arXiv subjects

Sushil Gorai

Publications and source records attributed to Sushil Gorai.

22 records · Page 2Linked to original sources

arXiv2013

Function theory and holomorphic maps on symmetric products of planar domains

We show that the $\bar{\partial}$-problem is globally regular on a domain in $\mathbb{C}^n$, which is the $n$-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary

math.CV↗

arXiv2011

On the polynomial convexity of the union of more than two totally-real planes in C^2

In this paper we shall discuss local polynomial convexity at the origin of the union of finitely many totally-real planes through $0 \in\mathbb{C}^2$. The planes, say $P_0,..., P_N$, satisfy a mild transversality condition that enables us to view them in Weinstock normal form, i.e. $P_0=\mathbb{R}^2$ and $P_j=M(A_j):=(A_j+i\mathbb{I})\mathbb{R}^2$, $j=1,...,N$, where each $A_j$ is a $2\times 2$ matrix with real entries. Weinstock has solved the problem completely for N=1 (in fact, for pairs of transverse, maximally totally-real subspaces in $\mathbb{C}^n\, \forall n\geq 2$). Using a characterization of simultaneous triangularizability of $2\times 2$ matrices over the reals, given by Florentino, we deduce a sufficient condition for local polynomial convexity of the union of the above planes at $0\in \mathbb{C}^2$. Weinstock's theorem for $\mathbb{C}^2$ occurs as a special case of our result. The picture is much clearer when N=2. For three totally-real planes, we shall provide an open condition for local polynomial convexity of the union. We shall also argue the optimality (in an appropriate sense) of the conditions in this case.

math.CV↗

arXiv2010

Uniform algebras generated by holomorphic and close-to-harmonic functions

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a nowhere-holomorphic harmonic function on D that is continuous up to the boundary --- equals the algebra of continuous functions on closure(D). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is "small" in a certain sense.

math.CV↗

arXiv2010

Local polynomial convexity of the union of two totally real surfaces at their intersection

We consider the following question: Let $S_1$ and $S_2$ be two smooth, totally-real surfaces in $\mathbb{C}^2$ that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is $S_1 \cup S_2$ locally polynomially convex at the origin? If $T_0S_1 \cap T_0S_2=\{0\}$, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dimension of $T_0S_1 \cap T_0S_2$ over the field of real numbers is 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.

math.CV↗