We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi–Hanson space.In particular,we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics...We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi–Hanson space.In particular,we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics on T S^(2),a Type II singularity modeled on the Eguchi–Hanson space develops in finite time.Furthermore,we show that for these Ricci flows the only possible blow-up limits are(i)the Eguchi–Hanson space,(ii)the flat R4/Z2 orbifold,(iii)the 4d Bryant soliton quotiented by Z2,and(iv)the shrinking cylinder R×RP^(3).As a byproduct of our work,we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection|k|≥3.展开更多
文摘We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi–Hanson space.In particular,we prove that starting from a class of asymptotically cylindrical U(2)-invariant initial metrics on T S^(2),a Type II singularity modeled on the Eguchi–Hanson space develops in finite time.Furthermore,we show that for these Ricci flows the only possible blow-up limits are(i)the Eguchi–Hanson space,(ii)the flat R4/Z2 orbifold,(iii)the 4d Bryant soliton quotiented by Z2,and(iv)the shrinking cylinder R×RP^(3).As a byproduct of our work,we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection|k|≥3.
