Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 17 min ago
PreprintarXiv licencearXiv

A Bernstein Theorem for the Self-Shrinking J-Equation and Some Generalizations

  • Yiyang Pan
  • Wenlong Wang
Open at sourcePDFDOICite

Abstract

We prove that every entire smooth strictly plurisubharmonic solution of the self-shrinking J-equation on ℂn is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to self-shrinking equations associated with a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the adjacent inverse complex Hessian quotient operators −σk−1/σk, 1≤ k≤ n. For the remaining quotients −σl/σk with l<k−1, we prove the same rigidity conclusion under the additional assumption that the solution is proper.

Related

  1. IC-rigidity and generalizations of a theorem of Mahler
    Bryna Kra, Scott Schmieding
    arXivPreprintarXiv licenceFull textDynamical systemsarXiv
  2. Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in ℝ⁴
    Xavier Cabré, Giovanni Catino, Luciano Mari +2
    Transactions of the American Mathematical SocietyPeer-reviewedLicence not statedFull text10 citedGeneral mathematicsOpenAlex
  3. Gauge invariant generalizations of the Proca equation and the Yang-Mills-Proca equation
    Nikolay Marchuk
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  4. The stable Bernstein theorem in ℝ7
    Han Hong, Haizhong Li, Gaoming Wang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  5. A Bernstein type theorem for a two dimensional translating soliton in ℝ4 with finite total curvature
    Xiaoli Han, Jiayu Li, Jun Sun +1
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{pan2026bernstein,
  title = {{A Bernstein Theorem for the Self-Shrinking $J$-Equation and Some Generalizations}},
  author = {Yiyang Pan and Wenlong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.29546},
  eprint = {2606.29546},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.29546},
}
RIS
TY  - UNPB
TI  - A Bernstein Theorem for the Self-Shrinking J-Equation and Some Generalizations
AU  - Yiyang Pan
AU  - Wenlong Wang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2606.29546
UR  - https://arxiv.org/abs/2606.29546
AB  - We prove that every entire smooth strictly plurisubharmonic solution of the self-shrinking J-equation on ℂ^n is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to self-shrinking equations associated with a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the adjacent inverse complex Hessian quotient operators −σ_k−1/σ_k, 1≤ k≤ n. For the remaining quotients −σ_l/σ_k with l<k−1, we prove the same rigidity conclusion under the additional assumption that the solution is proper.
ER  -