On the conditions to extend Ricci flow. II (Q2905258)

The author develops some estimates under the Ricci flow and uses them to study the blowup rates of curvatures at singularities. As applications, he obtains some gap theorems: \(\sup_X|Ric|\) and \(\sqrt{\sup_X|Rm|}.\sqrt{|R|}\) must blow up at least at the rate of type-I. These estimates also imply some gap theorems for shrinking Ricci solitons.NEWLINENEWLINENEWLINEIn 1995, \textit{R. S. Hamilton} [``The Formation of Singularities in the Ricci flow'', Surv. Differ. Geom., Suppl. J. Differ. Geom. 2, 7--136 (1995; Zbl 0867.53030)] showed that the Ricci flow can be extended over \(T\) if \(|Rm|\) is uniformly bounded on the space-time \(X\times[0,T)\). In other words, \(|Rm|\) blows up if \(T\) is a singular time. In [Am. J. Math. 127, No. 6, 1315--1324 (2005; Zbl 1093.53070)], \textit{N. Sesum} proved that \(|Ric|\) blows up at singular time. These theorems are fundamental. They were generalized in many directions by X. Cao; X. Cao and Q. S. Zhang; J. Emders, R. Muller and P.M. Topping; \textit{D. Knopf} [Proc. Am. Math. Soc. 137, No. 9, 3099--3103 (2009; Zbl 1172.53043)]; N. Q. Lee and N. Sesum; \textit{L. Ma} and \textit{L. Cheng} [Ann. Global Anal. Geom. 37, No. 4, 403--411 (2010; Zbl 1188.35033)]; \textit{B. Wang} [Int. Math. Res. Not. 2008, Article ID rnn012, 30 p. (2008; Zbl 1148.53050)]; \textit{R. Ye} [Calc. Var. Partial Differ. Equ. 31, No. 4, 439--455 (2008; Zbl 1142.53033)].NEWLINENEWLINEBefore the singular time \(T\) of a Ricci flow, an application of the maximum principle implies that \(|Rm|,\) not only blows up, but also blows up at a big rate (cf. [\textit{B. Chow, P. Lu} and \textit{L. Ni}, Hamilton's Ricci flow. Graduate Studies in Mathematics 77. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1118.53001)]): NEWLINENEWLINE\[NEWLINE\lim_{t\rightarrow T}|T-t|(\sup_X |Rm|)\geq\frac{1}{8}.NEWLINE\]NEWLINE NEWLINEA natural question is: Does similar behavior hold for \(|Ric|\)? In this paper, the author answers this question affirmatively.NEWLINENEWLINENEWLINETheorem 1.1. Suppose \(\{(X,g(t)),0\leq t< T\}\) is a Ricci flow solution, \(X\) is a closed manifold of dimension \(m\), and \(t=T\) is a singular time. Then NEWLINENEWLINE\[NEWLINE\limsup_{t\rightarrow T}|T-t|(\sup_X| Ric|_{g(t)})\geq\eta_1,\tag{\(*\)} NEWLINE\]NEWLINE where \(\eta_1=\eta_1(m,k)\), \(k\) is the noncollapsing constant of this flow.NEWLINENEWLINENEWLINEThe author notes that he does not assumes that the singularity is of type-I in Theorem 1.1. He says that if the singularity is of type-I, inequality (\(*\)) was implied by the major results in the works of X. Cao and Q. S. Zhang and J. Enders, R. Muller, and P. M. Topping (op. cit), and a gap theorem of gradient shrinking solitons in [{O. Munteanu} and \textit{M.-T. Wang}, ``The curvature of gradient Ricci solitons'', \url{arXiv:1006.3547}]. In this case \(\eta_1\) can be chosen as \(\frac{1}{100 m^2}\).NEWLINENEWLINENEWLINEOther interesting results of the paper are the following theorems:NEWLINENEWLINENEWLINETheorem 1.2. Suppose \(\{(X,g(t)), 0\leq t< T\}\) is a Ricci flow solution, \(X\) a closed manifold of dimension \(m\), and \(t=T\) a singular time. If \(\limsup_t\rightarrow |T-t|(\sup_X|Ric|_{g(t)})=C\), then \(\lim \sup_{t\rightarrow T}^{\lambda}(\sup_X|Rm|_{g(t)})=0\), whenever \(\lambda > \frac{C}{\epsilon_1}\), where \(\epsilon _1=\epsilon _1(m,k)\) depends on \(m\) and the noncollapsing constant \(k\) of this flow.NEWLINENEWLINETheorem 1.3. Suppose \(\{(X,g(t)), 0\leq t< T\}\) is a Ricci flow solution , \(X\) is a closed manifold of dimension \(m\), and \(t=T\) is a singular time. Then NEWLINENEWLINE\[NEWLINE\limsup_{t\rightarrow T}\left(\sqrt{\sup_X |Rm|_{g(t)}}.\sqrt{\sup_X |R|_{g(t)}}\right)\geq \eta_2, \text{ where }\eta_2=\eta_2(m,k).NEWLINE\]NEWLINE NEWLINETheorem 1.4. Suppose \(\{(X,g(t)), 0\leq t< T\}\) is a Ricci flow solution, \(X\) is a closed manifold of dimension \(m\), and \(t=T\) is a singular time. If \(\lim \sup _{t\rightarrow T}|T-t|(\sqrt{\sup_X|Rm|_{g(t)}}.\sqrt{\sup_X|R|_{g(t)}})=C\), then NEWLINENEWLINE\[NEWLINE\limsup_{t\rightarrow T}|T-t|^{\lambda}(\sup_{X}|Rm|_{g(t)})=0, \text{ whenever }\lambda> \frac{1}{\log_2(1+\frac{\epsilon_2}{C}},\text{ where }\epsilon _2=\epsilon_2(m,k).NEWLINE\]NEWLINE NEWLINETheorem 1.5. there exists a constant \(\eta_3=\eta_3(m,k)\) such that the following property holds.NEWLINENEWLINESuppose that \((X^m,g)\) is a complete, nonflat, \(k\)-noncollapsed Riemannian manifold. If \((X^m,g)\) satisfies the shrinking Ricci soliton equation \(Ric +\mathcal{L}_V g-\frac{g}{2}=0\), for some vector field \(V\) , \(\sup_X|Rm|<\infty\), then NEWLINE\[NEWLINE\min \left\{\sqrt{\sup_X|Rm|}.\sqrt{\sup_X|R|},\sup_X|Ric|\right\}\geq\eta_3>0.NEWLINE\]