TY - JOUR

T1 - Smooth structures, normalized Ricci flows, and finite cyclic groups

AU - Ishida, Masashi

AU - Şuvaina, Ioana

N1 - Funding Information:
Acknowledgements We would like to express our deep gratitude to Claude LeBrun for his warm encouragements. The first author is partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 20540090. The second author would like to thank the Institut des Hautes Études Scientifiques, where she was visiting as an IPDE postdoc, for its warm hospitality.

PY - 2009/5

Y1 - 2009/5

N2 - A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group Zd, where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group Zd, which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular ℤd -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of ℂℙ2s and d > 1/ℂℙ2s.

AB - A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group Zd, where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group Zd, which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular ℤd -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of ℂℙ2s and d > 1/ℂℙ2s.

KW - Exotic smooth structures

KW - Finite cyclic fundamental groups

KW - Normalized Ricci flows

KW - Seiberg-Witten theory

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U2 - 10.1007/s10455-008-9136-6

DO - 10.1007/s10455-008-9136-6

M3 - Article

AN - SCOPUS:63949088496

VL - 35

SP - 267

EP - 275

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 3

ER -