Speaker: Prof. Hervé Gaussier (Institut Fourier, France)
Time: 9h00, Friday, April 14, 2017
Location: Room 3, Builiding A14, Institute of Mathematics, Hanoi
Abstract: If $f : \mathbb C \rightarrow \mathbb C \mathbb P^n$ is a smooth map, the energy of $f$ is defined by:
$$E(f):=\limsup_{R \rightarrow \infty}\frac{1}{\pi R^2}\int_{D(0,R)}f^* \omega_{FS}$$
where $\omega_{FS}$ denotes the Fubini Study form on $\mathbb C \mathbb P^n$ and $D(0,R)$ is the disk centered at the origin with radius $R$.
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We prove that the energy of a uniformly continuous quasiconformal map in $\mathbb C \mathbb P^1$, avoiding two points, is equal to zero. As an application, we show that the energy of an entire pseudoholomorphic curve in $\mathbb C \mathbb P^2$, avoiding three J-lines in general position, is equal to zero. Finally, unlike the holomorphic case, we construct a one paramater family $(f_a)_{0 < a < 2}$ of uniformly continuous quasiconformal maps in $\mathbb C \mathbb P^1$, avoiding two points, such that for every $0 < a < 2$:
$$\lim_{R \rightarrow \infty}\frac{1}{\pi R^a}\int_{D(0,R)}(f_a)^* \omega_{FS} = \infty.$$
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