[Paper Review] Simons' type equation for $f$-minimal hypersurfaces and applications

Xu Cheng, Tito Mejia+1 more|arXiv (Cornell University)|May 10, 2013

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper derives a Simons-type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds, particularly in the cylinder shrinking soliton $\mathbb{S}^n(\sqrt{2(n-1)}) \times \mathbb{R}$ with $f = t^2/4$. It establishes a pinching theorem showing that closed $f$-minimal hypersurfaces with $|A|^2$ in a specific range must be the equatorial sphere $\mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$, and classifies those with $L_f$-index one as precisely this sphere.

ABSTRACT

We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold $\mathbb{S}^n(\sqrt{2(n-1)}) imes \mathbb{R}$ with $f=\frac {t^2}{4}$. Also we classify closed $f$-minimal hypersurfaces with $L_f$-index one immersed in $\mathbb{S}^n(\sqrt{2(n-1)}) imes \mathbb{R}$ with the same $f$ as above.

Motivation & Objective

• To derive a Simons-type equation for the weighted Laplacian $\Delta_f$ of $|A|^2$ on $f$-minimal hypersurfaces in weighted Riemannian manifolds.
• To apply this equation to classify closed $f$-minimal hypersurfaces in the cylinder shrinking soliton $\mathbb{S}^n(\sqrt{2(n-1)}) \times \mathbb{R}$ with $f = t^2/4$.
• To establish a pinching theorem for closed $f$-minimal hypersurfaces under bounds on $|A|^2$ involving the normal angle $\alpha = \langle \nu, \partial_t \rangle$.
• To classify $f$-minimal hypersurfaces with $L_f$-index one in the same ambient space.

Proposed method

• Derives a general Simons-type equation for $\Delta_f |A|^2$ on $f$-minimal hypersurfaces, involving the Bakry-Émery Ricci curvature $\overline{\text{Ric}}_f$.
• Applies the equation to the specific case of $M = \mathbb{S}^n(\sqrt{2(n-1)}) \times \mathbb{R}$ with $f = t^2/4$, computing relevant geometric quantities like $H$, $\alpha$, and $\nabla \alpha$.
• Uses integration of the Simons-type equation over the hypersurface to derive integral identities involving $|\nabla A|^2$, $|A|^2$, $H$, and $\alpha$.
• Employs the identity $\int_\Sigma |\nabla \alpha|^2 e^{-f} = \int_\Sigma \alpha^2 |A|^2 e^{-f}$ to constrain the geometry of $\Sigma$.
• Analyzes the $L_f$-index via spectral properties of the Jacobi operator $L_f = \Delta_f + |A|^2 + \text{Ric}_f(\nu,\nu)$.
• Combines pointwise estimates and integral identities to show that $|A|^2$ and $H$ must be constant, and $\alpha \equiv 0$ or $\alpha^2 \equiv 1$, leading to classification.

Experimental results

Research questions

• RQ1 Under what conditions on $|A|^2$ does a closed $f$-minimal hypersurface in $\mathbb{S}^n(\sqrt{2(n-1)}) \times \mathbb{R}$ with $f = t^2/4$ necessarily coincide with $\mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$?
• RQ2 What is the structure of $f$-minimal hypersurfaces with $L_f$-index one in the cylinder shrinking soliton?
• RQ3 How does the Simons-type equation for $\Delta_f |A|^2$ constrain the geometry of $f$-minimal hypersurfaces in weighted manifolds?
• RQ4 Can integral identities derived from the Simons-type equation rule out non-trivial $f$-minimal hypersurfaces under certain curvature and angle conditions?

Key findings

• The Simons-type equation for $\Delta_f |A|^2$ is derived for $f$-minimal hypersurfaces in smooth metric measure spaces, involving $\overline{\text{Ric}}_f$ and geometric quantities like $H$ and $\alpha$.
• A pinching theorem is established: if $|A|^2$ lies in the interval $\left[\frac{1}{4}\left(1 - \sqrt{1 - \frac{8}{n-1}\alpha^2(1 - \alpha^2)}\right), \frac{1}{4}\left(1 + \sqrt{1 - \frac{8}{n-1}\alpha^2(1 - \alpha^2)}\right)\right]$, then $\Sigma$ must be $\mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$.
• The integral identity $\int_\Sigma |\nabla \alpha|^2 e^{-f} = \int_\Sigma \alpha^2 |A|^2 e^{-f}$ holds for closed $f$-minimal hypersurfaces, implying strong constraints on $\alpha$ and $|A|^2$.
• It is shown that $\alpha \equiv 0$ or $\alpha^2 \equiv 1$ on $\Sigma$, and the case $\alpha^2 \equiv 1$ leads to $\Sigma$ lying in a horizontal slice, forcing $\Sigma = \mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$.
• The $L_f$-index of any closed $f$-minimal hypersurface in $\mathbb{S}^n(\sqrt{2(n-1)}) \times \mathbb{R}$ is at least one, with equality if and only if $\Sigma = \mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$.
• The classification of $L_f$-index one hypersurfaces is complete: only the equatorial sphere $\mathbb{S}^n(\sqrt{2(n-1)}) \times \{0\}$ satisfies this condition.