By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of 3-dimensional tori \(M_j\) that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form \(R_{g_{M_j}} \ge -1/j\). We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang–Lee, Huang–Lee–Sormani and Allen–Perales–Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus \((M, g_M)\) is replaced by a bound on the quantity \(-\int _T \min \{R_{g_M},0\} d{\mathrm {vol}_{g_T}}\), where \(M=\text {graph}(f)\), \(f: T \rightarrow \mathbb {R}\) and \((T,g_T)\) is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions \(n \ge 4\) as well.

通过Schoen–Yau和Gromov–Lawson的著作，具有非负标量曲率并且对圆环微分的黎曼流形与平面圆环等距。当标量曲率变为0时，Gromov推测Tori相对于弱Sobolev型度量的子收敛。对于3维tori \（M_j \）的非折叠序列，我们证明了平坦的和固有的平坦子收敛到平坦的圆环，可以实现为满足统一的上直径范围和标量曲率范围\（R_ {g_ {M_j}} \ ge -1 / j \）的平坦圆托上定义的某些函数的图形。我们还表明，收敛子序列的流形的体积收敛到极限空间的体积。我们这样做是为了适应Huang-Lee，Huang-Lee-Sormani和Allen-Perales-Sormani的结果。此外，当圆环\（（M，g_M）\）的标量曲率条件被数量\（-\ int _T \ min \ {R_ {g_M}，0 \ } d {\ mathrm {vol} _ {g_T}} \），其中\（M = \ text {graph}（f）\），\（f：T \ rightarrow \ mathbb {R} \）和\（（ T，g_T）\）是扁平圆环。使用这项工作完成后由Alaee，McCormick和第一位具名作者开发的参数，我们的结果也适用于维\（n \ ge 4 \）。