论文标题
在四维稳定梯度RICCI孤子上，尺寸会降低
On Four-dimensional Steady gradient Ricci solitons that dimension reduce
论文作者
论文摘要
在本文中，我们将研究在尺寸减少至$ 3 $ manifolds的条件下，4维稳定梯度RICCI孤子的渐近几何形状。我们将证明，这种4维稳定的梯度RICCI孤子尺寸要么降低到球形空间形式$ \ mathbb {s}^3/γ$或弱尺寸降低至$ 3 $维$维的Bryant Soliton。我们还表明，具有非负RICCI曲率的4维稳定梯度RICCI Siliton奇异性模型在紧凑型组合外部的曲率是ricci-flat ale $ 4 $ - manifolds，或者尺寸降低至$ 3 $ dibermensional歧管。作为应用程序，我们证明，在复杂表面上具有非负RICCI曲率的任何稳定梯度Kähler-icricci soliton奇异模型在紧凑的集合外部都必须是HyperkählerAleale ale ale ale ale ale ale ale ale ricc-flat $ 4 $ - manifolds。
In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form $\mathbb{S}^3/Γ$ or weakly dimension reduce to the $3$-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensional manifolds. As an application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE Ricc-flat $4$-manifolds.
