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Liangdi Zhang

Publications and source records attributed to Liangdi Zhang.

10 recordsLinked to original sources
arXiv2026

Higgs-Demailly System and Positivity of Higgs Bundles

We prove that the Higgs-Demailly system on a compact Riemann surface admits a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. This gives an independent analytic characterization of H-ampleness by Griffiths-positive Hitchin-Simpson curvature. The proof extends the Demailly-Pingali-Murakami approach using a priori estimates and Leray-Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.

math.DG↗
arXiv2026

Existence and geometry of Hermitian metrics with constant second scalar curvature

We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature within balanced Hermitian conformal classes, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a K\"ahler-Einstein metric.

math.DG↗
arXiv2025

First eigenvalue estimates on complete balanced Hermitian manifolds

We establish lower bounds for the first positive eigenvalue of the Laplace--de Rham operator on complete balanced Hermitian manifolds in terms of curvature of the Strominger--Bismut connection. Under a positive lower bound for its holomorphic Ricci curvature, we prove a Lichnerowicz--Obata type estimate and characterize the equality case in the K\"ahler setting. We also derive Li--Yau and Zhong--Yang type estimates from lower bounds on the same holomorphic Ricci curvature, including estimates under weaker assumptions only along a first eigendirection in the compact case. Finally, under a positive lower bound for the holomorphic sectional curvature of the Strominger--Bismut connection and a torsion-commutator condition along a first eigendirection, we obtain a lower bound for the first eigenvalue. In the compact case, the commutator condition follows from vanishing of the Strominger--Bismut torsion in that eigendirection. These results extend several classical K\"ahler and Riemannian spectral estimates to the balanced non-K\"ahler setting.

math.DG↗
arXiv2025

K\"ahlerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures

We prove several K\"ahlerness criteria for compact Hermitian surfaces under semi-definiteness assumptions on natural Ricci curvatures of the Strominger-Bismut connection. The key tools for proving these results are explicit identities relating these Ricci curvatures to the torsion of the Strominger-Bismut connection, together with corresponding Chern number identities on compact Hermitian surfaces. The results may be viewed as Strominger-Bismut analogues and reformulations of Yang's K\"ahlerness criteria for compact complex surfaces.

math.DG↗
arXiv2023

Gradient estimates for Donaldson's equation on a compact Kähler manifold

We prove a gradient estimate for Donaldson's equation \[ω\wedge(χ+\sqrt{-1}\partial\overline{\partial}φ)^{n-1}=e^F(χ+\sqrt{-1}\partial\overline{\partial}φ)^n\] (and its parabolic analog) on an $n$-dimensional compact Kähler manifold $(M,ω)$ with another Hermitian metric $χ$ directly from the uniform upper bounds for $tr_ωχ_φ$ and Alexandrov-Bakelman-Pucci (ABP) maximum principle.

math.DG↗
arXiv2023

Existence of the $(α,β)$-Ricci-Yamabe flow on closed manifolds

On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.

math.DG↗
arXiv2022

Comparison principles and dynamical stability for a logarithmic inverse-trace flow on compact Hermitian manifolds

We study a logarithmic inverse-trace flow on compact connected Hermitian manifolds. We first establish a parabolic comparison principle and the resulting oscillation nonexpansiveness for admissible potentials modulo constants. We then introduce common-drift sub- and supersolutions. Their combination yields a uniform drift-corrected zero-order estimate for arbitrary admissible initial data, while the lower barrier alone implies the strict cone condition required in the second-order estimate. Combining these observations with Sun's Hermitian second-order estimate, parabolic Evans-Krylov regularity, Schauder estimates, and a uniform Harnack inequality, we obtain long-time existence and exponential smooth convergence. In particular, whenever the associated elliptic equation admits a smooth solution, its class is the unique globally attracting fixed point of the induced semiflow, and any two trajectories synchronize exponentially modulo constants. We also derive a Hermitian cone-supersolution convergence criterion and an affine-renormalization criterion that uses Sun's convergence theorem to construct one elliptic solution and then propagates global dynamical stability to every admissible initial potential.

math.DG↗
arXiv2019

On the classification of four-dimensional gradient Ricci solitons

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. The same result can be obtained under the condition of $div^4W^\pm=0$. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.

math.DG↗
arXiv2017

Rigidity of Gradient Shrinking Ricci Solitons

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$ or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.

math.DG↗