Rigidity of critical metrics for some quadratic curvature functionals
speaker: Dario Monticelli (Politecnico di Milano)abstract:
In this talk I will present some recent results obtained in collaboration with G. Catino (Politecnico di Milano) and P. Mastrolia (Università degli Studi di Milano) concerning rigidity for complete, possibly non-compact, critical metrics of the quadratic curvature functionals F2t =
Ricg
2dVg + tR2gdVg, t ∈ R, and S2=R2gdVg. In particular, we showed that (i) flat surfaces are the only critical points of S2, (ii) flat three-dimensional manifolds are the only critical points of F2t for every t > −13, (iii) three-dimensional scalar flat manifolds are the only critical points of S2 with finite energy and (iv) n-dimensional, n > 4, scalar flat manifolds are the only critical points of S2 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M. T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.
timetable:
Tue 7 Jun, 14:00 - 15:00, Aula Dini
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