https://neckpinch.com/authors/hung-tran/New arXiv papers by Hung Tran, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.40994arxiv:2610.40994Thu, 29 Oct 2026 00:00:00 GMTErnani Ribeiro, Hung Tran — We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are U(2)-invariant nuts on ℝ⁴, including the Bryant and Cao solitons, and bolts on S² × ℝ² and on 𝓞(−m)→ℂP}¹. For each m≥3 there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.Ricci solitonsKähler–Ricci solitons(Non)collapsing4-manifoldsKähler geometryHomogeneous spacesBoundaries & non-compact https://arxiv.org/abs/2609.03332arxiv:2609.03332Thu, 03 Sep 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Hung Tran — We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds N := N₁ × ⋯ × Nᵣ, with metric constructed using the ansatz considered in, et. al. In the earlier work by the authors, we considered the "two-bolt" case where both ends of the foliation close with the "bolt" N. In this article, we continue our work on the more subtle "nut-bolt" and "two-nut" cases. The former has one end of the interval closes with a nut-type collapse (i.e. N' := N₂ × ⋯ × Nᵣ) and the other with a bolt (i.e. N). The compactification M is then a ℂP}^(m+1)-bundle over N'. The "two-nut" case is one that both ends close with nut-type collapses, necessarily two of the Nᵢ's must be ℂP}^(m₀) and ℂP}^(m_ℓ), and the compactification M is a ℂP}^(m₀+m_ℓ+1)-bundle over ∏_(k≥ 3)Nₖ. We proved that in all "two-bolt", "nut-bolt" and "two-nut" caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be (Σ^(m+1), g_Σ(t)) × (ℂ^(k), flat) with m, k ≥ 0, where Σ is one of the following: ℂP}^(m+1), Tot(𝓛^(⊕(m+1))), or a projectivization ℙ(𝓞^(⊕(m₀+1)) ⊕ 𝓛^(⊕(m_ℓ+1))) with m₀ + m_ℓ = m, and 𝓛 is a line bundle over the product of some of the N₁, ⋯, Nᵣ factors. The metric g_Σ(t) is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.Ricci flowKähler–Ricci flowRicci solitonsEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2605.01557arxiv:2605.01557Sat, 02 May 2026 00:00:00 GMTHa Tuan Dung, Catherine Searle, Hung Tran — This paper studies a non-trivial gradient Kähler-Ricci soliton, of complex dimension n, with an isometry group of dimension at least n²−1. We show that the isometry group acts by cohomogeneity one and, consequently, admits a special ansatz involving a Sasakian model. In complex dimension two, we can actually say more: namely, that every such soliton has maximal symmetry; that is, the isometry group is exactly of dimension 2². In addition, we prove that, if the isometry group acts by cohomogeneity one on a non-trivial gradient Ricci soliton (not necessarily Kähler), the potential function is invariant by the action.Ricci solitonsKähler–Ricci solitonsKähler geometryHomogeneous spacesSasakian & contact geometry https://arxiv.org/abs/2601.18931arxiv:2601.18931Mon, 26 Jan 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Hung Tran — In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2506.03545arxiv:2506.03545Wed, 04 Jun 2025 00:00:00 GMTHung Tran, Kazuo Yamazaki — This paper studies a complete gradient Ricci soliton with an isoparametric potential function. Our first theorem asserts that, for the steady case, there is a critical level set of codimension greater than one. This is consistent with construction of cohomogeneity one models with singular orbits. There is a partial result for the shrinking case. We also study a particular ansatz of popular interest and obtain asymptotic behaviors.Ricci solitonsHomogeneous spaces https://arxiv.org/abs/2409.13123arxiv:2409.13123Thu, 19 Sep 2024 00:00:00 GMTXiaodong Cao, Hung Tran — [Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.Ricci flowRicci solitonsSingularities4-manifoldsStability & uniqueness https://arxiv.org/abs/2406.06997arxiv:2406.06997Tue, 11 Jun 2024 00:00:00 GMTHa Tuan Dung, Hung Tran — We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local and global implications.Ricci solitons https://arxiv.org/abs/2404.18866arxiv:2404.18866Mon, 29 Apr 2024 00:00:00 GMTHung Tran — Let (M, g, ω, f, λ) be a Kähler gradient Ricci soliton in real dimension four. One first observes that it is an integrable Hamiltonian system in a classical sense. Indeed, all known complete examples are toric and the symmetry is intrinsically related to the potential function f and the scalar curvature SS. While another article addresses the case that these functions are functionally dependent, this one considers the independent case. The main result states that the soliton admits a toric action under a generic assumption. That is, one assumes that the system is non-degenerate and the potential function f is proper. Then there is an effective, completely integrable Hamiltonian toric T²- action on (M, ω).Ricci solitons4-manifoldsSurfacesKähler geometryScalar curvature https://arxiv.org/abs/2310.11328arxiv:2310.11328Tue, 17 Oct 2023 00:00:00 GMTHung Tran — All known examples of simply-connected gradient Kähler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function f and the scalar curvature SS. In this article, we consider the case that f and SS are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result.Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryScalar curvatureHomogeneous spacesSasakian & contact geometry https://arxiv.org/abs/2306.05787arxiv:2306.05787Fri, 09 Jun 2023 00:00:00 GMTHung Tran — Let (M, g, J, f) be an irreducible non-trivial Kähler gradient Ricci soliton of real dimension 2n. We show that its group of isometries is of dimension at most n² and the case of equality is characterized. As a consequence, our framework shows the uniqueness of U(n)-invariant Kähler gradient Ricci solitons constructed earlier. There are corollaries regarding the groups of automorphisms or affine transformations and a general version for almost Hermitian GRS. The approach is based on a connection to the geometry of an almost contact metric structure.Ricci solitonsKähler geometryStability & uniquenessSasakian & contact geometry https://arxiv.org/abs/2212.05267arxiv:2212.05267Sat, 10 Dec 2022 00:00:00 GMTXiaodong Cao, Ernani Ribeiro, Hung Tran — In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on S²× ℝ² (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.Ricci solitonsKähler–Ricci solitons4-manifoldsKähler geometryScalar curvatureStability & uniqueness