https://neckpinch.com/authors/detang-zhou/New arXiv papers by Detang Zhou, 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/2610.11935arxiv: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/2605.23199arxiv:2605.23199Fri, 22 May 2026 00:00:00 GMTXu Cheng, Franciele Conrado, Neilha Pinheiro, Detang Zhou — Let (M, g, f, τ) be a complete Ricci shrinker satisfying Ric+∇²f=g/2τ and let R denote its scalar curvature. For a confined function V on M, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator −Δ+R/4+V, expressed in terms of an integral quantity involving V and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's μ-functional. We also study the drifted Schrödinger operator −Δ_f+V on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if V is affine.Ricci solitonsScalar curvatureLimit spaces & RCDEigenvalues & spectrum https://arxiv.org/abs/2505.03499arxiv:2505.03499Tue, 06 May 2025 00:00:00 GMTFranciele Conrado, Detang Zhou — In this paper, we study the drifted Laplacian Δ_f on a hypersurface M in a Ricci shrinker (M,g,f). We prove that the spectrum of Δ_f is discrete for immersed hypersurfaces with bounded weighted mean curvature in a Ricci shrinker with a mild condition on the potential function. Next, we give a lower bound for the first nonzero eigenvalue of Δ_f when the hypersurface is an embedded f-minimal one. This estimate contains the case of compact minimal hypersurfaces in a positive Einstein manifold, in particular Choi and Wang's estimate for minimal hypersurfaces in a round sphere. The estimate also recovers the ones of Ding-Xin and Brendle-Tsiamis on self-shrinkers.Ricci solitonsEinstein metricsSelf-shrinkers & translatorsEigenvalues & spectrumMinimal & CMC surfaces https://arxiv.org/abs/2410.07372arxiv:2410.07372Wed, 09 Oct 2024 00:00:00 GMTHelton Leal, Matheus Vieira, Detang Zhou — In this paper, we study the spectrum of the drift Laplacian on Ricci expanders. We show that the spectrum is discrete when the potential function is proper, and we show that the hypothesis on the properness of the potential function cannot be removed. We also extend previous results concerning the asymptotic behavior of the potential function on Ricci expanders. This allows us to conclude that the drift Laplacian has discrete spectrum on Ricci expanders whose Ricci curvature is bounded below by a suitable constant, possibly negative. Further, we compute all the eigenvalues of the drift Laplacian on rigid expanders and rigid shrinkers. Lastly, we investigate the second eigenvalue of the drift Laplacian on rigid Ricci expanders whose Einstein factor is a closed hyperbolic Riemann surface.Ricci solitonsSurfacesEigenvalues & spectrum https://arxiv.org/abs/2309.16017arxiv:2309.16017Wed, 27 Sep 2023 00:00:00 GMTFranciele Conrado, Detang Zhou — Let (Mⁿ,g,f) be a Ricci shrinker such that Ric_f=1/2g and the measure induced by the weighted volume element (4π)^(−n/2)e^(−f)dv_g is a probability measure. Given a point p∈ M, we consider two probability measures defined in the tangent space TₚM, namely the Gaussian measure γ and the measure overlineν induced by the exponential map of M to p. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric g₀ between the measures overlineν and γ, and which also elucidates the rigidity implications resulting from this estimate.Ricci solitonsStability & uniqueness https://arxiv.org/abs/2203.14916arxiv:2203.14916Mon, 28 Mar 2022 00:00:00 GMTXu Cheng, Ernani Ribeiro, Detang Zhou — In this article, we investigate the geometry of 4-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a 4-dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.Ricci solitons4-manifolds https://arxiv.org/abs/2203.06270arxiv:2203.06270Fri, 11 Mar 2022 00:00:00 GMTHilário Alencar, Gregório Silva Neto, Detang Zhou — In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize the result of Hopf as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.Other curvature flowsRicci solitons3-manifoldsRotational symmetry & warped productsPhysics