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 -