https://neckpinch.com/authors/jian-ge/New arXiv papers by Jian Ge, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.19851arxiv:2609.19851Thu, 17 Sep 2026 00:00:00 GMTJian Ge, Chuanhuan Li, Ronggang Li — Let (Mⁿ, g) be a complete, simply connected Riemannian manifold without boundary, of dimension n≥3, with curvature operator at least that of the unit sphere. We prove that ∫_M rm scal(x) d rm Volₓ≤ n(n-1)ωₙ, where ωₙ is the volume of the unit n-sphere. Equality holds if and only if (M,g) is isometric to the unit round sphere. In fact, we obtain a stronger bound containing rm Vol(M, g). In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.Ricci solitonsScalar curvature https://arxiv.org/abs/2607.18216arxiv:2607.18216Mon, 20 Jul 2026 00:00:00 GMTJian Ge — Let V be an n-dimensional Euclidean vector space, ,where n≥ 4, and ℓ = lfloorn/2rfloor. We prove the sharp pointwise estimate q₂(E) ≥ −(2(ℓ −1))/3ℓ Scal(E) Id_(Λ²V^*) for every algebraic curvature tensor E on V with nonnegative sectional curvature. Applying this estimate to the decomposition Rm_g=KₘᵢₙI+E, we obtain the vanishing of H²(M; ℝ) under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields b₂(M)=0 in odd dimensions and b₂(M)≤ 1 in even dimensions. At even-dimensional endpoint, b₂(M)>0 forces (M, g) to be isometric, up to scaling, to ℂP}^(ℓ) with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion Kₘᵢₙ ≥ (n(n-1))/(n²−n+12)S₀ Longrightarrow PIC2. The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.Ricci flowCurvature pinchingScalar curvatureHomogeneous spaces