Four-manifolds with positive curvature operator

Mean curvature flow with surgeries of two–convex hypersurfaces

“…Hence, we can apply the maximum principle for tensors (see [10,Sect. 4]) to obtain that the inequality is preserved for any α 0 ≥ 0.…”

“…4]) to obtain that the inequality is preserved for any α 0 ≥ 0. If λ 1 + λ 2 were not strictly positive for positive times, the strong maximum principle in [10,Sect. 4] would imply that λ 1 + λ 2 = 0 everywhere on M 0 .…”

Classification of almost quarter-pinched manifolds

“…This was established by Hamilton in the proof of [11,Theorem 4.3] as a general property of solutions to heat flows. Specifically consider a heat flow…”

“…Finally, the sectional curvature bounds imply that the curvature tensors of g i are uniformly bounded. By the standard local existence theory for Ricci flow ( [10], [11], and [7]), we can thus run the Ricci flow for a fixed amount of time [0, t 0 ] for each of these metrics (M i , g i ), with the curvature tensor uniformly bounded in i on this time interval.…”

Quasi-convergence of the Ricci flow

“…Then since M+Z -^iZ -hZ 3 Again, we first obtain some preliminary results. then proves {Wj (•, 0)} is bounded and equicontinuous.…”