https://neckpinch.com/authors/guoqiang-wu/New arXiv papers by Guoqiang Wu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.37151arxiv:2610.37151Tue, 27 Oct 2026 00:00:00 GMTFengjiang Li, Wenqi Li, Jianyu Ou, Yuanyuan Qu, Guoqiang Wu, Detang Zhou — Let (Mⁿ,g,f) be a complete connected shrinking gradient Ricci soliton with constant scalar curvature, satisfying Ric+∇²f=1/2g. We prove that if the focal set of f is totally geodesic, then M is rigid. Equivalently, its universal cover is a product of a compact Einstein manifold and a Euclidean space.Ricci solitonsEinstein metricsScalar curvatureStability & uniqueness https://arxiv.org/abs/2610.17691arxiv:2610.17691Tue, 13 Oct 2026 00:00:00 GMTMijia Lai, Guoqiang Wu — We prove two constant rank theorems for the Ricci tensor in two geometric settings: manifolds with harmonic curvature and gradient Ricci solitons. In both cases, if the k-intermediate Ricci curvature satisfies Ricₖ≥0 for some 1≤ k≤ n-2, then the Ricci tensor has constant rank and its nullity distribution is parallel. In the complete case, the universal cover splits isometrically along the Ricci-null directions. We also show that the range 1≤ k≤ n-2 is sharp: in every dimension n≥4, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.Ricci solitons https://arxiv.org/abs/2610.11935arxiv:2610.11935Thu, 08 Oct 2026 00:00:00 GMTMatheus Horácio, Wenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete nonsteady gradient Kähler-Ricci soliton satisfying Ric+∇² f=λg, λ≠ 0. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as N^(2k)×ℂ^(m-k), where N^(2k) is Kähler-Einstein with Ric_(g_N)=λg_N. In the shrinking case the quotient is trivial; in the normalization λ=1/2 one has R≡ k. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if 𝓛_∇fRic is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching 0≤Ric≤λg forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.Ricci flowRicci solitonsKähler–Ricci solitonsEinstein metricsCurvature pinchingKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2609.36418arxiv:2609.36418Tue, 29 Sep 2026 00:00:00 GMTWenqi Li, Jianyu Ou, Guoqiang Wu, Detang Zhou — Let (M^(2m),g,f,J) be a complete gradient Kähler–Ricci shrinker satisfying Ric+∇² f=1/2 g. We prove that if the scalar curvature is constant, then the soliton is rigid. More precisely, there exists an integer k∈\0,…,m\ such that R≡ k and (M^(2m),g,J)≅ bigl(N^(2k)×ℂ^(m-k), g_N+g_Euc,J_N⊕ J₀bigr), where (N^(2k),g_N,J_N) is Kähler–Einstein and Ric_(g_N)=1/2 g_N. As part of the proof, we establish a Riemannian rigidity criterion: for a complete nonsteady gradient Ricci soliton with constant scalar curvature, the condition 𝓛_∇fRic=0 implies radial flatness and hence rigidity.Ricci solitonsEinstein metricsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2608.20040arxiv:2608.20040Thu, 20 Aug 2026 00:00:00 GMTFengjiang Li, Yuanyuan Qu, Guoqiang Wu — Let (Mⁿ, g, f) be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) Ric ≥ frac∇_∇fRicf on M∖ D, where D is a compact set over M; (ii) (Mⁿ, g, f) smoothly converges to ℝ² × Sⁿ⁻², we conclude that (Mⁿ, g, f) is isometric to ℝ² × Sⁿ⁻². Notably, condition (i) is weaker than the radial flatness condition in.Ricci solitonsScalar curvatureStability & uniqueness https://arxiv.org/abs/2604.26163arxiv:2604.26163Tue, 28 Apr 2026 00:00:00 GMTChen Wang, Guoqiang Wu — Let (M⁴, g, f) be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f= 1/2g. If its scalar curvature is 1, Cheng-Zhou proved that it is a finite quotient of ℝ²× S². In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.Ricci solitons4-manifoldsScalar curvatureBoundaries & non-compact https://arxiv.org/abs/2604.23939arxiv:2604.23939Mon, 27 Apr 2026 00:00:00 GMTChen Wang, Guoqiang Wu — Let (Mⁿ, g, f) be an n-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f= 1/2g. 1. If its scalar curvature is k/2, Ricci curvature is nonnegative and sectional curvature has upper bound 1/(2(k-1)), we prove that the Ricci shrinker is isometric to a finite quotient of ℝ^(n-k)× S^k. 2. If M has constant scalar curvature R=(n-2)/2, and each level set of f has vanishing Weyl curvature, we prove that it is a finite quotient of ℝ²× Sⁿ⁻². This can be seen a generalization of Cheng-Zhou's four dimensional result to high dimension, since the level set of the potential function f has vanishing Weyl curvature automatically when n=4.Ricci solitons4-manifoldsScalar curvatureStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2506.00887arxiv:2506.00887Sun, 01 Jun 2025 00:00:00 GMTFengjiang Li, Jianyu Ou, Yuanyuan Qu, Guoqiang Wu — Suppose (M, g, f) is a 5-dimensional complete shrinking gradient Ricci soliton with R=1. If it has bounded curvature, we prove that it is a finite quotient of ℝ³× S².Ricci solitonsStability & uniqueness https://arxiv.org/abs/2411.10712arxiv:2411.10712Sat, 16 Nov 2024 00:00:00 GMTFengjiang Li, Jianyu Ou, Yuanyuan Qu, Guoqiang Wu — Let (M, g, f) be a 5-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f= λg, where Ric is the Ricci tensor and ∇²f is the Hessian of the potential function f. We prove that it is a finite quotient of ℝ²× S³ if M has constant scalar curvature R=3 λ.Ricci solitonsScalar curvatureStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2411.06395arxiv:2411.06395Sun, 10 Nov 2024 00:00:00 GMTJianyu Ou, Yuanyuan Qu, Guoqiang Wu — Suppose (Mⁿ, g, f) is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature R=1 is isometric to a finite quotient of ℝ²× S², giving a new proof of the main results of Cheng-Zhou.Ricci solitonsScalar curvatureStability & uniqueness https://arxiv.org/abs/2404.16286arxiv:2404.16286Thu, 25 Apr 2024 00:00:00 GMTGuoqiang Wu, Jia-Yong Wu — In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.Ricci solitonsLimit spaces & RCDGeometric inequalities