The purpose of this article is to show that there exists a smooth real hypersurface germ (M p), of D''Angelo infinite type in C2 such that it does not admit any (singular) holomorphic curve that has infinite order contact with M at p .
VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 Original Article A Note on Infinite Type Germs of a Real Hypersurface in Nguyen Thi Kim Son1,*, Chu Van Tiep2 Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien, Bac Tu Liem, Hanoi Department of Mathematics, Da Nang University of Education at Da Nang, 459 Ton Duc Thang, Lien Chieu, Da Nang Received 02 April 2019 Revised 10 April 2019; Accepted 10 April 2019 Abstract: The purpose of this article is to show that there exists a smooth real hypersurface germ M , p of D'Angelo infinite type in such that it does not admit any (singular) holomorphic curve that has infinite order contact with M at p 2010 Mathematics Subject Classification Primary 32T25; Secondary 32C25 Key words and phrases: Holomorphic vector field, automorphism group, real hypersurface, infinite type point Introduction Let M , p be a germ at p of a real smooth hypersurface in n and let r be a local defining function for M near p The normalized order of contact of the curve with M at p is defined by M , , p : r , Where p and is the vanishing order of t at t 0, r is the vanishing order of r t at t The D'Angelo type of M at p is defined by Corresponding author Email address: kimsonnt.0611@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4345 82 N.T.K Son, C.V Tiep / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 M , p sup M , , p sup where the supremum is taken over all germs : with p Here and in what follows, z 83 r , n of non-constant holomorphic curves : z and : 1 We say that p is of D'Angelo finite type if M , p and of D'Angelo infinite type if otherwise Throughout the paper, we assume that M , p is of D'Angelo infinite type Then, there exists a sequence of non-constant holomorphic curves n such that r n as n It is natural n to ask whether there exists a variety that has infinite order contact with M , p This question pertains to the regularity issue of -Neumann problems over pseudoconvex domains (see [1, 2, 3, 4], and the references therein) If M , p is real-analytic, then by using the ideal theoretic method L Lempert and J P D'Angelo [5, 6] showed that M contains a nontrivial holomorphic curve passing through p For a germ of a real analytic hypersurface in result by using a geometric construction , we refer the interested reader to [7] for a proof of this For the case when M , p is a real smooth hypersurface in n , J E Fornæss, L Lee and Y Zhang [8] proved that if M , p , then there exists a formal complex curve in the hypersurface M through p However, Kang-Tae Kim and V T Ninh [9, Proposition 4] asserted independently that there is a formal curve t a t j 1 j j , t which has infinite order contact with M at p for the case M In [9], Kang-Tae Kim and V T Ninh pointed out that in general there is no such a regular holomorphic curve We ensure that this result still holds even for singular holomorphic curve Namely, our aim is to prove the following theorem Theorem There exists a hypersurface germ M , in with M , that does not admit any (singular) holomorphic curve that has infinite order contact with M at We now briefly sketch the idea of proof of Theorem As in the proof of Example in [9], we f n C0 with supp f n tending to 0 is harmonic in a sufficiently small disc in supp f n for each n * Moreover, the construct a certain sequence of smooth functions such that f n series f n 1 n converges uniformly on M can be defined by to a smooth function f z Then the desired hypersurface N.T.K Son, C.V Tiep / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 84 M z1 , z2 : Re z1 f z2 0 , which finishes the proof of Theorem In this paper, we only deal with a smooth real hypersurface in Theorem remains valid even for higher-dimensional hypersurfaces However, the statement of Proof of Theorem Proof of Theorem The proof of this theorem proceeds along the same lines as that of Example in [9] For the convenience of the reader, we shall provide some crucial arguments given in [9] First M n n1 of all, let n n1 be a sequence of real numbers such that M n 2n is a sequence in with n as n Let n n n 2 , n * , where be a strictly decreasing sequence of positive numbers with n as n such that, for each n * , there exists a holomorphic function g n on n satisfying that g n n and M g n j n 0 For instance, for every n M n 2n n 2 * , we define g n z : if j n, if n | j n n z n n n , where n : M 2n n and (see [9, Example 2] For each n 1, 2, , denote by f n z the C -smooth function on Re g n z fn z 0 such that if z n 1 , if z n Then, one can see that and f n n and Mn j fn j z 0 if j n, if n | j Now let n be an increasing sequence of positive numbers such that n max 1, k l f n z k z l : k , l , k l n , (1) N.T.K Son, C.V Tiep / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 represents the supremum norm Let us define a function where f n z : f n n z for each n n nn * 85 f n by setting Then, by the repeated use of the chain rule, we obtain k fn k fn z n z , k 0,1, z k n2nnk z k This together with (1) implies that Mn k fn 2n k z 0 by setting f z : Let us define a function f if k n, if n | k f z Then, for every n 1 n k, j , a direct computation shows that n 1 k l f n z z k z l k l n k l n 1 n n k l n 1 n nn k l k l f n z k l n k l 1 z z n k l 1 n n k l f n z z k z l n k l f n z k l z z n k l 1 n Hence, this ensures that f C Next, let us fix a sequence of prime numbers pn n1 with pn as n Then it is easy to see that pn pn f fk 0 pn z pn k z pn 1 j 2 pn pn pn f f f j 0 j p pn z pn z pn n j pn 1 z Mpn pn2 We now define a hypersurface germ M at 0, by setting M z1 , z2 : Re z1 f z2 0 N.T.K Son, C.V Tiep / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 86 We shall show that M ,0 To this, for each N , consider a holomorphic curve N z1 , z2 defined on t : t N 1 by N N n 1 n nn z1 t Then, we have N t g n nt ; z2 t t f t Furthermore, since f n for n 1, 2, n N 1 n n , it follows that N N , and hence M , We finally prove that there does not exist a (singular) holomorphic curve : ,0 ,0 , such that Note that, by a change of variables, we can assume that such a (singular) holomorphic curve is represented by a parametrization t h t , t m integer m , where h is a holomorphic function on a neighborhood of the origin in for some positive Indeed, suppose o t otherwise that such a holomorphic curve exists Then t Re h t f t m and thus 0 pn m z pn m Re h z f z m z 0 pn m pn m h z pn m z pn m f zm z 0 pn pn h m ! z pn m z pn pn m f z z 0 pn m p Mpn h m ! n pn m z pn2 Consequently, h pn m 0 m! Mpn n n 1 , and moreover, since n ! 2 pn 3 n pn M n 2nn n we have pn m 0 pn m ! h pn m pn m! M p m! m pn m p m pn m pn m !2 pn2 pn n n pn p 2mpn m mn 1 pn pn m 1 Therefore, we obtain limsup N N h N 0 N! limsup pn m pn h pn m pn m ! pn pn lim pn m 1 n and N.T.K Son, C.V Tiep / VNU Journal of Science: Mathematics – Physics, Vol 35, No (2019) 82-87 87 This implies that the Taylor series of h z at has radius of convergence 0, which is absurd since h is holomorphic in a neighborhood of the origin Hence, the proof is complete Acknowledgments It is a pleasure to thank Ninh Van Thu and Nguyen Ngoc Khanh for stimulating discussions on this material References [1] [2] [3] [4] [5] [6] [7] [8] [9] J P D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann of Math 115 (1982) 615-637 D Catlin, Necessary conditions for subellipticity of the -Neumann problem, Ann of Math 117 (1) (1983) 147-171 D Catlin, Boundary invariants of pseudoconvex domains, Ann of Math 120 (3) (1984) 529-586 D Catlin, Subelliptic estimates for the -Neumann problem on pseudoconvex domains, Ann of Math 126 (1) (1987) 131-191 J.P D'Angelo, Several complex variables and the geometry of real hypersurfaces, CRC Press, Boca Raton, 1993 L.Lempert, On the boundary regularity of biholomorphic mappings, Contributions to several complex variables, Aspects Math E9 (1986) 193-215 J.E Fornaess, L Lee, Y Zhang, Formal complex curves in real smooth hypersurfaces, Illinois J Math 58 (1) (2014) 1-10 J.E Fornaess, B Stensones, Infinite type germs of real analytic pseudoconvex domains in , Complex Var Elliptic Equ 57 (6) (2012) 705-717 K.T Kim, V.T Ninh, On the tangential holomorphic vector fields vanishing at an infinite type point, Trans Amer Math Soc 367(2) (2015) 867-885 ... Infinite type germs of real analytic pseudoconvex domains in , Complex Var Elliptic Equ 57 (6) (2012) 705-717 K.T Kim, V.T Ninh, On the tangential holomorphic vector fields vanishing at an infinite type. .. Theorem In this paper, we only deal with a smooth real hypersurface in Theorem remains valid even for higher-dimensional hypersurfaces However, the statement of Proof of Theorem Proof of Theorem... contact, and applications, Ann of Math 115 (1982) 615-637 D Catlin, Necessary conditions for subellipticity of the -Neumann problem, Ann of Math 117 (1) (1983) 147-171 D Catlin, Boundary invariants