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PreprintCC BY 4.0arXiv

Polynomial convexity and non-polynomial convexity of totally real discs in ℂ2

  • Golam Mostafa Mondal
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Abstract

We study the polynomial convexity of smooth totally real discs contained in the boundaries of strongly pseudoconvex domains in ℂ2. Using Wermer's classical example, we construct a bounded domain with smooth strongly pseudoconvex boundary containing a non-polynomially convex totally real disc. We also construct a nonsingular Levi-flat hypersurface in a non-Runge domain containing such a disc. This provides a partial negative answer, to a conjecture proposed in an earlier work. Furthermore, we prove that the Wermer graph over the closed disc of radius R is polynomially convex for every R∈ (0,1). Finally, we establish a sufficient condition for the polynomial convexity of totally real discs contained in the smoothly bounded strongly pseudoconvex domains in ℂ2.

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Cite

BibTeXRIS
BibTeX
@misc{mondal2026polynomial,
  title = {{Polynomial convexity and non-polynomial convexity of totally real discs in $\mathbbC^2$}},
  author = {Golam Mostafa Mondal},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02004},
  eprint = {2611.02004},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02004},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Polynomial convexity and non-polynomial convexity of totally real discs in ℂ^2
AU  - Golam Mostafa Mondal
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02004
UR  - https://arxiv.org/abs/2611.02004
AB  - We study the polynomial convexity of smooth totally real discs contained in the boundaries of strongly pseudoconvex domains in ℂ^2. Using Wermer's classical example, we construct a bounded domain with smooth strongly pseudoconvex boundary containing a non-polynomially convex totally real disc. We also construct a nonsingular Levi-flat hypersurface in a non-Runge domain containing such a disc. This provides a partial negative answer, to a conjecture proposed in an earlier work. Furthermore, we prove that the Wermer graph over the closed disc of radius R is polynomially convex for every R∈ (0,1). Finally, we establish a sufficient condition for the polynomial convexity of totally real discs contained in the smoothly bounded strongly pseudoconvex domains in ℂ^2.
ER  -