On convergence of the Kähler-Ricci flow
Let \(X\) be a Fano manifold of dimension \(n\). It is an important open problem to determine whether \(X\) admits a Kähler-Einstein metric.NEWLINENEWLINEThe central problem is the Yau-Tian-Donaldson conjecture. A Fano manifold \(X\) admits a Kähler-Einstein metric if and only if it is K-polystable.NEWLINENEWLINEOne way of attacking the problem is by means of the Ricci flow. Cao showed that the normalized Kähler-Ricci flow NEWLINE\[NEWLINE \frac \partial {\partial t}g_{i\bar{j}} =-\left( R_{i\bar{j}}-g_{i\bar{j} }\right) \qquad g_{i\bar{j}}\left( 0\right) \in c_1(X). NEWLINE\]NEWLINE exists for all time. If one tries to use the Kähler-Ricci flow to find a Kähler-Einstein metric, the key problem is to relate the K-polystability of \(X\) to the convergence of the flow. Many partial results have been obtained assuming e.g. that the Riemann curvature tensor remains uniformly bounded along the flow, \textit{D. H. Phong} and \textit{J. Sturm} [J. Differ. Geom. 72, No. 1, 149--168 (2006;Zbl 1125.53048)]; \textit{D.H. Phong} et al., [Invent. Math. 173, No. 3, 651--665 (2008; Zbl 1145.53050)]; \textit{G. Székelyhidi} [Am. J. Math. 132, No. 4, 1077--1090 (2010;Zbl 1206.53075)] and \textit{V. Tosatti} [J. Reine Angew. Math. 640, 67--84 (2010;Zbl 1189.53069)].NEWLINENEWLINEIn the paper under review the main assumption is that along the flow \(g(t)\) there is a uniform lower bound \(\lambda_t \geqslant \lambda > 0\) on the lowest positive eigenvalue \(\lambda_t\) of the \(\bar{\partial}\)-operator acting on vector fields. This condition was introduced by \textit{D.H. Phong} et al., [J. Differ. Geom. 81, No. 3, 631--647 (2009; Zbl 1162.32014)], who proved that if in addition the Mabuchi energy is bounded from below, then the flow converges to a Kähler-Einstein metric. The main result of the present paper is a weakening of the hypothesis on the Mabuchi energy.NEWLINENEWLINE{Theorem 3.} Suppose that along the Kähler-Ricci flow a uniform lower bound \(\lambda_t\geqslant\lambda > 0\) holds for the first eingenvalue, and in addition the Mabuchi energy satisfies NEWLINE\[NEWLINE \mathcal{M}(g(t)) > -C\log(1+t) - D, NEWLINE\]NEWLINE for some constants \(C,D>0\). Then the metrics \(g(t)\) converge exponentially fast in \(C^\infty\) to a Kähler-Einstein metric.
scientific article; zbMATH DE number 6052972