On type-I singularities in Ricci flow

By J. Enders, Reto Muller, P. Topping. 2010

181 citations recorded by the source corpus.

We dene several notions of singular set for Type I Ricci ows 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 [15]. As a byproduct we conclude that the volume of a nite-volume singular set vanishes at the singular time. We also dene a notion of density for Type I Ricci ows and use it to prove a regularity theorem reminiscent of White’s partial regularity result for mean curvature ow [22].

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d5607b1ebd5bdf768ca75b295dafcddfa9051a3a

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In Applied Mathematics

In Astronomy and Astrophysics

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Citations in the corpus

The conjugate heat equation and Ancient solutions of the Ricci flow (2010, background): “Topping [9], both of their works study the forward limit of singularities.”
New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow (2012, background): “12) to hold at every x in a definite neighborhood of x0 [18], which in [8] follows from their analogue of (1.”
Kähler currents and null loci (2013, background): “[24], we will call the complement of the set of points x ∈ X such that there exists an open neighborhoodU of x and a constant C > 0 with |Rm(t)|g(t) C on U × [0, T ).”
Geometry of shrinking Ricci solitons (2014, background): “Indeed, according to [11], the blow-ups around a type-I singularity point always converge to nontrivial gradient shrinking Ricci solitons.”
Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “We also mention that there exist many similarities between the formation of singularities in mean curvature flow and Ricci flow (see [32] for a nice overview on Type I singularities in Ricci flow).”
Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness (2015, background): “This is true for Type-I Ricci flow on compact manifolds by Enders–Müller– Topping [14] and Kähler–Ricci flow by Zhang [39], but it is still open in general and currently forms an active topic of research.”