Convergence of Compact Ricci Solitons

The Energy of a Smooth Metric Measure Space and Applications

“…Second, we want to understand to what extent the aforementioned families of quasi-Einstein metrics are "typical." To that end, we will prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, generalizing similar results for Einstein metrics [2,6,22] and for gradient Ricci solitons [42,44,45]. An important feature of our result is that it is a precompactness theorem for smooth metric measure spaces for which the dimensional parameter is allowed to vary within (1, ∞]; in particular, our result can be interpreted as stating that, after taking a subsequence if necessary, noncollapsing sequences of compact quasi-Einstein metrics with m → ∞ converge to shrinking gradient Ricci solitons.…”

Degeneration of Kähler–Ricci Solitons

“…For shrinking Ricci solitons, under certain curvature conditions, the degeneration property has been studied in [5,31,28,27,32]. These results gave generalizations of orbifold compactness theorem of Einstein manifolds [1,3,25].…”

Energy and asymptotics of Ricci-flat 4-manifolds with a Killing field

“…The next step is to establish a link between the curvature scale r R and the maximal function M | Rm | 2 , which is obtained via ǫ-regularity. [11], and [20].…”

“…Let R n = 2R − R 6 π 2 n i=1 1 i 2 , and R = 5R + R 6 π 2 n i=1 1 i 2 ; notice that R n ց R and R n ր 6R. First using (20) and then Hölder's inequality, we have A R n−1 ,R n−1 |W | 2 ≤ C(R/n 2 ) −4 |A R n ,Rn | + C(R/n 2 ) −1 |A R n ,Rn | where C ′ = C + C 4 4 is dimensional. But ∞ n=1 (3/4) n n 8 < ∞, so sending k → ∞,…”