I have some trouble solving the following problem:
Let $\gamma$ be a geodesic in a Riemann surface $(\Sigma,\lambda^2(z)|dz|^2)$ and $E$ be the energy of $\gamma$. Let $\gamma+s\eta$ be a smooth variation of $\gamma$ with $\eta(0)=\eta(1)=0$. Compute$$\frac{d^2}{ds^2}E(\gamma+s\eta)$$
I haven't learned Riemann Geometry, so I try to compute directly, here are some of my steps:
\begin{equation*} \begin{split} \frac{d^2}{ds^2}E(\gamma+s\eta)|_{s=0}&=\frac{d^2}{ds^2}\int_0^1(\lambda^2(\gamma+s\eta)|\gamma'+s\eta'|^2)d t\bigg|_{s=0}\\ &=\int_0^1\frac{d^2}{ds^2}(\lambda^2(\gamma+s\eta)|\gamma'+s\eta'|^2)\bigg|_{s=0}d t\\ &=\int_0^1\left(\frac{d^2}{ds^2}\lambda^2(\gamma+s\eta)\right)|\gamma'+s\eta'|^2+2\frac{d}{ds}\lambda^2(\gamma+s\eta)\frac{d}{ds}|\gamma'+s\eta'|^2\\ &\qquad+\lambda^2(\gamma+s\eta)\left(\frac{d^2}{ds^2}|\gamma'+s\eta'|^2\right)\bigg|_{s=0}d t\\ &=\int_0^12\eta|\gamma'|^2((\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\lambda_{\gamma}+\lambda(\lambda_{\gamma\gamma}\eta+\lambda_{\gamma\bar\gamma}\bar\eta))\\ &\qquad+2\bar\eta|\gamma'|^2((\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\lambda_{\bar\gamma}+\lambda(\lambda_{\bar\gamma\gamma}\eta+\lambda_{\bar\gamma\bar\gamma}\bar\eta))\\ &\qquad+4\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)(\gamma'\bar{\eta'}+\bar{\gamma'}\eta')+2\lambda^2|\eta'|^2d t\\ &=\int_0^12\eta|\gamma'|^2((\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\lambda_{\gamma}+\lambda(\lambda_{\gamma\gamma}\eta+\lambda_{\gamma\bar\gamma}\bar\eta))\\ &\qquad+2\bar\eta|\gamma'|^2((\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\lambda_{\bar\gamma}+\lambda(\lambda_{\bar\gamma\gamma}\eta+\lambda_{\bar\gamma\bar\gamma}\bar\eta))\\ &\qquad-2\bar\eta\big((\lambda_\gamma\gamma'+\lambda_{\bar\gamma}\bar\gamma')(\lambda_\gamma\eta+\lambda_{\bar\gamma}\bar\eta)\gamma'+\lambda((\lambda_{\gamma\gamma}\gamma'+\lambda_{\gamma\bar\gamma}\bar\gamma')\eta\\ &\qquad+\lambda_{\gamma}\eta'+(\lambda_{\bar\gamma\gamma}\gamma'+\lambda_{\bar\gamma\bar\gamma}\bar\gamma')\bar\eta+\lambda_{\bar\gamma}\bar\eta')\gamma'+\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\gamma''\big)\\ &\qquad-2\eta\big((\lambda_\gamma\gamma'+\lambda_{\bar\gamma}\bar\gamma')(\lambda_\gamma\eta+\lambda_{\bar\gamma}\bar\eta)\bar\gamma'+\lambda((\lambda_{\gamma\gamma}\gamma'+\lambda_{\gamma\bar\gamma}\bar\gamma')\eta\\ &\qquad+\lambda_{\gamma}\eta'+(\lambda_{\bar\gamma\gamma}\gamma'+\lambda_{\bar\gamma\bar\gamma}\bar\gamma')\bar\eta+\lambda_{\bar\gamma}\bar\eta')\bar\gamma'+\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)\bar\gamma''\big)\\ &\qquad+2\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)(\gamma'\bar{\eta'}+\bar{\gamma'}\eta')+2\lambda^2|\eta'|^2d t\\ &=\int_0^1-2\bar\eta\left((\lambda\lambda_{\gamma\gamma}-\lambda_{\gamma}^2)\gamma'^2\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\gamma'^2\bar\eta+\lambda\gamma'(\lambda_{\gamma}\eta'+\lambda_{\bar\gamma}\bar\eta')\right)\\ &\qquad-2\eta\left((\lambda\lambda_{\bar\gamma\bar\gamma}-\lambda_{\bar\gamma}^2)\bar\gamma'^2\bar\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\bar\gamma'^2\eta+\lambda\bar\gamma'(\lambda_{\bar\gamma}\bar\eta'+\lambda_{\gamma}\eta')\right)\\ &\qquad+2\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)(\gamma'\bar{\eta'}+\bar{\gamma'}\eta')+2\lambda^2|\eta'|^2d t\\ \end{split} \end{equation*}
From the last step that I post I’m stuck. Can anyone give me a hint? or point out what’s wrong in my calculation? Thank you very much!
OK, I find the way to compute. It seems I cannot post it in the comment since it exceeds the character limitation. So I put it in the Answer area. \begin{equation*} \begin{split} \qquad\qquad\qquad&=\int_0^1-2\bar\eta\left((\lambda\lambda_{\gamma\gamma}-\lambda_{\gamma}^2)\gamma'^2\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\gamma'^2\bar\eta+\lambda\gamma'(\lambda_{\gamma}\eta'+\lambda_{\bar\gamma}\bar\eta')\right)\\ &\qquad-2\eta\left((\lambda\lambda_{\bar\gamma\bar\gamma}-\lambda_{\bar\gamma}^2)\bar\gamma'^2\bar\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\bar\gamma'^2\eta+\lambda\bar\gamma'(\lambda_{\bar\gamma}\bar\eta'+\lambda_{\gamma}\eta')\right)\\ &\qquad+2\lambda(\lambda_{\gamma}\eta+\lambda_{\bar\gamma}\bar\eta)(\gamma'\bar{\eta'}+\bar{\gamma'}\eta')+2\lambda^2|\eta'|^2d t\\ &=\int_0^1-2\bar\eta\left((\lambda\lambda_{\gamma\gamma}-\lambda_{\gamma}^2)\gamma'^2\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\gamma'^2\bar\eta\right)\\ &\qquad-2\eta\left((\lambda\lambda_{\bar\gamma\bar\gamma}-\lambda_{\bar\gamma}^2)\bar\gamma'^2\bar\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\bar\gamma'^2\eta\right)\\ &\qquad+(2\lambda\lambda_{\gamma}\gamma'-2\lambda\lambda_{\bar\gamma}\bar\gamma')\eta\bar\eta'+(2\lambda\lambda_{\bar\gamma}\bar\gamma'-2\lambda\lambda_{\gamma}\gamma')\bar\eta\eta'+2\lambda^2|\eta'|^2d t\\ &=\int_0^1-2\bar\eta\left((\lambda\lambda_{\gamma\gamma}-\lambda_{\gamma}^2)\gamma'^2\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\gamma'^2\bar\eta\right)\\ &\qquad-2\eta\left((\lambda\lambda_{\bar\gamma\bar\gamma}-\lambda_{\bar\gamma}^2)\bar\gamma'^2\bar\eta+(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})\bar\gamma'^2\eta\right)\\ &\qquad+2\eta(\lambda_{\bar\gamma}\bar\gamma'\lambda_{\gamma}\gamma'\bar\eta+\lambda(\lambda_{\gamma\gamma}\gamma'+\lambda_{\gamma\bar\gamma}\bar\gamma')\gamma'\bar\eta-\lambda_{\gamma}^2(\gamma')^2\bar\eta+\lambda\lambda_{\gamma}\gamma'\bar\eta')\\ &\qquad+\overline{2\eta(\lambda_{\bar\gamma}\bar\gamma'\lambda_{\gamma}\gamma'\bar\eta+\lambda(\lambda_{\gamma\gamma}\gamma'+\lambda_{\gamma\bar\gamma}\bar\gamma')\gamma'\bar\eta-\lambda_{\gamma}^2(\gamma')^2\bar\eta+\lambda\lambda_{\gamma}\gamma'\bar\eta')}\\ &\qquad+4\lambda\Re(\lambda_{\gamma}\gamma'\eta\bar\eta')+2\lambda^2|\eta'|^2d t\\ &=\int_0^1(\lambda\lambda_{\gamma\bar\gamma}-\lambda_{\gamma}\lambda_{\bar\gamma})(\gamma'\bar\eta-\bar\gamma'\eta)(\bar\gamma'\eta-\gamma'\bar\eta)\\ &\qquad+8|\lambda_{\gamma}|^2|\gamma'|^2|\eta|^2+8\lambda\Re(\lambda_{\gamma}\gamma'\eta\bar\eta')+2\lambda^2|\eta'|^2d t\\ &=\int_0^1-K\lambda^4(|\gamma'|^2|\eta|^2-\Re(\gamma'^2\eta^2))\\ &\qquad+8|\lambda_{\gamma}|^2|\gamma'|^2|\eta|^2+8\lambda\Re(\lambda_{\gamma}\gamma'\eta\bar\eta')+2\lambda^2|\eta'|^2d t\\ \end{split} \end{equation*} I think the last formula is what I want because curvature $<0\Rightarrow$ it is nonnegative.