Simons' inequality in warped product manifolds and some applications

Ying Huang; Yong Yang; Wei Zhang; Ying Huang; Yong Yang; Wei Zhang

doi:10.3934/math.20261369

AIMS Mathematics

2026, Volume 11, Issue 10: 35238-35267. doi: 10.3934/math.20261369

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Research article

Simons' inequality in warped product manifolds and some applications

1.
School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China
2.
Department of Mathematics, Texas State University, San Marcos, TX 78666, USA
3.
School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
4.
Institute of Applied Mathematics, Yili Normal University, Yining 835000, China

Received: 26 August 2026 Revised: 07 October 2026 Accepted: 15 October 2026 Published: 27 October 2026
MSC : 53C42, 53C21, 53C24, 58J05

Let
$ (\overline M^{m+1}, \bar g) = \bigl(I\times_fF^m, \, dt^2+f(t)^2g_F\bigr), $
where $F$ has constant sectional curvature, and let $\Sigma^m\subset\overline M^{m+1}$ be a two-sided minimal hypersurface. We derived an explicit Simons identity in terms of
$ \alpha = -\frac{f''}{f}, \qquad \beta = \frac{k_F-(f')^2}{f^2}, \qquad \Xi = (\alpha-\beta)'-2\frac{f'}{f}(\alpha-\beta). $
When $\Xi = 0$, we determined the optimal angle-free algebraic lower coefficient for the curvature term and obtained a compact pinching theorem, recovering the classical spherical gap constant. For stable minimal hypersurfaces, a power Caccioppoli estimate, local Sobolev inequality, and Moser iteration yield, for $3\le m\le5$,
$ |A|^2(x_0) \le C\left( r^{-2} + \sup\limits_{B_{2r}(x_0)} \bigl(|\alpha|+|\beta|+|\Xi|^{2/3}\bigr) \right). $
Thus the classical Schoen–Simon–Yau (SSY) curvature-estimate mechanism extends to this warped-product setting with an explicit scale-correct curvature term.
- minimal hypersurface,
- Simons identity,
- warped product,
- second fundamental form,
- pinching,
- stability,
- curvature estimate
Citation: Ying Huang, Yong Yang, Wei Zhang. Simons' inequality in warped product manifolds and some applications[J]. AIMS Mathematics, 2026, 11(10): 35238-35267. doi: 10.3934/math.20261369

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Abstract

Let

$ (\overline M^{m+1}, \bar g) = \bigl(I\times_fF^m, \, dt^2+f(t)^2g_F\bigr), $

where $F$ has constant sectional curvature, and let $\Sigma^m\subset\overline M^{m+1}$ be a two-sided minimal hypersurface. We derived an explicit Simons identity in terms of

$ \alpha = -\frac{f''}{f}, \qquad \beta = \frac{k_F-(f')^2}{f^2}, \qquad \Xi = (\alpha-\beta)'-2\frac{f'}{f}(\alpha-\beta). $

When $\Xi = 0$, we determined the optimal angle-free algebraic lower coefficient for the curvature term and obtained a compact pinching theorem, recovering the classical spherical gap constant. For stable minimal hypersurfaces, a power Caccioppoli estimate, local Sobolev inequality, and Moser iteration yield, for $3\le m\le5$,

$ |A|^2(x_0) \le C\left( r^{-2} + \sup\limits_{B_{2r}(x_0)} \bigl(|\alpha|+|\beta|+|\Xi|^{2/3}\bigr) \right). $

Thus the classical Schoen–Simon–Yau (SSY) curvature-estimate mechanism extends to this warped-product setting with an explicit scale-correct curvature term.

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