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Liangming Shen

Publications and source records attributed to Liangming Shen.

12 recordsLinked to original sources
arXiv2026

The character of ideal circle patterns

Let $S$ be an oriented closed surface with a cellular decomposition $\mathcal{D}$ and a weight $\Phi\in(0, \pi)$. It is crucial to determine when $S$ supports an ideal $\mathcal{D}$-type circle pattern $\mathcal{P}$ with the exterior intersection angles given by $\Phi$. Rivin, Bobenko-Springborn and Ge-Hua-Zhou provided perfect solutions and gave wonderful criteria for the existence and uniqueness of ideal circle patterns. However, all criteria established by Rivin, Bobenko-Springborn and Ge-Hua-Zhou are extremely difficult to verify for the given cellular decomposition $\mathcal{D}$ and the weight $\Phi$. In this paper, we introduce the character $\mathcal{L}(\mathcal{D},\Phi)$ depends only on the data of the weighted cellular decomposition $(\mathcal D, \Phi)$ on $S$, and give some quite simple criteria for the existence of ideal circle patterns realizing $(\mathcal{D},\Phi)$. It seems that our character-type criteria are the first conditions totally different from criteria of Rivin, Bobenko-Springborn and Ge-Hua-Zhou, and provide more easily verifiable criteria. Our new character-type theorems may be of some independent interest. As an application, we give a new descriptions of the curvature image set $\mathbf{K}(\mathbb{R}^N_{>0})$. To approach our results, we shall use the combinatorial Ricci flows with ideal circle patterns introduced by Ge-Hua-Zhou as a fundamental tool. The main difficulty in the proof of our results is to establish the compactness of the solution to the flows. To circumvent the difficulty, we borrow the techniques developed by Ge and his collaborators.

math.DG↗
arXiv2022

The K"ahler-Ricci flow with Log Canonical Singularities

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Finally we also construct K"ahler-Ricci flow solutions performing divisorial contractions and flips with log canonical singularities.

math.DG↗
arXiv2019

The Partial $C^{0}$-estimate along a general continuity path and applications

We establish a new partial $C^{0}$-estimate along a continuity path mixed with conic singularities along a simple normal crossing divisor and a positive twisted $(1,1)$-form on Fano manifolds. As an application, this estimate enables us to show the reductivity of the automorphism group of the limit space, which leads to a proof of Yau-Tian-Donaldson Conjecture admitting some types of holomorphic vector fields.

math.DG↗
arXiv2018

Kähler-Ricci flow of cusp singularities on quasi projective varieties

Let $\overline{M}$ be a compact complex manifold with smooth Kähler metric $η$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hatω$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to Kähler-Ricci flow on $M$ which are comparable to $\hatω$, starting from a smooth initial metric $ω_0=η+i\partial \bar{\partial} ϕ_0$ where $ϕ_0\in C^{\infty}(M)$. When $ω_0\geq c \hatω$ on $M$ for some $c>0$ and $ϕ_0$ has zero Lelong number, we construct a smooth solution $ω(t)$ to Kähler-Ricci flow on $M\times [0, T_{[ω_0 ]})$ where $T_{[ω_0 ]}:= \sup \{ T: [η] +T (c_1(K_{\overline{M}}) + c_1(\mathcal{O}_D))\in \mathcal{K}_M \}$ so that $ω(t)\geq (\frac{1}{n} - \frac{4\hat{K}t}{c} )\hatω$ for all $t\leq \frac{c}{4n\hat{K}}$ where $\hat{K}$ is a non-negative upper bound on the bisectional curvatures of $\hatω$ (see Theorem 1.2). In particular, we do not assume $ω_0$ has bounded curvature. If $ω_0$ has bounded curvature and is asymptotic to $\hatω$ in an appropriate sense, we construct a complete bounded curvature solution on $M\times [0, T_{[ω_0 ]})$ (see Theorem 1.3). These generalize some of the results of Lott-Zhang in [15]. On the other hand if we only assume $ω_0\geq c η$ on $M$ for some $c>0$ and $ϕ_0$ is bounded on $M$, we construct a smooth solution to Kähler-Ricci on $M\times [0, T_{[ω_0 ]})$ which is equivalent to $\hatω$ for all positive times. This includes as a special case when $ω_0$ is smooth on $\overline{M}$ in which case the solution becomes instantaneously complete on $M$ under Kähler-Ricci flow (see Theorem 1.1).

math.DG↗
arXiv2018

On the deformation of ball packings

In this paper, we study the geometric aspects of ball packings on $(M,\mathcal{T})$, where $\mathcal{T}$ is a triangulation on a 3-manifold $M$. We introduce a combinatorial Yamabe invariant $Y_{\mathcal{T}}$, depending on the topology of $M$ and the combinatoric of $\mathcal{T}$. We prove that $Y_{\mathcal{T}}$ is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing Cooper-Rivin-Glickenstein functional. We then study the combinatorial Yamabe flow introduced by Glickenstein \cite{G0}-\cite{G2}. We first prove a small energy convergence theorem which says that the flow would converge to a constant curvature metric if the initial energy is close in a quantitative way to the energy of a constant curvature metric. We shall also prove: although the flow may develop singularities in finite time, there is a natural way to extend the solution of the flow so as it exists for all time. Moreover, if the triangulation $\mathcal{T}$ is regular (that is, the number of tetrahedrons surrounding each vertex are all equal), then the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.

math.DG↗
arXiv2018

The Kähler-Ricci flow on pseudoconvex domains

We establish the existence of Kähler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete Kähler-Einstein metric, which generalizes Topping's works on surfaces.

math.DG↗
arXiv2018

$C^{2,α}$-estimate for conical Kähler-Ricci flow

We establish a parabolic version of Tian's $C^{2,α}$-estimate for conical complex Monge-Ampere equations, which includes conical Kähler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler-Ricci flow in arXiv:1411.7284.

math.DG↗
arXiv2018

Conic Kähler-Einstein metrics along simple normal crossing divisors on Fano manifolds

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which generalizes Li-Rubinstein's estimate and derive high order estimates from this estimate.

math.DG↗
arXiv2018

Gradient flow of the norm squared of a moment map over Kahler manifolds

Inspired by Wilkin's work [23, 24] on Morse theory for the moduli space of Higgs bundles, we study the moduli space of gauged holomorphic maps by a heat flow approach in the spirit of Atiyah and Bott in a series of papers. In this paper, applying the method of Hong [9], we establish the global existence of smooth solutions of the gradient ow equations of the vortex functional over a compact Kahler manifold.

math.DG↗
arXiv2015

Smoothing conic Kähler metrics with uniformly upper bisectional curvature bound

Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may need to choose a new background Kähler metric in the same cohomology class of the original background metric. This setting will be helpful to the study of conical Kähler-Einstein metrics and conical Kähler-Ricci flow.

math.DG↗
arXiv2014

Unnormalize conical Kähler-Ricci flow

We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle $K_{M}+(1-β)[D]$ is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal crossing divisor.

math.DG↗