Infinitesimal CR automorphisms and stability groups of infinite-type models in C2

Abstract

The purpose of this article is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in C 2 . The decompositions of infinitesimal CR automorphisms are also given.

Infinitesimal CR automorphisms and stability groups of nonminimal infinite type models in $\mathbb C^2$

“…For a certain class of rigid hypersurfaces of finite type in the sense of D'Angelo in C 2 , we refer the reader to [10] which addresses the existence of infinitesimal CR automorphisms. However, if we move our attention to the case of rigid hypersurfaces of infinite type, then we necessarily encounter more complicated procedure to get such geometric object due to the computational difficulty and the lack of literatures in the setting of infinite type (see [3] and the references therein). As a significant result which has inspired the present paper, Hayashimoto and Ninh [3] investigated an infinite type model pM 1 P , 0q in C 2 which is defined by (2)…”

“…Moreover, a ruled hypersurface is known as a crucial prototype in considering local equivalence problem of nonminimal real analytic hypersurfaces in C 2 . We further say that a germ at p of a real hypersurface pN, pq in C 2 is m-nonminimal (m ě 1) at p if there exist local coordinates pz, wq P C 2 , p corresponds to 0, close by 0, such that N is given by an equation of the form (3) Im w " pRe wq m ψpz, z, Re wq, where ψpz, 0, Re wq " ψp0, z, Re wq " 0 and ψpz, z, 0q does not vanish identically (cf. [4] and [8]).…”

Rigid hypersurfaces and infinitesimal CR automorphisms

“…The following theorem shows that, conversely to Theorem 4.1, holomorphic vector fields determine the form of the defining function. Instead to give a proof of this theorem, we shall give two examples [4].…”

“…From these theorems, if the set of all points of infinite type is one point, the set of infinitesimal CR automorphisms and CR automorphism groups are small. If the connected component of the origin in S ∞ is not the origin, Aut(M P ) and aut(M P ) are still small [4]. Assume that M ⊂ C n is a real analytic connected hypersurface.…”

“…The main theorem is to determine all infinitesimal CR automorphisms of rigid hypersurfaces of infinite type (see Theorem 4.1) and, in some case, to determine defining functions of rigid hypersurfaces of infinite type by infinitesimal CR automorphisms (see Theorem 5.1). These are a joint work with Ninh Van Thu [4].…”