论文标题
在球体中的最小亚曼叶的音量差距
Volume gap for minimal submanifolds in spheres
论文作者
论文摘要
对于关闭的最小值submanifold $ f：m^n \ looparrowright \ mathbb {s}^{n} $在单位sphere $（n <n）$中，我们证明了$ {\ rm vol}（m^n） 1+φ_{p}^2 \ right）\ geq m {\ rm vol}（\ mathbb {s}^{n}），$$，其中$φ_{p}（p}（x）：= \ langle f（x），p \ rangle $在f（p \ rangle $ p \ p \ p p \ p $ p \ p $ p \ p $ p \ p $ p \ p $ p \ p $ p（$ p） f（m）$和$ {\ rm vol} $表示riemannian卷的功能，并且当且仅当$ m $完全是Geodesic时，每个平等才能保持。作为一个应用程序，如果$ m^n $的体积小于或等于任何$ n $维度最小的Clifford torus的体积，则必须嵌入$ m^n $，以验证Yau猜想的非安装案例。此外，在这种情况下，我们还获得了具有恒定标态曲率的最小超曲面的体积差距，从而改善了Cheng-Yau的经典体积间隙。还获得了其他一些音量间隙和相关的夹紧刚性。
For a closed minimal submanifold $f:M^n\looparrowright \mathbb{S}^{N}$ in the unit sphere $(n<N)$, we prove $${\rm Vol}(M^n) \geq\frac{n+1}{n+2}\int_{M}\left( 1+φ_{p}^2\right) \geq m{\rm Vol}(\mathbb{S}^{n}),$$ where $φ_{p}(x):=\langle f(x),p\rangle$ is the height function in direction $p\in f(M)$, $m$ denotes the multiplicity of $p\in f(M)$ and ${\rm Vol}$ denotes the Riemannian volume functional, and each equality holds if and only if $M$ is totally geodesic. As an application, if the volume of $M^n$ is less than or equal to the volume of any $n$-dimensional minimal Clifford torus, then $M^n$ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained.
