S²×S²上具有有界几何的几乎正截面曲率
Almost Positive Sectional Curvature with Bounded Geometry on $S^2\times S^2$
AI总结：
本文在S²×S²上构造满足有界几何的几乎正截面曲率度量，通过共形变形与微分同胚实现指定曲率条件，核心内容由ChatGPT 5.6生成并验证。
AI中文摘要：
设M=S²×S²，我们证明存在固定的光滑开球U⊂M，使得对任意ε>0，均可找到光滑度量g_ε与闭集S_ε，满足以下性质：在M\backslash S_ε上sec_{g_ε}>0；S_ε的相对体积小于ε；U上的所有截面曲率均大于1；存在与ε无关的常数Λ，使得-ε<sec_{g_ε}≤Λ；体积和直径有一致的正上下界；内射半径有一致的正下界。事实上，U∩S_ε=∅。构造从单位圆度量的乘积开始，通过对每个球上高度函数的绝对值进行平滑得到的小共形因子，在两个薄赤道带外具有负定Hessian，它使原本平坦的混合平面在任意小相对体积的集合外严格为正，而带内产生的负曲率仅为O(ε)。在极小区域内进行的第二次共形变形生成一个所有截面曲率大于1的核心，其过渡部分的Hessian上界为O(ε)。最后，微分同胚将固定拓扑球U压缩至该核心，拉回操作保留所有内在有界几何估计。本文主要内容由ChatGPT 5.6生成并经作者验证。
英文摘要：
Let $M=S^2\times S^2$. We prove that there is a fixed smooth open ball $U\subset M$ such that, for every $\varepsilon>0$, one can find a smooth metric $g_\varepsilon$ and a closed set $S_\varepsilon$ with the following properties: $\sec_{g_\varepsilon}>0$ on $M\setminus S_\varepsilon$; the relative volume of $S_\varepsilon$ is less than $\varepsilon$; every sectional curvature on $U$ is greater than $1$; $-\varepsilon<\sec_{g_\varepsilon}\leqΛ$ for one constant $Λ$ independent of $\varepsilon$; the volume and diameter have uniform positive lower and upper bounds; and the injectivity radius has a uniform positive lower bound. In fact, $U\cap S_\varepsilon=\varnothing$. The construction starts from the product of the unit round metrics. A small conformal factor, obtained by smoothing the absolute value of the height function on each sphere, has negative definite Hessian away from two thin equatorial bands. It makes every formerly flat mixed plane strictly positive outside a set of arbitrarily small relative volume, while the negative curvature created in the bands is only $O(\varepsilon)$. A second conformal deformation supported in a very small region produces a core on which all sectional curvatures are greater than $1$; its transition has Hessian bounded above by $O(\varepsilon)$. Finally, a diffeomorphism compresses the fixed topological ball $U$ into that core. Pullback preserves all intrinsic bounded-geometry estimates. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.
