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Einstein metrics on spheres. (English) Zbl 1093.53044
In this interesting paper the authors demonstrate existence theorems for many families of Einstein metrics on spheres and exotic spheres. More precisely, they prove:
(i) On \(S^5\) there exist 68 inequivalent families of Sasakian-Einstein metrics.
(ii) All 28 oriented diffeomorphism classes on \(S^7\) admit inequivalent families of Sasakian-Einstein structures.
(iii) For \(n\geq 2\), the \((4n+1)\)-dimensional standard and Kervaire spheres both admit many families of inequivalent Sasakian-Einstein metrics. Thus, the authors answer in an affirmative way the long standing open question about the existence of Einstein metrics on exotic spheres.
The main steps of the proof are the following. For a sequence \({\mathbf a}= (a_1,\ldots,a_m)\in\mathbb Z^m_+\) the authors consider the Brieskorn-Pham singularity
\[ Y({\mathbf a}):=\left\{\sum_{i=1}^m z_i^{a_i}=0\right\}\subset\mathbb C^m\quad \text{ and\;its\;link} \;\;L({\mathbf a}):=Y({\mathbf a})\cap S^{2m-1}(1), \]
which is a smooth, compact, \((2m-3)\)-dimensional manifold. Moreover, \(Y({\mathbf a})\) has a natural \(\mathbb C^\ast\)-action and \(L({\mathbf a})\) a natural \(S^1\)- action.
In the first step, basing on Kobayashi’s circle bundle construction, the authors observe that a positive Kähler-Einstein metric on the base space of a circle bundle gives an Einstein metric on the total space. This result was generalized to orbifolds by C. P. Boyer and K. Galicki [Int. J. Math. 11, No. 7, 873–909 (2000; Zbl 1022.53038)], giving Sasaki-Einstein metrics. Thus, a positive Kähler-Einstein orbifold metric on \((Y({\mathbf a})\setminus\{0\})/\mathbb C^\ast\) yields a Sasaki-Einstein metric on \(L({\mathbf a})\).
In the second step, for a sequence \(\mathbf a\) satisfying certain numerical conditions, the authors use the continuity method developed by Aubin, Siu and Tian to construct Kähler-Einstein metrics on orbifolds. The final step is to get examples, partly through computer searches, partly through writing down well chosen sequences.
The authors also formulate the conjecture that all odd-dimensional homotopy spheres which bound parallelizable manifolds admit Sasaki-Einstein metrics.

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32Q20 Kähler-Einstein manifolds
Citations:
Zbl 1022.53038
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