论文标题
有限指数CMC表面的几何形状
Geometry of CMC surfaces of finite index
论文作者
论文摘要
Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $ k_0 $，让$ m \ looparrowright x $成为[0，h_0] $的恒定平均曲率$ h \的完整沉浸式表面，并且最多有索引。由于[9]中的层次结构定理，我们将获得此类$ m \ looparrowright x $的几何估计。层次结构定理（请参见下面的定理2.2）将用于了解$ M \ looparrowright x $的全球属性，尤其是与该面积和直径相关的结果和$ m $。根据定理2.2的项目E，这种非紧凑型$ m \ looparrowright x $的区域是无限的。当连接$ M $时，我们将通过证明以下内容来改善该区域的结果； $ g（m）$表示$ m $的可定向封面属： 1。存在$ c_1 = c_1（i，r_0，k_0，h_0）> 0 $，因此面积$（m）\ geq c_1（g（m）+1）$。 2。存在$ c> 0，g（i）\ in \ mathbb {n} $独立于$ r_0，k_0，h_0 $和$ c $ $ i $ a $ $ $ $ $，以便如果$ g（m）\ geq g（i）$ \ frac {c} {（\ max \ {1，\ frac {1} {r_0}，\ sqrt {k_0}，h_0 \}）^2}^2}（g（m）+1）$。 3. If the scalar curvature $ρ$ of $X$ satisfies $3H^2+\frac{1}{2}ρ\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\leq A$ and Diameter$(M)\leq D$.因此，$ m $是紧凑的，按项目1，$ g（m）\ leq a/c -1 $。
Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M\looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ and with index at most $I$. We will obtain geometric estimates for such an $M\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [9]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of $M\looparrowright X$, especially results related to the area and diameter of $M$. By item E of Theorem 2.2, the area of such a non-compact $M\looparrowright X$ is infinite. We will improve this area result by proving the following when $M$ is connected; here $g(M)$ denotes the genus of the orientable cover of $M$: 1. There exists $C_1=C_1(I,r_0,K_0,H_0)>0$ such that Area$(M)\geq C_1(g(M)+1)$. 2. There exists $C>0,G(I)\in \mathbb{N}$ independent of $r_0,K_0,H_0$ and also $C$ independent of $I$ such that if $g(M)\geq G(I)$, then Area$(M)\geq \frac{C}{(\max\{1,\frac{1}{r_0},\sqrt{K_0}, H_0\})^2}(g(M)+1)$. 3. If the scalar curvature $ρ$ of $X$ satisfies $3H^2+\frac{1}{2}ρ\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\leq A$ and Diameter$(M)\leq D$. Hence, $M$ is compact and, by item 1, $g(M)\leq A/C -1$.
