Nonsymmetric Extensions of Dirichlet-Form Trace Constructions

Definitions and mathematical statement

Let $(\mathcal E,D(\mathcal E))$ denote a Dirichlet form, with form domain $D(\mathcal E)$, in the framework intended by the cited report. In a real bilinear convention, symmetry means \[ \mathcal E(u,v)=\mathcal E(v,u)\qquad\text{for all }u,v\in D(\mathcal E). \] This convention is explanatory; the supplied extract does not specify the scalar field, underlying space, or analytic hypotheses for the intended forms.

Question (as supplied). Does the trace construction for symmetric Dirichlet forms extend to nonsymmetric Dirichlet forms? The supplied background describes the intended extension more broadly as an extension of the trace and curvature method. The supplied original passage states: "It is an open problem whether this method still applies for nonsymmetric forms."

Unresolved formulation. The supplied record does not define the trace construction, identify the method referred to by "this method," or state its hypotheses and conclusions. It also does not specify the admissible class of nonsymmetric forms or the properties an extension must preserve. Consequently, this is a source-level extension question requiring clarification from the report before it can be expressed as a fully quantified mathematical proposition. No particular trace construction, curvature condition, or universal extension claim is imposed here.

Short English statement

The question asks whether the symmetric Dirichlet-form trace construction, together with the curvature method mentioned in the supplied background, can be extended to nonsymmetric forms. The construction, hypotheses, and required conclusions are missing from the extract, so the precise extension problem remains unresolved at the formulation level.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004942$, source problem code catalogue entry, title: Nonsymmetric Extensions of Dirichlet-Form Trace Constructions. No individual proposer is identified; the supplied proposed year is 2021.
• Supplied report citation: Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces (2022), Oberwolfach Reports, pp. 3135--3186, DOI 10.4171/owr/2021/58. https://doi.org/10.4171/owr/2021/58. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Mini-Workshop: Variable Curvature Bounds report. https://ems.press/content/serial-article-files/46936.
• Supplied literature source: Erbar--Rigoni--Sturm--Tamanini, "tamed spaces." https://doi.org/10.1016/j.matpur.2022.02.002.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the report explicitly leaves extension of the trace and curvature method to nonsymmetric Dirichlet forms open, with no general resolution supplied by later literature. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied verification claims; no source visit or independent verification is asserted in this draft.