Asian Journal of Mathematics

Volume 22 (2018)

Number 3

Special issue in honor of Ngaiming Mok (2 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

Jet vanishing orders and effectivity of Kohn’s algorithm in dimension 3

Pages: 545 – 568

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n3.a8

Authors

Sung-Yeon Kim (Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea)

Dmitri Zaitsev (School of Mathematics, Trinity College Dublin, Ireland)

Abstract

We propose a new class of geometric invariants called jet vanishing orders, and use them to establish a new selection algorithm in the Kohn’s construction of subelliptic multipliers for special domains in dimension 3, inspired by the work of Y.-T. Siu. In particular, we obtain effective termination of our selection algorithm with explicit bounds both for the steps of the algorithm and the order of subellipticity in the corresponding subelliptic estimates. Our procedure possesses additional features of certain stability under high order perturbations, due to deferring the step of taking radicals to the very end, see Remark 1.2 for more details.

We further illustrate by examples the sharpness in our technical results (in Section 3) and demonstrate the complete procedure for arbitrary high order perturbations of the Catlin–D’Angelo example in Section 5.

Our techniques here may be of broader interest for more general PDE systems, in the light of the recent program initiated by the breakthrough paper of Y.-T. Siu.

Keywords

finite type, multiplier ideals, Jacobian, subelliptic estimates, jets, germs of holomorphic functions, order of contact, $\overline{\partial}$-Neumann problem

2010 Mathematics Subject Classification

32B10, 32S05, 32T25, 32T27, 32V15, 32V35, 32V40, 32W05

Full Text (PDF format)

The first author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (grant number NRF-2015R1A2A2A11001367).

Received 9 February 2017

Accepted 7 March 2018

Published 8 August 2018