| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief communications Index of minimal surfaces in the 3-sphere E. A. Morozovab, A. V. Penskoica a HSE University b Independent University of Moscow c Lomonosov Moscow State University
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     IntroductionLet $\Sigma$ be an orientable surface without boundary and $\varphi\colon\Sigma\looparrowright\mathbb{S}^3\subset\mathbb{R}^4$ be a minimal immersion of $\Sigma$ in a sphere $\mathbb{S}^3$ of radius 1. Consider the map $\widetilde\varphi\colon\Sigma\to\mathbb{R}^4\mathrel{\wedge}\mathbb{R}^4\cong\mathbb{R}^6$ defined by $\widetilde\varphi(x)=\varphi(x)\wedge\nu(x)$, where $\nu$ is a unit normal field to $\varphi(\Sigma)$. It is known that $\widetilde\varphi(\Sigma)\subset\mathbb{S}^5$ and $\widetilde\varphi\colon\Sigma\looparrowright\mathbb{S}^5$ also is a minimal immersion ([2], § 11). Let $g$ and $\widetilde g$ denote the metrics on $\Sigma$ induced by the immersions $\varphi$ and $\widetilde\varphi$, respectively. In what follows we use the notation $\Sigma$ implying the metric $g$, and we write $\widetilde\Sigma$ implying $\widetilde g$. The second variation of the area functional on $\Sigma$ defines a Jacobi stability operator on normal vector fields. Since the normal bundle is trivial, we obtain the operator $J={\Delta-4+2K}$ acting on functions [6], where $\Delta$ and $K$ denote the Laplace–Beltrami operator and Gaussian curvature on $\Sigma$, respectively. Definition. The index $\operatorname{ind}\Sigma$ of the minimal surface $\Sigma$ is the number, counting with multiplicity, of negative eigenvalues of the operator $J$, and the nullity $\operatorname{null}\Sigma$ of the minimal surface $\Sigma$ is the dimension of the kernel of $J$. 1. The relationship between the index and nullity of the surface $\Sigma$ and the spectrum of $\widetilde\Sigma$Let $\widetilde\Delta$ be the Laplace–Beltrami operator on $\widetilde\Sigma$. It is known that $\widetilde g=(2-K)g$ and $\widetilde\Delta=(2-K)^{-1}\Delta$ ([2], § 11). Let $N_{\widetilde\Sigma}(\lambda)$ denote the number of eigenvalues less than $\lambda$ of $\widetilde\Delta$. Theorem 1. For each minimal immersion $\varphi\colon\Sigma\looparrowright\mathbb{S}^3$ the nullity $\operatorname{null}\Sigma$ is equal to the multiplicity of the eigenvalue 2 of the operator $\widetilde\Delta$, and the index $\operatorname{ind}\Sigma$ is equal to $N_{\widetilde\Sigma}(2)$. Proof. Let $\rho=2-K$; then $\widetilde\Delta=\rho^{-1}\Delta$ and $J=\Delta-2\rho$. The eigenfunctions with eigenvalue 0 of $J$ correspond to the eigenfunctions with eigenvalue 2 of $\widetilde\Delta$. This yields the assertion of Theorem 1 concerning the nullity.
We denote the $k$th eigenvalues of $\widetilde\Delta$ and $J$ by $\lambda_k$ and $\mu_k$, respectively. Let $R_{\widetilde\Delta}[f]=\displaystyle\int_\Sigma |\nabla f|^2\,dv_g\!\!\Bigm/\!\!\! \displaystyle\int_\Sigma \rho f^2\,dv_g$ and $R_J[f]=\displaystyle\int_\Sigma(|\nabla f|^2-2\rho f^2)\,dv_g\!\!\Bigm/\!\!\! \displaystyle\int_\Sigma f^2\,dv_g$ be the Rayleigh quotients for the operators $\widetilde\Delta$ and $J$ (here $f\in H^1(\Sigma,dv_g)$, and $dv_g$ is the volume form of the metric $g$). Suppose that $\lambda_k<2$ for some $k$. Let $\Phi\subset H^1(\Sigma,dv_g)$ be the subspace spanned by the eigenfunctions $\widetilde\Delta$ with eigenvalues $0=\lambda_0<\lambda_1\leqslant\cdots\leqslant\lambda_k$. Then for each function $f\in\Phi\setminus\{0\}$ we have $R_{\widetilde\Delta}[f]\leqslant\lambda_k<2$, so that $R_J[f]<0$. Hence $\mu_k\leqslant\sup_{f\in\Phi}R_J[f]<0$, where the second inequality is strict because the supremum is attained on a function from $\Phi$. Thus we have shown that if $\lambda_k<2$, then $\mu_k<0$. In a similar way, if $\mu_k<0$, then $\lambda_k<2$. Therefore, $\operatorname{ind}\Sigma=N_{\widetilde\Sigma}(2)$. $\Box$ Corollary. For Otsuki tori $O_{p/q}\subset\mathbb{S}^3$ (defined originally in [3]; we use the notation introduced in [5] and used in [1]) the equalities $\operatorname{ind} O_{p/q}=2q+4p-2$ and $\operatorname{null} O_{p/q}= 5$ hold. Proof. It follows from [1] that if $\Sigma=O_{p/q}$, then $N_{\widetilde\Sigma}(2)=2q+4p-2$ and the eigenvalue 2 has multiplicity 2 on $\widetilde\Sigma$, so this is a consequence of the theorem. $\Box$
2. The indices of Lawson $\tau$-surfacesUnfortunately, Theorem 1 does not ‘automatically’ return the index and nullity of an arbitrary minimal surface in $\mathbb{S}^3$. Apart from the fact that the calculation of $N_{\widetilde\Sigma}(2)$ is a difficult problem, in important examples $\Sigma$ is either non-orientable, or the map $\widetilde\varphi$ is a non-trivial covering. In particular, one comes up against such obstacles in investigating the important family of Lawson $\tau$-surfaces. These surfaces must be considered separately using the method of separation of variables presented in [4] and [5]. Definition. The image of the doubly periodic immersion $\Psi_{m,k}\colon\mathbb{R}^2\looparrowright\mathbb{S}^3\subset \mathbb{R}^4$ defined by $\Psi_{m,k}(x,y)=(\cos mx \cos y,\sin mx \cos y, \cos kx \sin y,\sin kx \sin y)$ is called the Lawson surface $\tau_{m,k}$ (see [2]). It is known [2] that for each unordered pair $(m,k)\in\mathbb{N}\times\mathbb{N}$, $\operatorname{\textrm{GCD}}(m,k)=1$, the surface $\tau_{m,k}$ is a compact minimal surface in $\mathbb{S}^3$ distinct from the other surfaces in the family. If $m$ and $k$ are odd, then $\tau_{m,k}$ is a torus, while if either $m$ or $k$ is even, then $\tau_{m,k}$ is a Klein bottle. We present a result for the Klein bottle $\tau_{2,1}$; other $\tau_{m,k}$ are considered in the forthcoming paper of the second-named author. Theorem 2. For the Lawson Klein bottle $\tau_{2,1}$ the equalities $\operatorname{null}\tau_{2,1}=5$ and $\operatorname{ind}\tau_{2,1}=7$ hold. Proof. The map $\Psi_{m,k}$ has periods $T_1=(2\pi,0)$ and $T_2=(0,2\pi)$. For the Lawson Klein bottles $\tau_{m,k}$ the torus $\mathbb{R}^2/\{aT_1+bT_2\colon a,b\in\mathbb{Z}\}$ with metric induced by the immersion $\Psi_{m,k}$ is a two-sheeted covering of $\tau_{m,k}$ two because $\Psi_{m,k}$ is also invariant under the transformation $(x,y)\mapsto(x+\pi,-y)$, so we can take $(x,y)\in[0,\pi)\times[-\pi,\pi)$ as coordinates on $\tau_{m,k}$. The spectral problem for the Jacobi operator reads
$$
\begin{equation*}
-\dfrac{1}{p(y)^2}\,\dfrac{\partial^2 f}{\partial x^2}- \dfrac{1}{p(y)}\,\dfrac{\partial}{\partial y} \biggl(p(y)\dfrac{\partial f}{\partial y}\biggr)-2f- \dfrac{2m^2k^2}{p(y)^4}f=\lambda f,
\end{equation*}
\notag
$$
where $p(y)=\sqrt{k^2+(m^2-k^2)\cos^2y}$ , with boundary conditions $f(x+\pi,-y)=-f(x,y)$ and $f(x,y+2\pi)=f(x,y)$. Since this Jacobi operator commutes with $\partial/\partial x$, we can reduce this spectral problem to one-dimensional ones using the approach from [4] and [5]. Then we obtain the family of one-dimensional spectral problems
$$
\begin{equation*}
-\dfrac{1}{p(y)}\,\dfrac{d}{dy}\biggl(p(y)\dfrac{d\varphi(y)}{dy}\biggr)+ \biggl(\dfrac{l^2}{p(y)^2}-2- \dfrac{2m^2k^2}{p(y)^4}-\lambda\biggr)\varphi(y)=0,\qquad \varphi(y+2\pi)\equiv\varphi(y),
\end{equation*}
\notag
$$
with spectrum $\lambda_i(l)$ and with solutions $\varphi_i(l,y)$. The required result is obtained by analyzing them using the methods from [4]. $\Box$ The authors are grateful to M. Karpukhin for useful discussions. 
Citation in format AMSBIB
\Bibitem{MorPen23} |
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