4
$\begingroup$

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where $X$ is a Kähler manifold. Do we have that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega),$$where $K(f^*\omega)$ denotes the Gaussian curvature of pullback of metric and $K(\omega)$ is the bisectional curvature?

$\endgroup$
3
  • $\begingroup$ If it is complex-differentiable then $C$ is saddle, by Gauss formula the intrinsic curvature of $C$ is smaller than the curvature of $X$ in the tangent direction. Hence you get $\le$ in your inequality, which implies $<$ in most of the cases. If the surface is just smooth then no hope. $\endgroup$ Jun 6, 2016 at 11:28
  • $\begingroup$ @AntonPetrunin I'm not sure if I follow. Could you elaborate a bit? $\endgroup$
    – user44803
    Jun 6, 2016 at 18:24
  • $\begingroup$ Well, what is the problem? $\endgroup$ Jun 7, 2016 at 6:42

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.