AI 中文总结
本文研究不同曲率条件下完备稳定极小超曲面的共形结构与刚性，证明非负数量曲率流形中不存在共形等价于非正数量曲率有界区域的非紧稳定极小超曲面，并给出紧致情形及扭曲积流形中的稳定性与刚性结果。
AI 中文摘要
我们研究了在各种曲率假设下黎曼流形中的完备稳定极小超曲面。在具有非负数量曲率的$(n+1)$维完备定向流形中，我们证明了不存在完备定向非紧稳定极小超曲面，使其共形等价于某个$n$维具有非正数量曲率流形中的有界区域。作为推论，在$\mathbb{R}^{n+1}$中不存在共形等价于$\mathbb{R}^n$中有界区域的完备稳定极小超曲面。我们还证明了具有非负数量曲率流形中的紧致稳定极小超曲面具有非负的光滑 Yamabe 不变量，并刻画了等号成立的情形。接下来，我们考虑截面曲率受夹挤的周围流形中的完备双侧稳定极小超曲面。在使得第二基本形式成为 Codazzi 张量的附加曲率条件下，我们在第二基本形式的$L^2$条件下获得了刚性结果，并导出了拉普拉斯算子第一特征值的上界。最后，我们证明了浸入扭曲积流形中的任何完备非紧双侧极小超曲面都是稳定的，前提是角度函数为正且扭曲函数的二阶导数非负。
英文摘要
We study complete stable minimal hypersurfaces in Riemannian manifolds under various curvature assumptions. In an $(n+1)$-dimensional complete oriented manifold with nonnegative scalar curvature, we prove that no complete oriented noncompact stable minimal hypersurface can be conformally equivalent to a bounded domain in an $n$-dimensional manifold with nonpositive scalar curvature. As a consequence, there exists no complete stable minimal hypersurface in $\mathbb{R}^{n+1}$ that is conformally equivalent to a bounded domain in $\mathbb{R}^n$. We also show that compact stable minimal hypersurfaces in manifolds with nonnegative scalar curvature have nonnegative smooth Yamabe invariant and we characterize the equality case. Next, we consider complete two-sided stable minimal hypersurfaces in ambient manifolds with pinched sectional curvature. Under an additional curvature condition that makes the second fundamental form a Codazzi tensor, we obtain a rigidity result under an $L^2$-condition on the second fundamental form and derive an upper bound for the first eigenvalue of the Laplacian. Finally, we establish that any complete noncompact two-sided minimal hypersurface immersed in a warped product manifold is stable, provided that the angle function is positive and the second derivative of the warping function is nonnegative.
Journal refJ. Geom. Anal. 36, 351 (2026)
DOI：10.1007/s12220-026-02604-9
