论文标题
在频谱中具有非负曲率的闭合表面上
On closed surfaces with nonnegative curvature in the spectral sense
论文作者
论文摘要
我们研究了满足光谱条件$λ_1（-Δ+βK）\geqλ\ geq0 $的闭合方向表面，其中$β$是一个正常数，而$ k $是高斯曲率。这种条件自然出现在稳定的最小表面中的3个脉冲曲率的稳定表面。我们显示了此类表面的等等仪不平等，区域生长定理和直径界限。这些不平等的有效性受到$β$的某些范围。与正值$Δφ\leqβkφ$相关，共形度量$φ^{2/β} g $具有侧重的非负曲率。利用新指标的几何形状，我们证明了Hölder的预发性和几乎刚性的结果。
We study closed orientable surfaces satisfying the spectral condition $λ_1(-Δ+βK)\geqλ\geq0$, where $β$ is a positive constant and $K$ is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for $β$. Associated to a positive super-solution $Δφ\leqβKφ$, the conformal metric $φ^{2/β}g$ has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition.
