The conjugate heat equation and Ancient solutions of the Ricci flow
By Xiaodong Cao, Qi S. Zhang. 2010
94 citations recorded by the source corpus.
Abstract We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ -solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelmanʼs previous result on backward limits of κ -solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23] , where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1af77254c247890cf2b6cd68c50f002200eb2671
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions (2015, result): “hen one would have u(x,t2) ≤ u(x,t1)(t2 t1 )k. Now suppose that M is an Einstein manifold Rij = ρgij with ρ>0 and u(x,t) = G(x,t;x0,0) the heat kernel under the Ricci flow. According to a result in [CZ], we have the Gaussian lower and upper bounds of G, i.e., Ct−n 2 e− cd2 t(x,x0) t ≤ G(x,t;x0,0) ”
- Space of Ricci Flows I (2009, background): “[13]), we obtain the on-diagonal bound 1 C |ti|−n < wi(yi, 0) < C|ti|−n for some uniform constant C.”
- On type-I singularities in Ricci flow (2010, background): “Nontrivial gradient shrinking solitons also arise as blow-down limits of certain ancient Ricci flow solutions (which are singularity models) as shown by Perelman [17] in 3 dimensions, and recently by Cao and Zhang [5] for higher dimensions in the Type I case.”