https://neckpinch.com/authors/kai-zheng/New arXiv papers by Kai Zheng, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.34340arxiv:2610.34340Sun, 25 Oct 2026 00:00:00 GMTJunhao Tian, Kai Zheng — In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any Tⁿ-invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2610.05217arxiv:2610.05217Sun, 04 Oct 2026 00:00:00 GMTHaozhao Li, Bing Wang, Kai Zheng — We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.Other curvature flowsSingularitiesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2606.06619arxiv:2606.06619Thu, 04 Jun 2026 00:00:00 GMTHaozhao Li, Yuanqing Ma, Kai Zheng — In this work, we establish compactness and regularity results for complete gradient Laplacian solitons of closed G₂-structures. Under a lower scalar-curvature bound and a distance-dependent bound on the gradient of the soliton potential, we prove pointed measured Gromov-Hausdorff compactness. With a uniform lower bound for the localised Perelman entropy, the Gromov-Hausdorff convergence improves to pointed C^(1,α) convergence. Our principal result shows that this C^(1,α) convergence upgrades to smooth convergence on the regular set. More precisely, after passing to a subsequence, the metrics, the defining positive 3-forms, and the soliton potentials converge smoothly on every compact subset of the regular set, and the limiting data define a gradient Laplacian soliton. The proof develops a local entropy method adapted to G₂-solitons. Since the available C^(1,α) control does not directly close an elliptic bootstrap for the G₂-soliton, and no suitable pseudolocality theorem is available in this setting, we instead use the localised Perelman's functionals. These yield an entropy ε-regularity theorem and a gap theorem for scalar-flat solitons. On the regular set, pointed C^(1,α) convergence gives an almost-Euclidean local isoperimetric inequality, which in turn verifies the required small-entropy condition automatically. The resulting curvature bounds are then combined with G₂-specific differential identities and quantitative interior estimates to control the soliton data. Finally, at the critical exponent in dimension seven, we show that a uniform weighted L^(7/2) -curvature bound then yields pointed C^∞ compactness.Generalized & G₂ flowsPerelman entropyScalar curvatureLimit spaces & RCDLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2312.05816arxiv:2312.05816Sun, 10 Dec 2023 00:00:00 GMTHaozhao Li, Linwei Zhang, Kai Zheng — In this paper, we show that the Calabi flow can be extended as long as the L^p scalar curvature is uniformly bounded for some p>n, and on a compact extremal Kähler manifold the Calabi flow with uniformly bounded L^p(p>n) scalar curvature exists for all time and converges exponentially fast to an extremal Kähler metric.Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2303.00631arxiv:2303.00631Wed, 01 Mar 2023 00:00:00 GMTJie He, Kai Zheng — Let (M,ω) be a compact symplectic manifold. We denote by ac the space of all almost complex structure compatible with ω. ac has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost Kähler metric in ac. We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to Kähler case. We also show weak parabolicity of the Hermitian Calabi flow.Other curvature flowsKähler geometryKähler–Einstein & K-stability