Instantaneously Complete Yamabe Flow in Higher Dimensions

Definitions and mathematical statement

A Yamabe flow, in the usual unnormalized convention, is a family of Riemannian metrics satisfying \[ \partial_t g(t)=-\operatorname{Scal}_{g(t)}\,g(t). \] The source does not explicitly specify this normalization. For a flow on a noncompact manifold $M$, defined for $0<t<T$, where $T>0$, instantaneous completeness means that $(M,g(t))$ is complete for every $t\in(0,T)$, regardless of whether the initial metric is complete.

To interpret the displayed bound, let $m=\dim M$ and let $g_H$ denote a hyperbolic reference metric. The conventional normalization is sectional curvature $-1$, hence scalar curvature $-m(m-1)$; this normalization and the completeness of $g_H$ require confirmation from the source. A conformally hyperbolic flow has \[ g(t)=u(\cdot,t)g_H,\qquad u:M\times(0,T)\longrightarrow(0,\infty). \] The tensor inequality in the source means \[ g(t)_x(v,v)\geq m(m-1)t\,(g_H)_x(v,v) \quad\text{for every }x\in M,\ v\in T_xM,\ 0<t<T. \]

Source question. Does the theory of instantaneously complete Yamabe flows on noncompact surfaces extend to noncompact manifolds of dimension $m\geq3$? In the intended class of conformally hyperbolic Yamabe flows, is the equivalence \[ \left[\,(M,g(t))\text{ is complete for every }t\in(0,T)\,\right] \quad\Longleftrightarrow\quad \left[\,g(t)\geq m(m-1)t\,g_H\text{ for every }t\in(0,T)\,\right] \] valid?

The intended class is not fully specified: the source does not identify the precise surface theory to extend, the admissible initial metrics and their limiting interpretation at $t=0$, curvature or growth assumptions, boundary conditions if boundaries are allowed, or whether the hyperbolic setting is hyperbolic space itself or a more general hyperbolic manifold. Thus the displayed equivalence records the requested target with its unresolved scope; it is not a fully specified universal conjecture.

Short English statement

The problem asks which results about Yamabe flows that become complete at every positive time extend from noncompact surfaces to higher dimensions. In a conformally hyperbolic setting, it also asks whether instantaneous completeness is equivalent to the stated linear lower bound for the metric. The flow normalization, reference geometry, initial data, and admissible class need clarification. The supplied record labels the problem partially solved; that status is not independently verified here.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30004197$, exact source problem code catalogue entry, title Instantaneously Complete Yamabe Flow in Higher Dimensions; collection: Oberwolfach Reports Open Problems. The record gives proposed year 2019 and does not name a proposer.
• Supplied source citation: Partial Differential Equations (2020), Oberwolfach Reports, pp. 2033--2097, DOI 10.4171/owr/2019/34. Original report: https://doi.org/10.4171/owr/2019/34. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Instantaneously Complete Yamabe Flow on Hyperbolic Space, https://doi.org/10.1007/s00526-019-1634-9.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem partially solved and reports that instantaneously complete Yamabe flow is proved for hyperbolic space in arbitrary dimension and for rotationally symmetric data, while the full higher-dimensional equivalence and generalization remain unresolved. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired for self-contained publication. This is supplied editorial metadata, not independent mathematical verification of this draft.