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A note on Einstein four-manifolds with positive curvature
Accepted Manuscript A note on Einstein four-manifolds with positive curvature Peng Wu
PII: DOI: Reference:
S0393-0440(16)30297-2 http://dx.doi.org/10.1016/j.geomphys.2016.11.017 GEOPHY 2881
To appear in:
Journal of Geometry and Physics
Received date : 23 February 2015 Accepted date : 19 November 2016 Please cite this article as: P. Wu, A note on Einstein four-manifolds with positive curvature, Journal of Geometry and Physics (2016), http://dx.doi.org/10.1016/j.geomphys.2016.11.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A NOTE ON EINSTEIN FOUR-MANIFOLDS WITH POSITIVE CURVATURE PENG WU Abstract. In this short note we prove that an oriented Einstein four-manifold with √ 1 Ric = g and sectional curvature K ≥ 30 (19 − 271) is isometric to (S 4 , g0 ) or (CP 2 , gF S ) up to rescaling.
1. Introduction A Riemannian manifold (M n , g) is Einstein if its Ricci curvature is a multiple of the metric, i.e., Ric = λg for some constant λ. The classification of Einstein four-manifolds with positive sectional curvature is one of the basic problems in differential geometry. The first result goes back to Berger more than a half century ago, Berger [1] proved that Einstein four-manifolds with 14 -pinched sectional curvature are isometric to (S 4 , g0 ). Hitchin (Theorem 13.30 in [2]) proved that half conformally-flat Einstein four-manifolds with positive scalar curvature are isometric to either (S 4 , g0 ) or (CP 2 , gF S ) up to rescaling. Gursky and LeBrun [8] proved an interesting gap theorem for the L2 -norm of (anti-)self-dual Weyl curvature, using which they proved that Einstein four-manifolds with nonnegative sectional curvature and positive intersection form are isometric to (CP 2 , gF S ) up to rescaling. Yang [13] proved that sectional curvature √ oriented Einstein four-manifolds with4 Ric = g and 1 2 K ≥ 120 ( 1249 − 23) ≈ 0.102 are isometric to (S , g0 ) or (CP , gF S ) up to rescaling. √ Costa [5] later relaxed Yang’s condition to K ≥ 61 (2 − 2) ≈ 0.097. In this short note, we further relax the condition on the sectional curvature, precisely we prove, Theorem 1.1. An √ oriented Einstein four-manifold4 with Ric = g2 and sectional curva1 ture K ≥ 30 (19 − 271) ≈ 0.0846 is isometric to (S , g0 ) or (CP , gF S) up to rescaling.
Remark 1.1. The curvature in Theorem 1.1 can be further relaxed to K ≥ 0.08433, see Remark 3.1. The rest of the paper is organized as follows. In Section 2, we recall the Weitzenb¨ock formula and estimates for the (anti-)self-dual Weyl curvature. In Section 3, we prove Theorem 1 by combining the gap theorem of Gursky and LeBrun [8] and the argument of Yang [13]. Acknowledgement. The author thanks his advisors Professors Xianzhe Dai and Guofang Wei for their guidance, encouragement, and constant support. The author was partially supported by an AMS-Simons postdoctoral travel grant. The result in this Date: December 2, 2016. 2010 Mathematics Subject Classification. Primary 53C24, 53C25. Key words and phrases. Einstein four-manifolds, positive sectional curvature, Weitzenb¨ ock formula. 1
paper was proved in September 2012, when the author was preparing talks on his Ph.D. thesis [10] and related paper of Gursky and LeBrun [8], and Yang [13]. The paper was written in Winter 2014. ¨ ck formula and estimates for half Weyl curvature 2. The Weitzenbo The Hodge star operator ? : ∧∗ T M → ∧∗ T M induces a natural decomposition of the vector bundle of 2-forms ∧2 T M on an oriented four-manifold (M, g), ∧2 T M = ∧+ M ⊕∧− M , where ∧± M are eigenspaces of ±1 respectively, sections of which are called self-dual and anti-self-dual 2-forms respectively. It further induces a decomposition for the curvature operator R : ∧2 T M → ∧2 T M , ! ◦ R + g + W Ric 12 R= , ◦ R − g + W Ric 12 ◦
where Ric is the traceless Ricci curvature, and R is the scalar curvature. If (M 4 , g) is Einstein, then we get R g + W+ 0 12 . R= R g + W− 0 12 Derdzi´ nski [6] proved the following Weitzenb¨ock formula by using the harmonicity of ± W (see also Wu [11, 12] for an alternative proof), Theorem 2.1. Let (M, g) be an Einstein four-manifold, then ∆|W ± |2 = 2|∇W ± |2 + R|W ± |2 − 36 det W ± .
Gursky and LeBrun [8]proved a refined Kato inequality (see also Yang [13], Wu [11] for alternative proofs), which was proven to be optimal by Branson [3] and Calderbank, Gauduchon, and Herzlich [4], Theorem 2.2. Let (M, g) be an Einstein four-manifold, then 5 |∇W ± |2 ≥ |∇|W ± ||2 . 3 By applying a conformal transformation argument, Gursky and LeBrun [8] proved an optimal gap theorem for the L2 -norm of (anti-)self-dual Weyl curvature, Theorem 2.3. Let (M, g) be an Einstein four-manifold with positive scalar curvature. If g is not half conformally flat, then Z Z R2 ± 2 |W | dvg ≥ dvg , M M 24 with equality if and only if ∇W ± ≡ 0.
Finally we recall that, Gursky and LeBrun [8] and Yang [13] derived an upper bound for |W + | + |W − | when the sectional curvature is bounded below by a nonnegative number, Theorem 2.4. Let (M, g) be an Einstein four-manifold with Ric = g. Suppose that K ≥ δ ≥ 0, then r 8 (1 − 3δ). |W + | + |W − | ≤ 3 2
3. Proof of Theorem 1.1 We follow the argument of Yang [13] and prove by contradiction, if (M 4 , g) is not half conformally flat, then there exists t > 0 such that Z (|W + | − t|W − |)dvg = 0. M
Apply the Weitzenbock formula to |W + |2 +t2 |W − |2 and take the integration, we have
(1)
0=
Z
ZM
∆(|W + |2 + t2 |W − |2 )dvg
2|∇W + |2 + 2t2 |∇W − |2 + (4|W + |2 − 36 det W + ) + t2 (4|W − |2 − 36 det W − ). M R R Denote V = Vol(M, g), a = V −1 M |W + |, b = V −1 M |W − |, so we have a = tb. Apply the refined Kato inequality and the Poincar´e inequality, we have Z 2|∇W + |2 + 2t2 |∇W − |2 ZM 10 ≥ (|∇|W + ||2 + t2 |∇|W − ||2 ) 3 M Z 10 ≥ λ1 (|W + | − a)2 + t2 (|W − | − b)2 (2) 3 M Z 2 Z 10 20 + + 2 2 − 2 −1 |W | = λ1 (|W | + t |W | ) − λ1 V 3 3 M M Z 160 t2 10 (|W + |2 + t2 |W − |2 ) − λ (1 − 3δ)2 V. ≥ λ1 2 1 3 9 (1 + t) M =
where λ1 = 43 is a lower bound for the first positive eigenvalue of the Laplace operator [9], and in the last upper bound estimate in Theorem 2.4. √ step we± used the ± 3 Recall that 3 6| det W | ≤ |W | . Plugging (2) to (1), we obtain Z 10 0≥ λ1 (|W + |2 + t2 |W − |2 ) + (4|W + |2 − 36 det W + ) + t2 (4|W − |2 − 36 det W − ) M 3 160 t2 − λ1 (1 − 3δ)2 V 9 (1 + t)2 Z √ √ 40 40 ≥ ( + 4 − 2 6|W + |)|W + |2 + t2 ( + 4 − 2 6|W − |)|W − |2 9 M 9 2 640 t (1 − 3δ)2 V − 27 (1 + t)2 Z 4 4 640 t2 ≥ ( + 24δ)|W + |2 + t2 ( + 24δ)|W − |2 − (1 − 3δ)2 V 2 9 9 27 (1 + t) M By the gap theorem of Gursky and LeBrun in Theorem 2.3, we get 2 4 640 t2 0 ≥ (1 + t2 )( + 24δ) − (1 − 3δ)2 . 2 3 9 27 (1 + t) 3
Notice that (1 + t2 )(1 + t)2 t−2 ≥ 8. Therefore we will get a contradiction if 4 40 ( + 24δ) − (1 − 3δ)2 ≥ 0, 9 9 √ 1 which is equivalent to δ ≥ 30 (19 − 271) ≈ 0.0846.
Remark 3.1. On the other hand, Gallot [7] proved that for an Einstein four-manifold with Ric = g and K ≥ δ, we have λ1 ≥ 8δ + 32 . Using this estimate we will get a contradiction if (1 + t2 )(1 + t)2 (57δ − 2) − 20t2 (12δ + 1)(1 − 3δ)2 ≥ 0, which holds if δ ≥ 0.08433. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Berger, M., Sur quelques vari´et´es d’Einstein compactes, Ann. Mat. Pura Appl. 53 (1961), 89–95. Besse, A., Einstein manifolds, Berlin-Heidelberg, Springer-Verlag, 1987. Branson, T., Kato constants in Riemannian geometry, Math. Res. Let. 7 (2000), 245-261. Calderbank, D., Gauduchon, P., and Herzlich, M., Refined Kato inequalities and conformal weights in Riemannian geometry, J. of Functional Analysis 173 (2000), 214–255. ´ On Einstein four-Manifolds, J. of Geometry and Physics 51 (2004), 244–255. Costa, E, Derdzi´ nski, A., Self-dual K¨ ahler manifolds and Einstein manifolds of dimension four, Com. Math. 49 (1983), 405–433. Gallot, S., Vari´et´es dont le spectre ressemble ` a celui d’une sph`ere, Ast´erisque 80 (1980), 33–52. Gursky, M., LeBrun, C., On Einstein manifolds of positive sectional curvature, Ann. Glob. An. Geom. 17 (1999), 315–328. Lichnerowicz, A., G´eometrie des groupes de transformations, Dunod, Paris, 1958. Wu, P., Studies on Einstein manifolds and gradient Ricci solitons, Ph. D. dissertation, University of California, Santa Barbara, 2012. Wu, P., Einstein four-manifolds with three-positive curvature operator are half conformally flat, preprint, 2013. Wu, P., A Weitzenb¨ ock formula for canonical metrics on four-manifolds, Trans. Amer. Math. Soc. 369 (2017), 1079–1096. Yang, D., Rigidity of Einstein 4-manifolds with positive curvature, Invent. Math. 142 (2000), 435–450.
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China E-mail address: [email protected]
4
PII: DOI: Reference:
S0393-0440(16)30297-2 http://dx.doi.org/10.1016/j.geomphys.2016.11.017 GEOPHY 2881
To appear in:
Journal of Geometry and Physics
Received date : 23 February 2015 Accepted date : 19 November 2016 Please cite this article as: P. Wu, A note on Einstein four-manifolds with positive curvature, Journal of Geometry and Physics (2016), http://dx.doi.org/10.1016/j.geomphys.2016.11.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A NOTE ON EINSTEIN FOUR-MANIFOLDS WITH POSITIVE CURVATURE PENG WU Abstract. In this short note we prove that an oriented Einstein four-manifold with √ 1 Ric = g and sectional curvature K ≥ 30 (19 − 271) is isometric to (S 4 , g0 ) or (CP 2 , gF S ) up to rescaling.
1. Introduction A Riemannian manifold (M n , g) is Einstein if its Ricci curvature is a multiple of the metric, i.e., Ric = λg for some constant λ. The classification of Einstein four-manifolds with positive sectional curvature is one of the basic problems in differential geometry. The first result goes back to Berger more than a half century ago, Berger [1] proved that Einstein four-manifolds with 14 -pinched sectional curvature are isometric to (S 4 , g0 ). Hitchin (Theorem 13.30 in [2]) proved that half conformally-flat Einstein four-manifolds with positive scalar curvature are isometric to either (S 4 , g0 ) or (CP 2 , gF S ) up to rescaling. Gursky and LeBrun [8] proved an interesting gap theorem for the L2 -norm of (anti-)self-dual Weyl curvature, using which they proved that Einstein four-manifolds with nonnegative sectional curvature and positive intersection form are isometric to (CP 2 , gF S ) up to rescaling. Yang [13] proved that sectional curvature √ oriented Einstein four-manifolds with4 Ric = g and 1 2 K ≥ 120 ( 1249 − 23) ≈ 0.102 are isometric to (S , g0 ) or (CP , gF S ) up to rescaling. √ Costa [5] later relaxed Yang’s condition to K ≥ 61 (2 − 2) ≈ 0.097. In this short note, we further relax the condition on the sectional curvature, precisely we prove, Theorem 1.1. An √ oriented Einstein four-manifold4 with Ric = g2 and sectional curva1 ture K ≥ 30 (19 − 271) ≈ 0.0846 is isometric to (S , g0 ) or (CP , gF S) up to rescaling.
Remark 1.1. The curvature in Theorem 1.1 can be further relaxed to K ≥ 0.08433, see Remark 3.1. The rest of the paper is organized as follows. In Section 2, we recall the Weitzenb¨ock formula and estimates for the (anti-)self-dual Weyl curvature. In Section 3, we prove Theorem 1 by combining the gap theorem of Gursky and LeBrun [8] and the argument of Yang [13]. Acknowledgement. The author thanks his advisors Professors Xianzhe Dai and Guofang Wei for their guidance, encouragement, and constant support. The author was partially supported by an AMS-Simons postdoctoral travel grant. The result in this Date: December 2, 2016. 2010 Mathematics Subject Classification. Primary 53C24, 53C25. Key words and phrases. Einstein four-manifolds, positive sectional curvature, Weitzenb¨ ock formula. 1
paper was proved in September 2012, when the author was preparing talks on his Ph.D. thesis [10] and related paper of Gursky and LeBrun [8], and Yang [13]. The paper was written in Winter 2014. ¨ ck formula and estimates for half Weyl curvature 2. The Weitzenbo The Hodge star operator ? : ∧∗ T M → ∧∗ T M induces a natural decomposition of the vector bundle of 2-forms ∧2 T M on an oriented four-manifold (M, g), ∧2 T M = ∧+ M ⊕∧− M , where ∧± M are eigenspaces of ±1 respectively, sections of which are called self-dual and anti-self-dual 2-forms respectively. It further induces a decomposition for the curvature operator R : ∧2 T M → ∧2 T M , ! ◦ R + g + W Ric 12 R= , ◦ R − g + W Ric 12 ◦
where Ric is the traceless Ricci curvature, and R is the scalar curvature. If (M 4 , g) is Einstein, then we get R g + W+ 0 12 . R= R g + W− 0 12 Derdzi´ nski [6] proved the following Weitzenb¨ock formula by using the harmonicity of ± W (see also Wu [11, 12] for an alternative proof), Theorem 2.1. Let (M, g) be an Einstein four-manifold, then ∆|W ± |2 = 2|∇W ± |2 + R|W ± |2 − 36 det W ± .
Gursky and LeBrun [8]proved a refined Kato inequality (see also Yang [13], Wu [11] for alternative proofs), which was proven to be optimal by Branson [3] and Calderbank, Gauduchon, and Herzlich [4], Theorem 2.2. Let (M, g) be an Einstein four-manifold, then 5 |∇W ± |2 ≥ |∇|W ± ||2 . 3 By applying a conformal transformation argument, Gursky and LeBrun [8] proved an optimal gap theorem for the L2 -norm of (anti-)self-dual Weyl curvature, Theorem 2.3. Let (M, g) be an Einstein four-manifold with positive scalar curvature. If g is not half conformally flat, then Z Z R2 ± 2 |W | dvg ≥ dvg , M M 24 with equality if and only if ∇W ± ≡ 0.
Finally we recall that, Gursky and LeBrun [8] and Yang [13] derived an upper bound for |W + | + |W − | when the sectional curvature is bounded below by a nonnegative number, Theorem 2.4. Let (M, g) be an Einstein four-manifold with Ric = g. Suppose that K ≥ δ ≥ 0, then r 8 (1 − 3δ). |W + | + |W − | ≤ 3 2
3. Proof of Theorem 1.1 We follow the argument of Yang [13] and prove by contradiction, if (M 4 , g) is not half conformally flat, then there exists t > 0 such that Z (|W + | − t|W − |)dvg = 0. M
Apply the Weitzenbock formula to |W + |2 +t2 |W − |2 and take the integration, we have
(1)
0=
Z
ZM
∆(|W + |2 + t2 |W − |2 )dvg
2|∇W + |2 + 2t2 |∇W − |2 + (4|W + |2 − 36 det W + ) + t2 (4|W − |2 − 36 det W − ). M R R Denote V = Vol(M, g), a = V −1 M |W + |, b = V −1 M |W − |, so we have a = tb. Apply the refined Kato inequality and the Poincar´e inequality, we have Z 2|∇W + |2 + 2t2 |∇W − |2 ZM 10 ≥ (|∇|W + ||2 + t2 |∇|W − ||2 ) 3 M Z 10 ≥ λ1 (|W + | − a)2 + t2 (|W − | − b)2 (2) 3 M Z 2 Z 10 20 + + 2 2 − 2 −1 |W | = λ1 (|W | + t |W | ) − λ1 V 3 3 M M Z 160 t2 10 (|W + |2 + t2 |W − |2 ) − λ (1 − 3δ)2 V. ≥ λ1 2 1 3 9 (1 + t) M =
where λ1 = 43 is a lower bound for the first positive eigenvalue of the Laplace operator [9], and in the last upper bound estimate in Theorem 2.4. √ step we± used the ± 3 Recall that 3 6| det W | ≤ |W | . Plugging (2) to (1), we obtain Z 10 0≥ λ1 (|W + |2 + t2 |W − |2 ) + (4|W + |2 − 36 det W + ) + t2 (4|W − |2 − 36 det W − ) M 3 160 t2 − λ1 (1 − 3δ)2 V 9 (1 + t)2 Z √ √ 40 40 ≥ ( + 4 − 2 6|W + |)|W + |2 + t2 ( + 4 − 2 6|W − |)|W − |2 9 M 9 2 640 t (1 − 3δ)2 V − 27 (1 + t)2 Z 4 4 640 t2 ≥ ( + 24δ)|W + |2 + t2 ( + 24δ)|W − |2 − (1 − 3δ)2 V 2 9 9 27 (1 + t) M By the gap theorem of Gursky and LeBrun in Theorem 2.3, we get 2 4 640 t2 0 ≥ (1 + t2 )( + 24δ) − (1 − 3δ)2 . 2 3 9 27 (1 + t) 3
Notice that (1 + t2 )(1 + t)2 t−2 ≥ 8. Therefore we will get a contradiction if 4 40 ( + 24δ) − (1 − 3δ)2 ≥ 0, 9 9 √ 1 which is equivalent to δ ≥ 30 (19 − 271) ≈ 0.0846.
Remark 3.1. On the other hand, Gallot [7] proved that for an Einstein four-manifold with Ric = g and K ≥ δ, we have λ1 ≥ 8δ + 32 . Using this estimate we will get a contradiction if (1 + t2 )(1 + t)2 (57δ − 2) − 20t2 (12δ + 1)(1 − 3δ)2 ≥ 0, which holds if δ ≥ 0.08433. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Berger, M., Sur quelques vari´et´es d’Einstein compactes, Ann. Mat. Pura Appl. 53 (1961), 89–95. Besse, A., Einstein manifolds, Berlin-Heidelberg, Springer-Verlag, 1987. Branson, T., Kato constants in Riemannian geometry, Math. Res. Let. 7 (2000), 245-261. Calderbank, D., Gauduchon, P., and Herzlich, M., Refined Kato inequalities and conformal weights in Riemannian geometry, J. of Functional Analysis 173 (2000), 214–255. ´ On Einstein four-Manifolds, J. of Geometry and Physics 51 (2004), 244–255. Costa, E, Derdzi´ nski, A., Self-dual K¨ ahler manifolds and Einstein manifolds of dimension four, Com. Math. 49 (1983), 405–433. Gallot, S., Vari´et´es dont le spectre ressemble ` a celui d’une sph`ere, Ast´erisque 80 (1980), 33–52. Gursky, M., LeBrun, C., On Einstein manifolds of positive sectional curvature, Ann. Glob. An. Geom. 17 (1999), 315–328. Lichnerowicz, A., G´eometrie des groupes de transformations, Dunod, Paris, 1958. Wu, P., Studies on Einstein manifolds and gradient Ricci solitons, Ph. D. dissertation, University of California, Santa Barbara, 2012. Wu, P., Einstein four-manifolds with three-positive curvature operator are half conformally flat, preprint, 2013. Wu, P., A Weitzenb¨ ock formula for canonical metrics on four-manifolds, Trans. Amer. Math. Soc. 369 (2017), 1079–1096. Yang, D., Rigidity of Einstein 4-manifolds with positive curvature, Invent. Math. 142 (2000), 435–450.
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, China E-mail address: [email protected]
4