Weak-Topology Dependence for Geometric Flows

Definitions and mathematical statement

The supplied question asks whether geometric flows depend continuously on their initial data in a weak topology, with Gromov--Hausdorff and $C^{\alpha}$ topologies given as examples. It does not specify a geometric evolution equation or a class of admissible initial data, so it does not determine a single mathematical assertion.

A conditional formulation of the requested property is as follows. Let $X$ be a specified space of admissible initial data for a specified geometric flow. Assume that each $x\in X$ determines a unique solution $u_x(t)$ on a specified interval $I_x$ containing $0$, with $u_x(0)=x$. For a specified time $t>0$, put \[ D_t=\{x\in X:t\in I_x\},\qquad S_t:D_t\longrightarrow Y_t,\quad S_t(x)=u_x(t), \] where $Y_t$ is a specified space of solution states. After equipping $D_t$ and $Y_t$ with specified topologies, fixed-time continuous dependence means that, for every $x\in D_t$ and every neighborhood $V$ of $S_t(x)$, there is a neighborhood $U$ of $x$ in $D_t$ such that $S_t(U)\subseteq V$. For metrizable choices of topology this is equivalently \[ x_n\longrightarrow x\ \text{in }D_t \quad\Longrightarrow\quad u_{x_n}(t)\longrightarrow u_x(t)\ \text{in }Y_t, \] where the expression $u_{x_n}(t)$ is intended on the left side of the second convergence.

Gromov--Hausdorff topology compares metric spaces up to isometry; its compact and pointed noncompact versions require different specifications. A $C^{\alpha}$ topology requires an exponent, a class of fields or geometric objects, and a means of comparing them on identified domains. None of these choices is supplied. The formulation above is only a template: existence, uniqueness, admissible geometries, boundary conditions, common existence times, the input and output topologies, and whether convergence should hold at fixed times or uniformly on time intervals remain unresolved. The source does not specify whether it seeks a theorem for a particular flow or a general principle for a class of flows.

Short English statement

The question asks whether small changes in initial geometry, measured in a weak topology such as Gromov--Hausdorff or $C^{\alpha}$ topology, lead to small changes in the resulting geometric flow. The flow, admissible data, comparison topology, and time dependence are unspecified, so a precise theorem requires additional hypotheses. The supplied record labels the problem open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003355$, title Weak-Topology Dependence for Geometric Flows, exact source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. The citation's 2018 designation and the record's 2017 year require reconciliation.
• Supplied literature source, labeled "Oberwolfach report on nonnegative surfaces": https://doi.org/10.4171/owr/2017/3. The literature assessment was checked on 2026-08-21, according to the supplied dated curation, and has not been independently verified here. That assessment describes the extracted statement as reporting that the relevant Gromov--Hausdorff or $C^{\alpha}$ problem is believed open, and says that no later primary result located in that curation supplies a resolution. The supplied background requests independent verification of time-sensitive status and cited claims.