论文标题
有限指数的分支最小表面的几何形状
Geometry of branched minimal surfaces of finite index
论文作者
论文摘要
给定$ i，b \ in \ mathbb {n} \ cup \ {0 \} $，我们调查了最多$ i $ y $ i $ y Morse Index的完全有限分支最小的最小表面$ m $在$ \ mathbb {r}^3 $中，最多是$ i $ $ b $。 Fischer-Colbrie和Ros的先前作品解释说，此类表面恰恰是$ \ Mathbb {R}^3 $中有限的总曲率和有限的总分支顺序的完整最小表面。除其他事项外，我们为这样的$ m $得出了规模不变的弱和弦类型结果，其估计值是$ i $和$ b $。 In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for $m$-dimensional submanifolds $Σ$ of an $n$-dimensional Riemannian manifold $X$, where these area estimates depend on the geometry of $X$ and upper bounds on the lengths of the mean curvature vectors of $Σ$.我们还描述了一个完整的，有限分支的最小表面的家族，$ \ mathbb {r}^3 $是稳定且不可定向的。这些例子概括了经典的Henneberg最小表面。
Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of Fischer-Colbrie and Ros explain that such surfaces are precisely the complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an $M$ with estimates that are given in terms of $I$ and $B$. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for $m$-dimensional submanifolds $Σ$ of an $n$-dimensional Riemannian manifold $X$, where these area estimates depend on the geometry of $X$ and upper bounds on the lengths of the mean curvature vectors of $Σ$. We also describe a family of complete, finitely branched minimal surfaces in $\mathbb{R}^3$ that are stable and non-orientable; these examples generalize the classical Henneberg minimal surface.
