Several Complex Variables Problem List — Problem 82

Definitions and mathematical statement

Let $M$ and $N$ be strongly pseudoconvex CR manifolds of real dimension $3$. Write $H_M\subset TM$ and $H_N\subset TN$ for their real rank-two CR distributions, with complex structures $J_M$ and $J_N$. Strong pseudoconvexity makes these distributions contact distributions. Choose smooth positive-definite inner products on the distributions compatible with their complex structures. The associated Carnot--Carathéodory distances are defined by \[ d_M(p,q)=\inf_{\gamma}\int_0^1\|\dot\gamma(t)\|_{H_M}\,{\,\mathrm{d}} t, \] where the infimum is over absolutely continuous curves from $p$ to $q$ satisfying $\dot\gamma(t)\in H_M$ almost everywhere; define $d_N$ analogously.

The source calls a homeomorphism $f:M\to N$ conformal when it maps infinitesimal Carnot--Carathéodory spheres to infinitesimal spheres. One possible precise metric interpretation of this condition is the following. For each $x\in M$, set \[ L_f(x,r)=\sup_{d_M(x,y)=r}d_N(f(x),f(y)),\qquad \ell_f(x,r)=\inf_{d_M(x,y)=r}d_N(f(x),f(y)), \] and require \[ \limsup_{r\downarrow0}\frac{L_f(x,r)}{\ell_f(x,r)}=1 \qquad\text{for every }x\in M. \] This is an explicit candidate interpretation, not a quantified definition supplied by the source.

For a $C^1$ map, the CR condition is \[ {\,\mathrm{d}} f_x(H_{M,x})\subseteq H_{N,f(x)},\qquad {\,\mathrm{d}} f_x(J_Mv)=J_N{\,\mathrm{d}} f_x(v) \quad(x\in M,\ v\in H_{M,x}). \] Question (Puqi Tang; source Problem 82). Is every conformal homeomorphism between strongly pseudoconvex CR $3$-manifolds CR?

Unresolved qualifications. The source does not quantify its infinitesimal-sphere condition or specify the regularity intended in the conclusion. It also imposes no orientation or CR-orientation condition and does not state whether anti-CR maps are to be excluded or included in the conclusion. These issues must be resolved before the candidate metric interpretation above can be treated as a definitive formulation. The source's broader introductory assertion that a Carnot--Carathéodory metric can always be defined on a CR $3$-manifold also requires qualification: the horizontal length construction needs an accessibility hypothesis to give finite distances between relevant points; strong pseudoconvexity supplies the local contact hypothesis in the actual question.

Short English statement

Puqi Tang asks whether a homeomorphism between strongly pseudoconvex CR manifolds of real dimension $3$ that preserves infinitesimal Carnot--Carathéodory spheres must preserve the CR structures. The meaning of infinitesimal sphere preservation, the required regularity, and the treatment of anti-CR maps remain unspecified in the supplied source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800282$, exact source code catalogue entry, titled Several Complex Variables Problem List --- Problem 82. The registry labels the problem open; this supplied label is not an independently verified mathematical or literature assessment.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 82. Supplied source URL: https://arxiv.org/abs/math/9509202. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; neither access nor source contents have been independently verified here. The extracted entry bears the label "(New)" and describes the question as a Liouville type assertion on CR $3$-manifolds.
• The registry's proposer field names Buzzard, Fornaess, Krantz, whereas the extracted problem itself is signed Puqi Tang, with supplied contact tang@math.purdue.edu. The question is therefore attributed here to Tang within that collected problem list.