Intended Deturck Flow Geometry Analysis Based On Quasi-constant Curvature Metric

C.Guenther uses center manifold and maximal regularity theory to study stability of DeTurck flowin constant curvature space. We expand this case to Quasi constant curvature space, and get the similar result. That isto say, let (M^", g₀) be a quasi constant curvature manifold, and the main Ricci curvature T with respect to ζ, satisfies: T ≥ n -1, the variation of g is a second symmetric covariant tensor, then the solutions whose initial data lie sufficiently near g₀ are attracted at anexponential rate to the center manifold who contains g₀.On the other hand, we use the same method to discuss the stability for Einstein metric under some Killing condition. The result is: let Mⁿ be a closed connected compactEinstein manifold, and ||Rm|| ≤∧, let % denotes the closure of S₂^μ ((?) S₂^μ+ ) with respectto the ||·||_2+ρ Holder norm, then for each r∈N, there is a C^r center manifold M_loc^c existing ina neighborhood O_r of g₀ , and the solutions whose initial data lie sufficiently near g₀ are attracted at an exponential rate to that center manifold.