arXiv research

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

New metrics found on orbifold resolutions with specific curvature.

The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.

Study finite time singularities in Ricci flow with bounded scalar curvature.

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

The main result of this paper is: Given any constant C, there is $(ε,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(ε,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the…

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in d…

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study t…

Constructs a new type of metric for elliptic surfaces.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, us…

Study classifies bubbles of Type I singularities in Kähler-Ricci flow on compact surfaces.

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumptio…

Ancient solutions of Ricci flow with Type I growth are classified.

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

Study on ancient Ricci flows with positive curvature, proving noncollapsedness.

The purpose of this article is to examine the possible shapes of type I singularities that form in the mean curvature flow of submanifolds of arbitrary codimension, assuming that the initial submanifold satisfies a particular curvature pinching condition.

Study of Lagrangian mean curvature flow with equivariant symmetry.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Proves properties of free boundary stable MOTS in spacetimes.

We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose e…

Let $(M,g,φ)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $φ$. We show that a complete, $κ$-noncollapsed solution $(M,g,φ)$ to this coupled Ricci flow with a Type I singularity at time $T<\infty$ will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base po…

Ancient convex solutions to flow equations are limited to simple shapes.

In this short paper, we show there do not exist three-dimensional noncompact $κ$-solutions of Ricci flow that have positive curvature and satisfy a Type-I bound. This represents progress towards the proof of Perelman's conjecture that the only complete noncompact three-dimensional $κ$-solution with positive curvature i…

We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott. To do this we adapt the technique of quantitative s…

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

The study classifies manifolds based on their geometric properties and invariants.

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular s…

Motivated by recent proposals for a de Sitter version of the AdS/CFT correspondence, we give some topological restrictions on spacetimes of de Sitter type, i.e., spacetimes with $Λ>0$, which admit a regular past and/or future conformal boundary. For example we show that if $M^{n+1}$, $n \ge 2$, is a globally hyperbolic…

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

Study of splitting maps in Type I Ricci flows for understanding singular set structure.

In a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role o…

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

Study shows curvature behavior for Kähler-Ricci flow with finite singularities.

Study ancient solutions on noncompact steady Ricci solitons, proving types of ancient solutions.

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions ar…

The curvature-dimension condition implies a new weighted scalar curvature.

Study curvature properties of $G_2$ connections with skew-symmetric torsion.

Paper establishes a relation between Berwald scalar curvature and S-curvature.

The paper examines Randers metrics with isotropic scalar curvature properties.

The paper studies Kropina metrics with a specific curvature property.

The paper studies Berwald scalar curvature properties in Finsler geometry.

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Small Weyl infimum on 4-manifolds with positive scalar curvature.

New scalar curvature defined from Ollivier-Ricci curvature for graphs.

The aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of $H_{p}$-scalar curvature and of $H_{p}\,$-constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of $H_{p}$-scalar curvature to be of perpendicular scalar curvature i…