We discuss two topics in this talk. First we study compact Ricci-flat 4- manifolds without boundary and obtain pointwise restrictions on curvature (not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat Khler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.

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Last updated: 22 May 2019