### Mathematical Colloquium - On a conjecturte of H. Cartan

00:00 01-11-2010
&amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;Abstract:&amp;amp;nbsp;&amp;amp;nbsp;&amp;lt;/font&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;&amp;lt;font size="2" face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/span&amp;gt;&amp;amp;nbsp;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;In 1928 H. Cartan proved an extension of Montel&amp;amp;#39;s normality criterion &amp;lt;/span&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;to holomorphic curves in complex projective plane \$P^2\$. He also conjectured &amp;lt;/span&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;that a similar result is true for holomorphic curves in \$P^n\$ for every \$n\$. &amp;lt;/span&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;I give a counterexample to this conjecture for any \$n&amp;amp;gt;2\$ and show how to modify Cartan&amp;amp;#39;s&amp;lt;br /&amp;gt;conjecture so that it becomes true, at least for \$n=3\$.&amp;lt;/span&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/p&amp;gt; &amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;font size="2"&amp;gt;&amp;amp;nbsp;&amp;lt;/font&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/p&amp;gt;