Ricci Flow on Surfaces with Conical Singularities

Yin, Hao

doi:10.1007/s12220-010-9136-1

Cited by 35 publications

(35 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One expects uniqueness of the flow to fail without specifying further boundary restrictions. Not only is it possible to obtain a Ricci flow that remains singular (see Mazzeo‐Rubinstein‐Sesum and Yin ), but one may also obtain a solution to the flow starting at a singular metric that becomes instantaneously complete (see Giesen‐Topping , ) or smooths out the cone angle (see Simon ).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Yamabe flow on manifolds with edges

Bahuaud

Vertman

2013

Mathematische Nachrichten

View full text Add to dashboard Cite

Abstract. Let (M, g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schaudertype estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Yamabe flow on manifolds with edges

Bahuaud

Vertman

2013

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…We state now precisely our results. As in [52,53], we fix a metric g β with conic singularities on the pair (S 2 , β), which is smooth away from the points {p 1 , · · · , p k }, given by g β = |z| −2β i |dz| 2 in a holomorphic coordinate z near each point p i , and normalized to that S 2 g β = 2. Note that this normalization coincides with g ∈ c 1 (S 2 ).…”

Section: Introductionmentioning

confidence: 99%

“…Because of the conical singularities, it is important to specify the regularity of the metrics g(t), t ≥ 0. In [52,53], the key notion of weighted Schauder spaces C ℓ,α (S 2 , β) for a surface with conical singularities was introduced, for each ℓ ∈ N, α ∈ (0, 1). Their precise definition will be recalled in Section §2 below.…”

Section: Introductionmentioning

confidence: 99%

The Ricci flow on the sphere with marked points

Phong

Song

Sturm

et al. 2020

J. Differential Geom.

View full text Add to dashboard Cite

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a unique conical shrinking gradient Ricci soliton in the unstable case.

show abstract

“…Again the Ricci flow is an elegant way to solve the problem on conical surfaces. Yin [42,43,44] established a basic theory in this regards, and proved long time existence and convergence of the flow when χ(Σ, β) ≤ 0. The convergence in the case χ(Σ, β) > 0 was studied by Phong-Song-Sturm-Wang [32,33].…”

Section: Introductionmentioning

confidence: 99%