Related papers: Eguchi-Hanson singularities in U(2)-invariant Ricc…
We study the Ricci flow out of spaces with edge type conical singularities along a closed, embedded curve. Under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively…
Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in…
We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.
We use the Ricci flow with surgery to study four-dimensional SU(2) x U(1)-symmetric metrics on a manifold with fixed boundary given by a squashed 3-sphere. Depending on the initial metric we show that the flow converges to either the…
We introduce and study a new general flow of $\mathrm{G}_2$-structures which we call the Ricci-harmonic flow of $\mathrm{G}_2$-structures. The flow is the coupling of the Ricci flow of underlying metrics and the isometric flow of…
We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the…
We prove the existence of a unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two…
In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci…
We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb R^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as the Ricci flow. In…
In this paper, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder $\mathbb{R}^k \times S^{n-k}$ or the…
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery…
Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton…
We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all…
We consider a normalization of the Ricci flow on a closed Riemannian manifold given by the evolution equation $\partial_{t}g(t)=-2(Ric(g(t))-\frac{1}{2\tau}g(t))$ where $\tau$ is a fixed positive number. Assuming that a solution for this…
In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity…
We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$ on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If $g_{0}$ has bounded Hopf-fiber, curvature…
This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as…
In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly…
In this paper we give an explicit bound of $\Delta_{g(t)}u(t)$ and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature…
We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all…