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## 3. Mappings

1. #### Problem 3.1.

[D’Angelo] Given proper holomorphic maps $f,g: \mathbb{B}^n\to \mathbb{B}^N$, how does one show that $f$ and $g$ are not homotopic? Find homotopy invariants for proper holomorphic maps between balls.

$\bullet$ Special case (Lebl): Is the Faran map from $\mathbb{B}^2$ to $\mathbb{B}^4$ given by $$(z,w) \to (z^3,\sqrt{3}zw, w^3,0)$$ homotopic to the map $(z,w)\to (z,w, 0, 0)$?
It is known that the Faran map is homotopic to this embedding when the target dimension is 5, and it is not homotopic when the target dimension is 3.
• #### Problem 3.2.

[D’Angelo] Let $R(z,\overline{z})$ be a bihomogeneous polynomial. Find necessary and sufficient conditions such that there exists an integer $N$ with $$(R(z,\overline{z}))^N=\sum_{j=1}^{m}|p_j(z)|^2.$$ Here $p_j(z)$’s are linearly independent holomorphic polynomials. See [MR1682713] and [MR2770459].
• #### Problem 3.3.

[Ebenfelt] Let $R(z,\overline{z})$ be a Hermitian polynomial that is a sum of squares (SOS) $$R(z,\overline{z})=\sum_{j=1}^m|p_j(z)|^2,$$ where $p_j(z)$’s are linearly independent holomorphic polynomials. Suppose that $R(z,\overline{z})$ is divisible by $||z||^2$ (i.e. $R(z,\overline{z})=||z||^2A(z,\overline{z})$). What are the possible values of $m$?

$\bullet$ Huang’s lemma [MR1703603] implies that $m$ is either 0 or at least $n$.

$\bullet$ See [MR2869101] for a generalization of the Huang’s lemma when $||z||^2$ is replaced by $||z||^{2d}$. Consider the same question in this case.
• #### Problem 3.4.

[Zaitsev and Mok] Let $D_1$ and $D_2$ be two bounded symmetric domains such that neither is a ball. Let $F:D_1\to D_2$ be a proper holomorphic map. Is $F$ a trivial map, that is $F(z)=(z,g(z))$, in suitable coordinates, where $g(z)$ is a vector valued holomorphic function? By a result of Tsai, the answer is yes when the rank (as a bounded symmetric domain) of $D_2$ is not greater than the rank of $D_1$.
• #### Problem 3.5.

[Epstein] It was proven by Eliashberg that any embeddable CR-structure on $\mathbb{S}^3$ bounds a Stein manifold $X\simeq \mathbb{B}^4$ with a strictly plurisubharmonic exhaustion function with a single critical point. Can $X$ always be embedded into $\mathbb{C}^2$?

Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.