Related papers: Ricci expanders and type III Ricci flow

In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate…

We discuss a natural form of Ricci--flow conjugation between two distinct general relativistic data sets given on a compact $n\geq 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the…

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of…

We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow.…

This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the…

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…

We establish a point-wise gradient estimate for $all$ positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions.…

We review recent results about heat kernel estimates based on Kato conditions on the negative part of the Ricci curvature.

Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some…

We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on…

In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant…

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow…

In this note we give a heat kernel lower bound in term of integral Ricci curvature, extending Cheeger-Yau's estimate.

Following Donaldson's oppenness theorem on deforming a conical K\"ahler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical K\"ahler-Ricci Flow exists for…

We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.

We develop a refined singularity analysis for the Ricci flow by investigating curvature blow-up rates locally. We first introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible…

In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci…

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions.…

In this paper we prove local results for solutions to the Ricci flow (heat flow) whose speed (height) is bounded by $\frac c t$ for some time interval $ t \in (0,T)$. These results are contained in chapter 7 of the author's habilitation…