Several Complex Variables Problem List — Problem 83

Definitions and mathematical statement

Let $M$ and $N$ be strongly pseudoconvex CR manifolds of real dimension $2n+1$, with contact bundles $H_M\subset TM$ and $H_N\subset TN$, and CR complex structures $J_M$ and $J_N$ on these bundles. Fix a diffeomorphism $f:M\to N$ that is quasiconformal in the source's sense: its differential preserves the contact bundles, \[ {\,\mathrm{d}} f_x(H_{M,x})=H_{N,f(x)}\qquad(x\in M), \] and distorts the CR structures boundedly. The source does not give a numerical definition of this boundedness independent of the metric choices.

A Hermitian metric on a contact bundle means a positive definite real inner product invariant under its CR complex structure. Given such metrics $g_M$ and $g_N$, write $A_x={\,\mathrm{d}} f_x|_{H_{M,x}}$. The ratio of the longest to shortest semiaxes of the image of the unit sphere is \[ K_{g_M,g_N}(x) = \frac{\displaystyle\max_{g_{M,x}(v,v)=1} \sqrt{g_{N,f(x)}(A_xv,A_xv)}} {\displaystyle\min_{g_{M,x}(v,v)=1} \sqrt{g_{N,f(x)}(A_xv,A_xv)}}. \] This is one explicit interpretation of the source's sphere-to-ellipsoid measurement. The source emphasizes that the measurement depends on the chosen Hermitian metrics when $n>1$.

Problem. For the fixed quasiconformal diffeomorphism $f$, can Hermitian metrics on the two contact bundles be chosen so that the distortion is minimal?

Unresolved formulation. The source does not specify whether minimality is pointwise or concerns a global distortion functional, nor does it specify the regularity or normalization of the admissible metrics. For example, under the additional interpretation that all smooth Hermitian metric pairs are admissible and that distortion means \[ K_\infty(f;g_M,g_N)=\sup_{x\in M}K_{g_M,g_N}(x), \] the question would ask whether there exists an admissible pair $(g_M^*,g_N^*)$ satisfying \[ K_\infty(f;g_M^*,g_N^*) = \inf_{(g_M,g_N)}K_\infty(f;g_M,g_N). \] This is a conditional precise formulation, not a specification supplied by the source. No compactness assumption on either manifold is stated, and a global supremum may be infinite for some metric pairs.

Short English statement

For a fixed quasiconformal diffeomorphism between strongly pseudoconvex CR manifolds, can one choose Hermitian metrics on their contact bundles that minimize its distortion? The source highlights metric dependence for $n>1$, but leaves the distortion functional and the meaning of minimization unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800283$, exact source code catalogue entry, title "Several Complex Variables Problem List --- Problem 83." The supplied registry labels the problem open; this label is not an independently verified literature assessment.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 83. Supplied source URL: https://arxiv.org/abs/math/9509202. The registry attributes the proposal to Buzzard, Fornaess, and Krantz and gives the year 1995; the quoted problem itself is signed by Puqi Tang, tang@math.purdue.edu.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This is supplied dated provenance, not independent verification of access, source contents, or current mathematical status. The source's "(New)" marker is retained as historical source wording, not a present-day novelty claim.