1 · The essentials
These manuscripts ask whether regions with complicated holes can be represented using only perfectly round holes and single missing points. The setting is the complex plane together with a point at infinity, viewed as a sphere. The representation must preserve angles locally, rather than preserve lengths or areas. The second manuscript states that every connected open region on this sphere has such a representation. The main manuscript addresses a different question: how much freedom remains when changing one round-hole representation into another? It establishes that, under a particular condition on the boundary, only a restricted family of transformations of the whole sphere is possible. This main result has a Lean formalization, and it places no restriction on the number of holes.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine redrawing a map so that every missing patch becomes a round hole or a single missing dot, while preserving the angles at which paths meet. The existence question asks whether such a map can always be drawn; the rigidity question asks how much two such maps can differ when an additional boundary condition holds.
Why it matters
The existence statement gives a common geometric form for every domain in the Riemann sphere. The rigidity result complements it by controlling the possible conformal changes between circle domains when the starting boundary is conformally removable. The source identifies this as the removability-to-rigidity direction of the He–Schramm conjecture, not as the entire conjecture.
2 · Going further
A circle domain is a domain in the Riemann sphere whose complementary components are round disks or points. The existence problem asks whether every domain in that sphere is conformally equivalent to a circle domain. The companion manuscript, “Koebe's Circle-Domain Conjecture,” states an affirmative answer, described in the repository summary as resolving the existence part of Koebe's circle-domain conjecture.
The main manuscript, “Removable Boundaries and Rigidity of Circle Domains,” concerns the possible conformal equivalences after a circle-domain representation is available. Its result says that if a circle domain has conformally removable boundary, every conformal equivalence from it to another circle domain is a Möbius transformation. Crucially, the stated result imposes no restriction on the number of complementary components. The source identifies this implication as the removability-to-rigidity direction of the He–Schramm conjecture.
The two manuscripts therefore address existence and conditional rigidity, respectively: one supplies a circle-domain representation, while the other restricts changes between such representations under a boundary hypothesis. The supplied abstracts state the conclusions but do not describe the arguments, so they do not support an account of the construction or the mechanism connecting removability to rigidity. The main rigidity result has a Lean formalization; the supplied material does not detail its coverage of companion claims.
Useful notions
- The complex plane and Riemann sphere
- Open connected regions and their boundaries
- Conformal maps and conformal equivalence
- Connected components of a complement
- Möbius transformations
What was known before / what is new
Before
The source frames the existence question as Koebe's circle-domain conjecture. It also names the He–Schramm conjecture as the context for the relationship between boundary removability and rigidity. It supplies no account of earlier partial results or methods.
New
The companion manuscript states that every domain in the Riemann sphere is conformally equivalent to a circle domain. The main manuscript establishes rigidity for circle domains with conformally removable boundary, without restricting the number of complementary components. This main result has a Lean formalization.
Limits and assumptions
The existence statement concerns domains in the Riemann sphere and allows complementary components that are points as well as round disks. The rigidity statement requires conformal removability of the boundary; it is not a claim that all circle domains are rigid. Only the removability-to-rigidity direction of the He–Schramm conjecture is established here, and the supplied source does not report the status of the reverse direction. The main result has a Lean formalization, but no detailed formalization scope is supplied, so verification of the companion existence statement or every supporting claim should not be inferred. The abstracts provide no argument details.
Mini glossary
- Riemann sphere —
- The complex plane together with a point at infinity, viewed as a sphere.
- Domain —
- An open connected region.
- Circle domain —
- A domain in the Riemann sphere whose complementary components are round disks or points.
- Conformal equivalence —
- An invertible conformal map between domains, preserving angles locally.
- Conformal rigidity —
- For a circle domain, the property that every conformal equivalence to another circle domain is a Möbius transformation.
- Conformally removable boundary —
- The boundary condition in the main result that is sufficient to force conformal rigidity; the supplied abstracts do not give its technical definition.
Keywords : Circle domains · Conformal removability · Conformal rigidity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Removable Boundaries and Rigidity of Circle Domains
23/09/2026We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
Koebe's Circle-Domain Conjecture
23/09/2026We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.