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Mathematical Colloquium - On a conjecturte of H. Cartan

00:00 01-11-2010

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

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