arXiv subjects

Yongjia Zhang

Publications and source records attributed to Yongjia Zhang.

At least 19 recordsLinked to original sources

arXiv2026

A gap theorem for metric solitons and its applications

In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\varepsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).

math.DG↗

arXiv2026

Rigidity and gap theorems for Ricci shrinkers

We prove local versions of the Ricci curvature and $ν$-entropy gap theorems for Ricci shrinkers, which respectively generalize a previous result of Munteanu-Wang and a prior result of the authors with Ma. The key point is that these local gaps depend only on the dimension and not on the global entropy or any other geometric information of the Ricci shrinker. As an application, we provide a local criterion for removable Type~I singularities of the Ricci flow.

math.DG↗

arXiv2026

Ancient Ricci flows with nonnegative Ricci curvature

In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity -- a noncollapsed $\mathbb{F}$-limit metric soliton [Bam23,CMZ23]. We first prove some estimates for noncollapsed $\mathbb{F}$-limit metric solitons with nonnegative Ricci curvature, and then obtain two dichotomy theorems for ancient Ricci flows. In particular, we show that: (1) for a noncollapsed ancient Ricci flow with nonnegative Ricci curvature, either its asymptotic volume ratio is always zero, or every tangent flow at infinity is a Ricci flat cone; (2) for a noncollapsed ancient Ricci flow with positively pinched Ricci curvature ($\operatorname{Ric}\ge \varepsilon R g$), either it is compact, or every tangent flow at infinity is a Ricci flat cone.

math.DG↗

arXiv2026

Correction to: Curvature growth of some 4-dimensional gradient Ricci soliton singularity models

This note corrects an error in the proof of Proposition 13 in arXiv:1903.09181 and simultaneously establishes a more general result. We prove that if $M $ is a compact connected oriented $4$-manifold with connected boundary $\partial M$, and if an unbounded number of disjoint copies of $M$ embed topologically and locally flatly in the interior of a compact $4$-manifold $N,$ then $\operatorname{Tor}H_1(\partial M;\mathbb{Z})$ is a direct double, i.e., $\operatorname{Tor}H_1(\partial M;\mathbb{Z})\cong A \oplus A$, with the linking pairing vanishing identically on the first summand, i.e., the linking pairing is split metabolic. This partially generalizes Hantzsche's theorem stating that the linking pairing for a closed $3$-manifold that embeds in $S^4$ is hyperbolic.

math.DG↗

arXiv2025

An $ε$-regularity theorem for Perelman's reduced volume

In this article, we prove an $ε$-regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to $1$, then the curvature radius at the base point cannot be too small.

math.DG↗

arXiv2024

On noncollapsed $\mathbb{F}$-limit metric solitons

A noncollapsed $\mathbb{F}$-limit metric soliton is a self-similar singularity model that inevitably arises when studying the Ricci flow with the tool of $\mathbb{F}$-convergence [Bam20a,Bam20b,Bam20c]. In this article, we shall present a systematic study of the noncollapsed $\mathbb{F}$-limit metric soliton, and show that, apart from the known results in [Bam20c], it satisfies many properties of smooth Ricci shrinkers. In particular, we show a quadratic lower bound for the scalar curvature, a local gap theorem, a global Sobolev inequality, and an optimal volume growth lower bound.

math.DG↗

arXiv2023

Local smooth convergence of $\mathbb{F}$-limit flows

The metric flow is introduced and extensively studied by Bamler [Bam20b, Bam20c], especially as an $\mathbb{F}$-limit of a sequence of smooth Ricci flows with uniformly bounded Nash entropy, in which case each regular point on the limit is a point of smooth convergence. In this note, we shall consider the $\mathbb{F}$-convergence of a sequence of $\mathbb{F}$-limit flows, and, like Bamler, show that each regular point on the limit is also a point of smooth convergence. The main result will be applied in a forthcoming work of the authors [CMZ23].

math.DG↗

arXiv2023

Dimension Reduction for Positively Curved Steady Solitons

We consider noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature. We show that such solitons always dimension reduce at infinity. This generalizes an earlier result in [CDM22] to higher dimensions. In dimension four, we classify possible reductions at infinity, which lays foundation for possible classifications of steady solitons. Moreover, we show that any tangent flow at infinity of a general noncollapsed steady soliton must split off a line. This generalizes an earlier result in [BCDMZ21] to higher dimensions. While this article is under preparation, we realized that part of our main results are proved independently in a recent post [ZZ23] under different assumptions.

math.DG↗

arXiv2023

New vision of convection induced freckle formation theory in Nickel-based superalloys by electron microscopy

Freckles, one of the common defects in blades used in heavy duty gas turbines, hugely deteriorates blades mechanical properties and liability under service conditions. Thermal-solutal convection theory is a widely adopted formation mechanism but few solid experimental evidences were reported. Here for the first time we systematically studied the microstructure of 117 grains in freckle chains from four different Nickel-based superalloys of either single crystal or directionally solidified alloys. The relationship between the internal stress and the misorientation throughout the freckle chains is studied by means of state-of-the-art electron microscopy. All results give new experimental proof to the theory of thermal-solutal convection, which is further supported by the fact that borides at the boundary are randomly orientated to alloys. Our results enrich the methodology of freckle study, providing a new sight of the formation mechanism of casting defects.

cond-mat.mtrl-sci↗

arXiv2022

A local gap theorem for Ricci shrinkers

We prove a local gap theorem for Ricci shrinkers, which states that if the local $μ$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global $μ$-functional to be close enough to $0$. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].

math.DG↗

arXiv2022

Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

In this paper, we prove a pseudolocality-type theorem for $\mathcal L$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the $\mathcal L$-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L.~Chen.

math.DG↗

arXiv2022

Lower bounds for the scalar curvatures of Ricci flow singularity models

In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger--Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of $4$-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.

math.DG↗

arXiv2022

Kählerity of Einstein four-manifolds

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) Kähler-Einstein, provided that $λ_2 \geq -\frac{S}{12}$, where $λ_2$ is the middle eigenvalue of the self-dual Weyl tensor $W^+$ and $S$ is the scalar curvature. The same conclusion holds for closed oriented four-manifolds with $δW^+=0$.

math.DG↗

arXiv2022

Four-Dimensional Steady Gradient Ricci Solitons with $3$-Cylindrical Tangent Flows at Infinity

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $\mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general.

math.DG↗

arXiv2022

Volume growth estimates of gradient Ricci solitons

In this paper, we survey the volume growth estimates for shrinking, steady, and expanding gradient Ricci solitons. Together with the known results, we also prove some new volume growth estimates for expanding gradient Ricci solitons.

math.DG↗

arXiv2022

The rate of $\mathbb{F}$-convergence for Ricci flows with closed and smooth tangent flows

This article is a continuation of [CMZ21b], where we proved that a Ricci flow with a closed and smooth tangent flow has unique tangent flow, and its corresponding forward or backward modified Ricci flow converges in the rate of $t^{-β}$ for some $β>0$. In this article, we calculate the corresponding $\mathbb{F}$-convergence rate: after being scaled by a factor $λ>0$, a Ricci flow with closed and smooth tangent flow is $|\log λ|^{-θ}$ close to its tangent flow in the $\mathbb{F}$-sense, where $θ$ is a positive number, $λ\gg 1$ in the blow-up case, and $λ\ll 1$ in the blow-down case.

math.DG↗

arXiv2022

Canonical surgeries in rotationally invariant Ricci flow

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

math.DG↗

arXiv2021

Hamilton-Ivey estimates for gradient Ricci solitons

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\leq cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for $4$-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a $3$-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any $3$-dimensional gradient Ricci expander $C^2$ asymptotic to $\left(C(\mathbb S^2), dt^2+αt^2 g_{\mathbb{S}^2}\right)$ is rotationally symmetric, where $α\in (0,1]$ is a constant and $g_{\mathbb{S}^2}$ is the standard metric on $\mathbb{S}^2$ with constant curvature $1$.

math.DG↗