Almost Positive Sectional Curvature with Bounded Geometry on $S^2\times S^2$

Abstract

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.