On the regularity of Ricci flows coming out of metric spaces (Q2135430)

The authors investigate the following problem: consider a possibly incomplete Ricci flow \((M,g(t))_{t \in (0,T)}\) satisfying \[ |\mathrm{Rm}|\leq c_0/t \tag{1} \] and whose associated metric spaces \((M,d_{g(t)})\) Gromov-Hausdorff converge to a metric space \((X,d_X)\) as \(t \searrow 0\). What additional assumptions on \((X,d_X)\) and \((M,g(t))_{t \in (0,T)}\) guarentee that \(g(t)\) converges locally smoothly (or continuously) to a smooth (or continous) metric as \(t \searrow 0\)? The authors give a positive answer under the additional assumption that for all \(t \in (0,T)\) \[ \mathrm{Ric}_{g(t)}\geq -1, \tag{2}\] \(B_{g(t)}(x_0,1) \Subset M\) and that \((X,d_X)\) is \emph{smoothly (or continously) \(n\)-Riemannian}. One says that \((X,d_X)\) is \emph{smoothly (respectively continously) \(n\)-Riemannian} if for any \(x_0 \in X\) there are \(0<\tilde r<r\) with \(\tilde r<r/5\) and points \(a_1,\ldots,a_n\in B_{d_X}(x_0,r)\) such that the map \[F(x) = (d_X(a_1,x),\ldots,d_X(a_n,x)),\quad x \in B_{d_X}(x_0,r)\] is an \((1+\varepsilon_0)\)-bilipschitz homeomorphism on \(B_{d_X}(x_0,5\tilde r)\), and the pushed forward of \(d_X\) via \(F\) is on \(\mathbb{B}_{4\tilde r}(F(x_0))\Subset F(B_{d_X}(x_0,5\tilde r))\) induced by a smooth (respectively continous) Riemannian metric. The smooth definition corresponds to \((X,d_X)\) being locally isometric to smooth \(n\)-Riemannian manifolds. Before stating the main results let us recall a bit of context. When all \(g(t)\) are complete and satisfy (1) and (2), \textit{M. Simon} and \textit{P. M. Topping} [Geom. Topol. 25, No. 2, 913--948 (2021; Zbl 1470.53083)] show that \(d_{g(t)} \to d_0\) a metric on \(M\) as \(t \searrow 0\), and that \((M,d_0)\) is isometric to \((X,d_X)\). Without the completeness assumption, a localized version of this holds assuming that \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\). Then \(X:=\cap_{s \in (0,T)} B_{g(s)}(x_0,1/2)\) is non empty and is endowed with a well defined limiting metric \(d_0=\lim d_{g(t)}\) as \(t \searrow 0\). Moreover \(B_{g(t)}(x_0,r) \Subset \mathcal{X} \subset X\) for all \(r \leq R(c_0,n)\) and \(t \leq S(c_0,n)\), where \(\mathcal{X}\) is the connected component of \(X\) containing \(x_0\), and the topology of \(B_{d_0}(x_0,r)\) induced by \(d_0\) agrees with that of \(M\). The main results of the authors are the following. Theorem 1.6 : Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is smoothly \(n\)-Riemannian for some \(r<R(c_0,n)\). Then there exists a smooth Riemannian metric \(g_0\) on \(B_{d_0}(x_0,s)\) for some \(s<r\) such that \(g(t)\) extends to a smooth solution \((B_{d_0}(x_0,s), g(t))_{t \in [0,T)}\) by defining \(g(0)=g_0\). Theorem 1.7: Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is continuously \(n\)-Riemannian for some \(r<R(c_0,n)\). Then for any stricty monotone sequence \(t_i \searrow 0\), there exists \(v>0\) and a continuous Riemannian metric \(\tilde g_0\) defined on \(\mathbb{B}_v(p) \subset \mathbb{R}^n\) and a family of smooth diffeomorphisms \(Z_i : B_{d_0}(x_0,2v) \to \mathbb{R}^n\) such that \((Z_i)_\ast(g(t_i))\) converges in the \(C^0\)-sense to \(\tilde g_0\) as \(t_i \searrow\) on \(\mathbb{B}_v(p)\).