Short-Time PIC1 Ricci Flow

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$, $n\geq3$, and let $(M,g_0)$ be a smooth complete $n$-dimensional Riemannian manifold. To describe the curvature condition, for a Riemannian metric $h$ set $\widehat h=h+{\,\mathrm{d}} s^2$ on $M\times{\mathbb{R}}$. Use the curvature convention in which $R(u,v,u,v)$ is positive on orthonormal pairs on a round sphere. For every $\widehat h$-orthonormal four-frame $(e_1,e_2,e_3,e_4)$, write $\widehat R_{ijkl}=R_{\widehat h}(e_i,e_j,e_k,e_l)$ and define its isotropic curvature expression by \[ \mathcal I_{\widehat h}(e_1,e_2,e_3,e_4) =\widehat R_{1313}+\widehat R_{1414}+\widehat R_{2323}+\widehat R_{2424}-2\widehat R_{1234}. \] The weak $\mathrm{PIC1}$ condition means that this expression is nonnegative at every point of $M\times{\mathbb{R}}$ for every such frame. The strict version requires it to be positive. This product formulation applies also when $n=3$.

Unresolved qualification. The supplied statement uses $\mathrm{PIC1}$ without defining whether it means the weak or strict condition. Its literature assessment separately mentions "suitable weak PIC1 pinching" without specifying that pinching. Thus the sign convention intended in the question must be checked against the cited report; the weak and strict readings must not be silently identified.

Question, retaining the source's intended curvature condition. For every integer $n\geq3$ and every smooth complete $(M,g_0)$ satisfying the intended $\mathrm{PIC1}$ condition, do there exist $T>0$ and a smooth family of Riemannian metrics $g(t)$ on $M$, $0\leq t<T$, such that \[ \frac{\partial}{\partial t}g(t)=-2\operatorname{Ric}_{g(t)},\qquad g(0)=g_0, \] and $g(t)$ satisfies that same $\mathrm{PIC1}$ condition for every $t\in[0,T)$? The existence time may depend on $(M,g_0)$. No bounded-curvature hypothesis, uniform pinching constant, or completeness requirement for $g(t)$ at positive times is explicitly supplied; none is added here.

Short English statement

Does every smooth complete Riemannian manifold of dimension at least three satisfying the source's $\mathrm{PIC1}$ condition admit a smooth Ricci flow for some positive time that starts at the given metric and preserves that condition? The weak versus strict meaning of $\mathrm{PIC1}$ remains unresolved. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30004176$, title Short-Time PIC1 Ricci Flow, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied report citation: Differentialgeometrie im Großen (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/30, pp. 1791--1839. https://doi.org/10.4171/owr/2019/30. The background attributes the problem to workshop open-problem discussions documented in this report.
• Supplied literature source: Ricci flow and PIC1. https://arxiv.org/abs/2309.00596.
• Supplied literature source labelled "original OWR report": https://doi.org/10.4171/owr/2019/30.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem partially solved and reports existence with suitable weak $\mathrm{PIC1}$ pinching in dimensions at least four, with the corresponding conjecture remaining open in dimension three. This is supplied dated curation, not independently verified here; the precise hypotheses and time-sensitive status require independent checking.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This verification claim is supplied provenance and has not been independently verified here.