The canonical expanding soliton and Harnack inequalities for Ricci flow
The paper under review concerns the Ricci flow theory. The authors introduce a notion of ``canonical expanding Ricci soliton'' whose definition and basic properties are presented in the following statement:NEWLINENEWLINETheorem. Let \(g(t)\) be a Ricci flow, i.e., a solution of \(\frac{\partial g}{\partial t} = -2 \operatorname{Ric}(g(t))\), defined on a manifold \(M^n\), for \(t\) within a time interval \([0,T]\), with uniformly bounded curvature. Suppose \(N>0\), and define a metric \(\breve{g}\) on \(\breve{M} = M\times (0,T]\) by NEWLINE\[NEWLINE \breve{g}_{ij}=\frac{1}{t}g_{ij},\quad \breve{g}_{00} = \frac{1}{2t^3}N+ \frac{1}{2t^2}n+ \frac{1}{t}R,\quad \breve{g}_{0j}=0, NEWLINE\]NEWLINE where \(i\), \(j\) are coordinate indices on the \(M\) factor, \(0\) represents the index of the time coordinate \(t\in (0,T]\), and \(R\) is the scalar curvature of \(g\).NEWLINENEWLINEThen, up to errors of order \(\frac{1}{N}\), the metric \(\breve{g}\) is a gradient expanding Ricci soliton on the manifold \(\breve{M}\): NEWLINE\[NEWLINE E_N:=Ric(\breve{g})+Hess_{\breve{g}}\left(-\frac{N}{2t}\right)+\frac{1}{2}\breve{g}\simeq 0, NEWLINE\]NEWLINE which means that for any \(k\in\{0, 1, 2, \dots\}\) the quantity \(N\left[\breve{\nabla}^kE_N\right]\) is bounded uniformaly locally on \(\breve{M}\) (independently on \(N\)), where \(\breve{\nabla}\) is the Levi-Civita connection corresponding to \(\breve{g}\).NEWLINENEWLINEIt is shown that the full curvature tensor \({\mathcal R}(\breve{g})\) of \(\breve{g}\), viewed as a section of \(\otimes^4T^*\breve{M}\), converges smoothly to a limit algebraic curvature tensor \(R_\infty\) as \(N\to \infty\), and it turns out that this curvature tensor may be used to rephrase some known Harnack inequalities [\textit{R.~Hamilton}, J. Differ. Geom. 24, 153-179 (1986; Zbl 0628.53042); \textit{S.~Brendle}, J. Differ. Geom. 82, No. 1, 207--227 (2009; Zbl 1169.53050)] as well as to produce new ones.
scientific article; zbMATH DE number 6038807
scientific article