Conformal descent on a stable minimal hypersurface

Let \((N^{n+1},g_N)\) be a closed Riemannian manifold and let \(\Sigma^n\subset N\) be a closed, two-sided, stable minimal hypersurface. Denote the induced metric on \(\Sigma\) by \(g\). The stability inequality is

\begin{equation} \tag{6.2.1} \label{eq:psc-stability-hypersurface} \int_\Sigma |\nabla\phi|^2\,d\mu_g \geq \int_\Sigma \bigl(|A|^2+\mathrm{Ric}_N(\nu,\nu)\bigr)\phi^2\,d\mu_g, \qquad \phi\in C^\infty(\Sigma). \end{equation}

The Gauss equation, using \(H=0\), gives

\begin{equation} \tag{6.2.2} \label{eq:psc-gauss-scalar} R_N = R_\Sigma+2\mathrm{Ric}_N(\nu,\nu)+|A|^2. \end{equation}

Combining (6.2.1) and (6.2.2),

\begin{equation} \tag{6.2.3} \label{eq:psc-sy-rearrangement} \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \int_\Sigma (R_N+|A|^2)\phi^2\,d\mu_g . \end{equation}

This is the basic Schoen–Yau rearrangement.

Proposition 6.2.1

Assume \(R_N\gt{}0\) and \(n\geq3\). Then every closed, two-sided, stable minimal hypersurface \(\Sigma^n\subset N^{n+1}\) admits a metric of positive scalar curvature.

Proof. Since \(\Sigma\) is compact and \(R_N\gt{}0\) on \(N\), there is a number \(\kappa\gt{}0\) such that \(R_N\geq\kappa\) on \(\Sigma\). From (6.2.3),

\[ \int_\Sigma 2|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Because \(c_n=4(n-1)/(n-2)\geq2\), we also have

\[ \int_\Sigma c_n|\nabla\phi|^2+R_\Sigma\phi^2\,d\mu_g \geq \kappa\int_\Sigma \phi^2\,d\mu_g . \]

Thus the first eigenvalue of the conformal Laplacian \(L_g\) is positive. The claim follows from Lemma 6.1.1. ◻

When \(n=2\), the same rearrangement gives useful topological information directly. Taking \(\phi\equiv1\) in (6.2.3) gives

\[ \int_\Sigma R_\Sigma\,d\mu_g = 2\int_\Sigma K_\Sigma\,d\mu_g \gt{}0. \]

Hence \(\chi(\Sigma)\gt{}0\) by Gauss–Bonnet. In particular a closed orientable stable minimal surface in a three-manifold with \(R_N\gt{}0\) is a union of two-spheres.

Conformal descent on a stable minimal hypersurface

Gaoming Wang

Assistant Professor