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Fengjiang Li

Publications and source records attributed to Fengjiang Li.

10 recordsLinked to original sources
arXiv2026

Rigidity of shrinking gradient ricci soliton with constant scalar curvature

Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$, we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.

math.DG↗
arXiv2026

Estimates on scalar curvature of self-shrinkers

In this paper, we study $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form. We partially resolve the conjecture on $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form. Furthermore, if the scalar curvature of an $n$-dimensional self-shrinker is constant, then we prove that the scalar curvature $R$ satisfies $R\leq n-1$. We also classify $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with non-negative constant scalar curvature.

math.DG↗
arXiv2025

Quasi-Einstein manifolds with Harmonic Weyl curvature

In this paper, we classify $n$-dimensional ($n\geq 5$) quasi-Einstein manifolds with harmonic Weyl curvature, thus extending the work of Shin \cite{Shin} in dimension four for quasi-Einstein manifolds and refining the work of He-Petersen-Wylie \cite{HPW}. As a consequence, we provide new examples of quasi-Einstein manifolds which are neither locally conformally flat nor D-flat in the sense of \cite{CC12}.

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.

math.DG↗
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.

math.DG↗
arXiv2021

Vacuum Static Spaces with Harmonic Curvature

In this paper, we classify $n$-dimensional ($n\geq 5$) vacuum static spaces with harmonic curvature, thus extending the $4$-dimensional work by Kim-Shin \cite{KS}. As a consequence, we provide new counterexamples to the Fischer-Marsden conjecture on compact vacuum static spaces.

math.DG↗
arXiv2021

Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

Our main aim in this paper is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\geq 5$) complete noncompact gradient steady Ricci soliton with harmonic Weyl tensor and multiply warped product metric is either Ricci flat or isometric to the Bryant soliton up to scaling. Meanwhile, for $n\ge 5$, we provide a local structure theorem for $n$-dimensional connected (not necessarily complete) gradient Ricci solitons with harmonic Weyl curvature and multiply warped product metric.

math.DG↗
arXiv2019

Monotonicity of functionals along conformal Ricci flow

The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.

math.DG↗