论文标题
Donaldson功能和平均曲率表面的最小化和平均曲率的最小化器的渐近学
Asymptotics for minimizers of a Donaldson functional and mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds
论文作者
论文摘要
Huang-lucia-tarantello [17]在给定的$ \ vert c \ vert <1 $中，是持续平均曲率（CMC）$ C $的模量空间的封闭式封闭式表面，可定位的表面$ \ \ \ \ \ \ \ \ \ \ egeq {g} \ geq 2 $ 3 $ -3 $ -3的词汇量的均可出现的杂物均可构成$ 3-元素的词组， Teichmüller空间。这是通过显示管理高斯方程（CMC）C-Immersions的独特可溶性来实现的。相应的唯一解决方案被确定为Donaldson功能$ d_t $（在[11]中引入）的全局最小值（唯一的关键点），其中（1.3）中给出了$ t = 1-c^{2} $。 当$ \ vert c \ vert \ geq 1 $（即$ t \ leq 0 $）时，到目前为止，对类似（CMC）C-Immersions的存在一无所知。确实，对于$ t \ leq 0 $，功能$ d_ {t} $可能不再从下面进行界定，并且确实发生了明显的不存在的情况。 $ \ vert c \ vert = 1 $（即$ t = 0 $）出现了相当涉及，实际上（CMC）只能以$ \ vert c \ vert c \ vert c \ vert c \ longrightArrow 1^{ - } $的“限制”（cmc）C-iMmersions的“限制”。为了处理这种情况，我们在这里分析$ d_ {t} $的最小化器的渐近行为为$ t \ longrightArrow 0^{+} $。 我们使用准确的渐近分析来描述可能的爆破现象。通过这种方式，我们可以将（CMC）1-放入的存在与Kodaira地图联系起来。结果，我们获得了（cmc）属$ \ mathfrak {g} {g} = 2 $的（CMC）1-蚀刻的第一个存在和独特性结果。
It has been shown in by Huang-Lucia-Tarantello [17] that, for given $\vert c \vert <1$, the moduli space of constant mean curvature (CMC) $c$-immersions of a closed orientable surface of genus $\mathfrak{g} \geq 2$ into a hyperbolic $3$-manifold can be parametrized by elements of the tangent bundle of the corresponding Teichmüller space. This is attained by showing the unique solvability of the Gauss-Codazzi equations governing (CMC) c-immersions. The corresponding unique solution is identified as the global minimum (and only critical point) of the Donaldson functional $D_t$ (introduced in [11]) given in (1.3) with $t=1-c^{2}$. When $\vert c \vert \geq 1$ (i.e. $t\leq 0$), so far nothing is known about the existence of analogous (CMC) c-immersions. Indeed, for $t\leq 0$ the functional $D_{t}$ may no longer be bounded from below and evident non-existence situations do occur. Already the case $\vert c \vert =1$ (i.e. $t=0$) appears rather involved and actually (CMC) 1-immersions can be attained only as "limits" of (CMC) c-immersions for $\vert c \vert \longrightarrow 1^{-}$. To handle this situation, here we analyse the asymptotic behaviour of minimizers of $D_{t}$ as $t \longrightarrow 0^{+}$. We use an accurate asymptotic analysis to describe possible blow-up phenomena. In this way, we can relate the existence of (CMC) 1-immersions to the Kodaira map. As a consequence, we obtain the first existence and uniqueness result about (CMC) 1-immersions of surfaces of genus $\mathfrak{g}=2$ into hyperbolic 3-manifolds.
