arXiv subjects

Shu-Cheng Chang

Publications and source records attributed to Shu-Cheng Chang.

At least 19 recordsLinked to original sources

arXiv2026

On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian $5$-manifold

In this paper, we first confirm the Hamilton-Tian conjecture for the Sasaki-Ricci flow in a compact transverse Fano quasi-regular Sasakian $5$-manifold with klt foliation singularities. Secondly, we derive the compactness theorem of Sasaki-Ricci solitons on transverse Fano quasi-regular Sasakian $5$-manifolds. Then,by the second Sasakian structure theorem, we confirm the Hamilton-Tian conjecture for a compact transverse Fano Sasakian $5$-manifold. With its applications, we show that the gradient Sasaki-Ricci soliton orbifold metric on a compact Sasakian $5$-manifold is Sasaki-Einstein if $M$ is transverse $K$-stable.

math.DG↗

arXiv2026

On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature

In this paper, we concern with the Sasaki analogue of Yau uniformization conjecture in a complete noncompact Sasakian manifold with nonnegative transverse bisectional curvature. As a consequence, we confirm that any $5$-dimensional complete noncompact Sasakian manifold with positive transverse bisectional curvature and the maximal volume growth must be CR-biholomorphic to the standard Heisenberg group $\mathbb{H}_{2}$ which can be stated as the standard contact Euclidean $5$-space $\mathbb{R}^{5}$.

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arXiv2025

Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor

We establish integral curvature estimates for complete gradient shrinking Sasaki-Ricci solitons. As an application, we show that any such soliton with harmonic Weyl tensor must be a finite quotient of a sphere. This result can be regarded as the Sasaki analogue of the work of Munteanu and Sesum [15] on Ricci solitons.

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arXiv2025

Rigidity and Classification of Legendrian Self-Shrinkers

In this article, we first classify Legendrian self-shrinkers in $\mathbb{R}% ^{3}$ and $\mathbb{R}^{5}$. We then proved a Legendrian rigidity theorem, which can be regarded as an analogue of the result of Li-Wang \cite{lw}. More precisely, let $F(\Sigma)\subset\mathbb{R}^{5}$ be an orientable Legendrian self-shrinker, if $\Vert A\Vert_{g}^{2}\leq2$ and the associated Legendrian immersion $\bar{F}\subset\mathbb{R}^{4}\times\mathbb{S}^{1}$ is compact, then $\bar{F}$ must be a flat minimal generalized Legendrian Clifford torus in $\mathbb{S}^{5}$, whose cone $\mathcal{C}(\bar{F}(\Sigma))$ is the Harvey-Lawson special Lagrangian cone in $\mathbb{C}^{3}$.

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arXiv2025

Geometry and Topology of Gradient Shrinking Sasaki-Ricci Solitons

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking K\"ahler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

math.DG↗

arXiv2025

The Rigidity Theorem of Legendrian self-shrinkers

By estimating the weighted volume, we obtain the optimal volume growth for Legendrian self-shrinkers. This, in turn, yields a rigidity theorem for entire smooth Legendrian self-shrinkers in the standard contact Euclidean (2n+1)-space.

math.DG↗

arXiv2025

Transverse Rigidity of Shrinking Sasaki-Ricci Solitons

In this paper, we study several properties of Sasaki-Ricci solitons as singularity models of the Sasaki-Ricci flow. First, we establish several fundamental equations for Sasaki-Ricci solitons, which enable us to derive potential estimates and prove the positivity of the scalar curvature. Then we present two criteria for the transverse rigidity of Sasaki-Ricci solitons. As essential applications, we prove that any low-dimensional Sasaki-Ricci soliton with constant scalar curvature must be Sasaki-Einstein, and that any Sasaki-Ricci soliton with harmonic Weyl tensor is a finite quotient of the sphere.

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arXiv2024

On the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five

In this paper, we derive the uniform L^{4}-bound of the transverse conic Ricci curvature along the conic Sasaki-Ricci flow on a compact transverse log Fano Sasakian manifold M of dimension five and the space of leaves of the characteristic foliation is not well-formed. Then we first show that any solution of the conic Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold conic Sasaki-Ricci soliton on M_{infinite} which is a S^{1}-orbibundle over the unique singular conic Keahler-Ricci soliton on a log del Pezzo orbifold surface. As a consequence, there exists a Keahler-Ricci soliton orbifold metric on its leave space which is a log del Pezzo orbifold surface. Second, we show that the conic Sasaki-Ricci soliton is the conic Sasaki-Einstein if M is transverse log K-polystable. In summary, we have the existence theorems of orbifold Sasaki-Ricci solitons and Sasaki-Einstein metrics on a compact quasi-regular Sasakian manifold of dimension five.

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arXiv2023

The Smale Conjecture and Minimal Legendrian Graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$

In this article, we recapture the Smale conjecture on a Sasakian $3$-sphere via the Legendrian mean curvature flow. More precisely,~we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian $3$-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in $\mathbb{S}^{2}\times \mathbb{S}^{3}$. Finally, we will address the rigidity theorem of $2$-dimensional Legendrian self-shrinkers in $\mathbb{R}^{5}$. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in $\mathbb{C}^{3}$ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.

math.DG↗

arXiv2023

Legendrian mean curvature flow in $η$-Einstein Sasakian manifolds

Recently, there are a great deal of work done which connects the Legendrian isotopic problem with contact invariants. The isotopic problem of Legendre curve in a contact 3-manifold was studies via the Legendrian curve shortening flow which was introduced and studied by K. Smoczyk. On the other hand, in the SYZ Conjecture, one can model a special Lagrangian singularity locally as the special Lagrangian cones in C^{3}. This can be characterized by its link which is a minimal Legendrian surface in the 5-sphere. Then in these points of view, in this paper we will focus on the existence of the long-time solution and asymptotic convergence along the Legendrian mean curvature flow in higher dimensional η-Einstein Sasakian (2n+1)-manifolds under the suitable stability condition due to the Thomas-Yau conjecture.

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arXiv2022

Gradient Shrinking Sasaki-Ricci Solitons on Sasakian Manifolds of Dimension Up to Seven

In this paper, we show that the uniform L^4-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian (2n+1)-manifold M. When M is dimension up to seven and the space of leaves of the characteristic foliation is well-formed, we first show that any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton on the limit space which is a S^1-orbibundle over the unique singular Kaehler-Ricci soliton on a normal projective variety with codimension two orbifold singularities. Secondly, for n=1, we show that there are only two nontrivial Sasaki-Ricci solitons on a compact quasi-regular Fano Sasakian three-sphere with its leave space a teardrop-like and football-like space, respectively. For n=2,3, we show that the Sasaki-Ricci soliton is trivial one if M is transverse K-stable.

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arXiv2022

Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

Let (M,η,ξ,Φ,g) be a compact quasi-regular Sasakian 5-manifold with finite cyclic quotient foliation singularities of type (1/r)(1,a). First, we derive the foliation minimal model program by applying the resolution of cyclic quotient foliation singularities. Secondly, based on the study of local model of resolution of foliation singularities, we prove the foliation canonical surgical contraction or the foliation extremal ray contraction under the Sasaki-Ricci flow. As a consequence, we prove a Sasaki analogue of analytic minimal model program with the Keahler-Ricci flow due to Song-Tian and Song-Weinkove.

math.DG↗

arXiv2022

Convergence of the Sasaki-Ricci flow on Sasakian 5-manifolds of general type

In this paper, we show that the uniform L^{4}-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular Sasakian (2n+1)-manifold M of general type. As an application, any solution of the normalized Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular Sasaki η-Einstein metric on the transverse canonical model M_{can} of M if n is less than or equal to 3. In particular for n equal to 2, M_{can} is a S^{1}-orbibundle over the unique Keahler-Einstein orbifold surface (Z_{can},ω_{KE}) with finite point orbifold singularities. The floating foliation (-2)-curves in M will be contracted to orbifold points by the Sasaki-Ricci flow as t goes to infinite.

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arXiv2019

On the CR analogue of Frankel conjecture and a smooth representative of the first Kohn-Rossi cohomology group

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we obtain the CR Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold with the vanishing first Kohn-Rossi cohomology group. In particular, this conjecture holds in a spherical boundary of the Stein manifold.

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arXiv2019

Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian Manifolds

In this paper, we show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group.

math.DG↗

arXiv2019

On the Obata Theorem in a weighted Sasakian manifold

In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obtain the CR Obata theorem in a compact weighted Sasakian manifold with or without boundary.

math.DG↗

arXiv2019

Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed strictly pseudoconvex CR $3$-manifold $(M,\ J,[θ_{0}])$ of the vanishing first Chern class $c_{1}(T_{1,0}M)$. Suppose that $M$ is embeddable and the CR Paneitz operator $P_{0}$ is nonnegative with kernel consisting of the CR pluriharmonic functions. We show that the solution of CR $Q$-curvature flow exists for all time and has smoothly asymptotic convergence on $M\times \lbrack 0,\infty ).$\ As a consequence, we are able to affirm the Conjecture in a closed strictly pseudoconvex CR $3$-manifold of the vanishing first Chern class and vanishing torsion.

math.DG↗

arXiv2019

Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem

In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian. Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz operator and obtain the CR uniformization theorem for a class of CR 3-manifolds. At the end, under the positivity assumption of the pseudohermitian curvature, we derive the existence theorem for pseudo-Einstein contact forms and uniformization theorems in a closed strictly pseudoconvex CR 3-manifold of nonnegative CR Paneitz operator with kernel consisting of the CR-pluriharmonic functions and the CR Q-curvature is CR-pluriharmonic.

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