Thời gian: 9h Thứ 6 ngày 14/4/2017
Địa điểm: Phòng 3 nhà A14, Viện Toán học 18 Hoàng Quốc Việt
Tóm tắt: 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 $0 a 2$:
$$lim_{R rightarrow infty}frac{1}{pi R^a}int_{D(0,R)}(f_a)^* omega_{FS} = infty.$$