https://neckpinch.com/authors/frederick-tsz-ho-fong/New arXiv papers by Frederick Tsz-Ho Fong, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.21934arxiv:2610.21934Thu, 15 Oct 2026 00:00:00 GMTFrederick Tsz-Ho Fong, Junbang Liu — Let X be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class [ω], Jian–Shi–Song proved that there exists a unique cscK metric in the class 2πc₁(K_X)+ε[ω] when ε∈ (0,ε₀], using the properness criterion of Chen–Cheng. We prove that the cscK metrics in 2πc₁(K_X)+ε[ω] converge, as εdownarrow0, in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of X as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on X.Ricci flowKähler–Ricci flowEinstein metricsKähler geometryScalar curvatureLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program 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/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/2601.01863arxiv:2601.01863Mon, 05 Jan 2026 00:00:00 GMTTsz-Kiu Aaron Chow, Frederick Tsz-Ho Fong — In this article, we introduce an energy functional on closed Riemannian spin manifolds which unifies Perelman's W- and F-functionals, Baldauf-Ouzch's E-functional, and Dirchlet energy for spinors. We compute its first variation formula, and show that its critical points under natural constraints are twisted Ricci solitons and eigen-spinsors of the weighted Dirac operator. We introduce a negative L^2-gradient flow of this functional, and establish its short-time existence and uniqueness via contraction mapping methods.Ricci solitonsPerelman entropyStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2411.08198arxiv:2411.08198Tue, 12 Nov 2024 00:00:00 GMTFrederick Tsz-Ho Fong — In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree −1 (including the inverse mean curvature flow) in warped product spaces I ×_φ Mⁿ, where Mⁿ is a compact homogeneous manifold and φ" ≥ 0, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than −1 and with an extra assumption φ' ≥ 0. Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.Mean curvature flowOther curvature flowsInverse mean curvature flowSelf-shrinkers & translatorsHomogeneous spacesStability & uniquenessBoundaries & non-compactRotational symmetry & warped products