0
$\begingroup$

Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:

  • Does $g_\varepsilon$ converge to the flat metric on the interval $\theta\in [0,\pi]$ as $\varepsilon\to 0\text{?}$
  • Do the eigenfunctions $\lambda_k(-\Delta_{g_\varepsilon})\to \lambda_k([0,\pi]),$ where $\lambda_k$ are Neumann eigenvalues and $\lambda_k([0,\pi])$ denotes the eigenvalues on the interval $[0,\pi]\text{?}$
Improve this question
$\endgroup$
3
  • $\begingroup$ $\bar g$ is not a Riemannian metric. How do you define its Laplace operator? $\endgroup$
    – Moishe Kohan
    Commented Mar 23 at 0:26
  • $\begingroup$ i meant the flat metric on the interval $[0,\pi]$. $\endgroup$
    – Student
    Commented Mar 23 at 0:50
  • $\begingroup$ You mean convergence in the Gromov-Hausdorff sense? $\endgroup$
    – Ben McKay
    Commented Mar 23 at 13:45

0

Reset to default

You must log in to answer this question.