带弱凸边界流形的谱体积比较
A spectral volume comparison for manifolds with weakly convex boundary
AI总结：
该研究建立了带弱凸边界流形的谱意义下的Bonnet-Myers定理与Bishop-Gromov体积比较定理，推导了流形直径的上界及体积的最优上界，明确了等号成立的条件。
摘要：
我们建立了带弱凸边界流形的谱意义下的Bonnet-Myers定理和Bishop-Gromov体积比较定理。对于n≥3，设(Mⁿ,g)为带弱凸边界∂M的单连通紧光滑n维流形，若存在正函数w∈C^∞(M)满足：在M内有-(n-1)/(n-2)Δw + Λ_Ric w ≥ (n-1)w，在∂M上有∂w/∂η=0，其中Λ_Ric表示Ricci张量的最小特征值，η为∂M在M中的单位余法向量场，则M的直径满足diam(M) ≤ (max w / min w)^((n-3)/(n-1)) π。若w还在边界∂M上取得最小值，我们得到M体积的最优上界：Vol(M) ≤ Vol(𝕊ⁿ₊)，等号成立当且仅当Mⁿ等距于单位半球面𝕊ⁿ₊。
英文摘要：
We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For $n\geq 3$, let $(M^n,g)$ be a simply connected compact smooth $n$-manifold with weakly convex boundary $\partial M$. If there exists a positive function $w\in C^{\infty}(M)$ that satisfies: \begin{equation*} \begin{cases} -\frac{n-1}{n-2}Δw+Λ_{\Ric} w\geq (n-1)w, \enspace in \enspace M, \frac{\partial w}{\partial η}=0, \enspace\enspace\enspace\enspace \enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace \enspace\enspace \enspace\enspace on \enspace\partial M, \end{cases} \end{equation*} where $Λ_{\Ric}$ denotes the smallest eigenvalue of the Ricci tensor, $η$ is the unit co-normal vector field of $\partial M$ in $M$, then the diameter of $M$ satisfies $\diam(M)\leq (\frac{\max w}{\min w})^{\frac{n-3}{n-1}}π$.\par If, in addition, $w$ attains its minimum on the boundary $\partial M$, we obtain a sharp upper bound for the volume of $M$: $\Vol(M)\leq \Vol(\bS^n_{+})$, with equality holding if and only if $M^n$ is isometric to the unit round hemisphere $\bS^{n}_{+}$.
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