Riemannian Control of Bergman Geometry

Definitions and mathematical statement

Let $M$ be an open Riemann surface and let $g=ds^2$ be a complete Riemannian metric compatible with its complex structure. Here "open Riemann surface" is understood in its usual sense of a noncompact Riemann surface without boundary. No curvature, volume, isoperimetric, or topological restrictions beyond the supplied assumptions are imposed.

For an intrinsic formulation, adopt the following convention, which the supplied record does not explicitly specify. Let \[ \mathcal H(M)=\left\{\omega:\omega\text{ is a holomorphic }(1,0)\text{-form on }M,\quad \|\omega\|^2:=\frac{i}{2}\int_M\omega\wedge\overline\omega<\infty\right\}. \] For an orthonormal basis $(\omega_j)$ of this Hilbert space, define its Bergman kernel by \[ \mathcal K_M(p,\overline q)=\sum_j\omega_j(p)\otimes\overline{\omega_j(q)}. \] In a local coordinate $z$, write $\omega_j=f_j(z)\,dz$ and put \[ B_z(z)=\sum_j|f_j(z)|^2 =\sup_{\substack{\omega=f(z)\,dz\in\mathcal H(M)\\\|\omega\|\leq1}}|f(z)|^2. \] Where $B_z>0$, the associated Bergman metric expression, with one standard normalization, is \[ ds_B^2=\frac{\partial^2\log B_z}{\partial z\,\partial\overline z}\,|dz|^2. \] It defines a Riemannian metric only where its coefficient is positive. The stated assumptions do not guarantee that $\mathcal H(M)\neq\{0\}$, that $B_z$ is everywhere positive, or that this expression is nondegenerate. Such existence and positivity conditions must therefore be included in any conclusion involving a Bergman distance or Bergman completeness.

Research question. How does the Riemannian geometry of the complete compatible metric $g$ control the Bergman kernel $\mathcal K_M$ and, when defined and nondegenerate, the Bergman metric $ds_B^2$? Seek precise geometric hypotheses and corresponding estimates or qualitative conclusions. This preserves the supplied question as a research program: the record specifies no particular inequality, classification, necessary-and-sufficient condition, or quantified convergence assertion to prove.

The supplied background mentions positive isoperimetric constants together with curvature or volume hypotheses, and conclusions concerning Bergman completeness, distance comparison, kernel decay, and convergence. It does not give the definitions of those constants, the exact hypotheses, the comparison quantities, or the objects and topology of convergence. Consequently, those remarks cannot be promoted here to precise theorem statements.

Short English statement

For an open Riemann surface equipped with a complete compatible Riemannian metric, determine how geometric assumptions yield information about its intrinsic Bergman kernel and, when it exists as a nondegenerate metric, its Bergman metric. The source poses a broad research question and leaves the desired hypotheses and conclusions unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30002770$, exact source problem code catalogue entry, title "Riemannian Control of Bergman Geometry." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2015$; no individual proposer is supplied.
• Supplied report citation: Geometric Methods of Complex Analysis (2015), Oberwolfach Reports, pp. 235--283, DOI 10.4171/owr/2015/4. Supplied source URL: https://doi.org/10.4171/owr/2015/4. This is a report-wide page range; no page locating the question is supplied.
• Supplied literature source and attribution: "Chen, Oberwolfach Report," https://doi.org/10.4171/owr/2015/4. The record supplies no full name or separate article-level citation for this attribution.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and states that, under positive isoperimetric constants and curvature or volume hypotheses, the cited work proves Bergman completeness, distance comparison, kernel decay, and convergence, while no general geometric classification is known. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and that the question, definitions, and notation were restored without strengthening the claim. These are supplied provenance statements; no independent source inspection is claimed in this draft.