# First eigenvalue in Ricci flow.

In the picture below, why is $\lambda(g_{ij})$ the first eigenvalue of $-4\Delta+R$?

The first eigenvalue is strange to me. After reading the wiki about it, I find

$$\lambda_1=\inf\limits_{u\ne 0}\frac{\int_\Omega|\nabla u|^2}{\int_\Omega|u|^2}$$

where $u$ is the solution to $$\begin{cases} \Delta u + \lambda u = 0 & \text{in \Omega} \\ \left. u\right|_{\partial \Omega} = 0 \end{cases}$$

But seemingly, it is not the first eigenvalue in pictures 1,2 and 3. Also, why is the first eigenfunction $u_0>0$ in picture 3 bigger than $0$?

Pictures 1, 2 and 3 are from the 203rd page of this paper.

• Do you really need Picture 1? You could write the Dirichlet problem yourself. – Silvia Ghinassi Nov 24 '15 at 14:30
• @SilviaGhinassi Sorry, I don't know how to write the big $\{$ – lanse7pty Nov 25 '15 at 1:10
• \begin{cases} \Delta u + \lambda u = 0 & \text{in $\Omega$} \\ \left. u\right|_{\partial \Omega} = 0 \end{cases} between double \$. – Silvia Ghinassi Nov 25 '15 at 1:22
• @SilviaGhinassi Thanks ,I have edit it . – lanse7pty Nov 25 '15 at 1:27