In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k+o(|z|^{k})\}$ or $M_t := \left\{ (z,w)\in\mathbb{C}^2 : w=(z+t\overline{z})^k+o(|z|^k) \right\}$, for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\frac{\pi}{k}\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min\left\{\sin\left(\frac{\pi}{k}\right),\frac{1}{k-2}\right\}$.
<jats:p>
The surfaces in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>ℂ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:math>
with an isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, under certain geometric condition, can be reduced — up to biholomorphism of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>ℂ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
</mml:math>
— to a one-parameter family of the form
</jats:p>
<jats:p>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block">
<mml:mrow>
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
<mml:mo>:</mml:mo>
<mml:mo>=</mml:mo>
<mml:mfenced close="}" open="{" separators="">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>∈</mml:mo>
<mml:msup>
<mml:mi>ℂ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>:</mml:mo>
<mml:mi>w</mml:mi>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mi>z</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mover>
<mml:mi>z</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mo>+</mml:mo>
<mml:mi>t</mml:mi>
<mml:mi>z</mml:mi>
<mml:msup>
<mml:mover>
<mml:mi>z</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>+</mml:mo>
<mml:mfrac>
<mml:msup>
<mml:mi>t</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mn>3</mml:mn>
</mml:mfrac>
<mml:msup>
<mml:mover>
<mml:mi>z</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
<mml:mn>3</mml:mn>
</mml:msup>
<mml:mo>+</mml:mo>
<mml:mi>o</mml:mi>
<mml:mfenced close=")" open="(">
<mml:msup>
<mml:mrow>
<mml:mo>|</mml:mo>
<mml:mi>z</mml:mi>
<mml:mo>|</mml:mo>
</mml:mrow>
<mml:mn>3</mml:mn>
</mml:msup>
</mml:mfenced>
</mml:mfenced>
<mml:mo>,</mml:mo>
<mml:mspace width="1em"/>
<mml:mi>t</mml:mi>
<mml:mo>∈</mml:mo>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>∞</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</jats:p>
<jats:p>
near the origin. We prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:math>
is not locally polynomially convex if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo><</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
. The local hull contains a ball centred at the origin if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo><</mml:mo>
<mml:msqrt>
<mml:mn>3</mml:mn>
</mml:msqrt>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:math>
. We also prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:math>
is locally polynomially convex for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>≥</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:math>
. We show that, for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msqrt>
<mml:mn>3</mml:mn>
</mml:msqrt>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>≤</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo><</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
, the local hull of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:math>
contains a one-parameter family of analytic discs passing through the origin. We also prove that, if we remove the higher order terms from the graphing function of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>M</mml:mi>
<mml:mi>t</mml:mi>
</mml:msub>
</mml:math>
, it is locally polynomially convex for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>≥</mml:mo>
<mml:mfrac>
<mml:msqrt>
<mml:mrow>
<mml:mn>15</mml:mn>
<mml:mo>-</mml:mo>
<mml:msqrt>
<mml:mn>33</mml:mn>
</mml:msqrt>
</mml:mrow>
</mml:msqrt>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:msqrt>
<mml:mn>2</mml:mn>
</mml:msqrt>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
. Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.
</jats:p>
Sushil Gorai· Annales de l'Institut Fourie...· 0 citations
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