac{121}{2}B^{2} − r^{2}B − \frac{2r^{2}B_{3}^{∗}}{\sqrt{B}} + \frac{rB_{3}^{∗}}{2}

≥ 9r^{2}B − \frac{121}{2}B^{2} − \frac{2r^{2}B}{\sqrt{6}} − \frac{|r|B^{3/2}}{2\sqrt{6}}

= (9 − \frac{2}{\sqrt{6}} − \frac{1}{4})r^{2}B + (\frac{|r|\sqrt{B}}{2} − \frac{B}{2\sqrt{6}})^{2} − \frac{121}{2}B^{2} − \frac{B^{2}}{24}

> 7 × 11B^{2} − 61B^{2} > 0.

Therefore, we have, from (3.43) and the above inequality,

x < −\frac{1}{6}(B + r \frac{B_{3}^{∗}}{B} )−\frac{1}{6}\sqrt{(r\sqrt{B} + r \frac{B_{3}^{∗}}{B} )^{2}}

= −\frac{1}{6}B + \frac{1}{6}r\sqrt{B}.

Hence, by substituting the above inequality into (3.44), we infer, from B > 0, r < 0 and 0 < μ_{2}^{∗}^{2} < \frac{B}{6} ,

0 ≤ −\frac{3}{B} (μ_{2}^{∗}^{2} − \frac{2}{3}B)(−\frac{1}{6}B + \frac{1}{6}r\sqrt{B}) + 3μ_{2}^{∗}^{2} + \frac{1}{2}rμ_{2}^{∗} − B

= (\frac{7}{2} − \frac{r}{2\sqrt{B}})μ_{2}^{∗}^{2} + \frac{1}{2}rμ_{2}^{∗} − \frac{4}{3}B + \frac{1}{3}r\sqrt{B}

< (\frac{7}{2} − \frac{r}{2\sqrt{B}})\frac{B}{6} − \frac{1}{2}r \begin{matrix}√ & \overline{B} \\ √ & \overline{6} \end{matrix} − \frac{4}{3}B + \frac{1}{3}r\sqrt{B}

= −\frac{9}{12}B + (\frac{1}{3} − \frac{1}{12} − \frac{1}{2\sqrt{6}})r\sqrt{B} < 0.

But this is impossible. Thus the proof of Theorem 3 is completed. □

References

1. Qing-Ming Cheng, Susumu Ishikawa and Katsuhiro Shiohama, Conformally flat 3-manifolds with constant scalar curvature, J. Math. Soc. Japan 51 (1999), 209-226.

Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan