Related papers: Revisiting ancient noncollapsed flows in $\mathbb{…

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar…

We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a…

In this paper we generalize the neck-stability theorem of Kleiner-Lott to a special class of four-dimensional nonnegatively curved Type I $\kappa$-solutions, namely, those whose asymptotic shrinkers are the standard cylinder…

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on $\mathbb{R}^{n+1}$ that are noncollapsed at infinity, without assuming bounded…

We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the…

In this paper we continue the study on intrinsic Harnack inequality for non- homogeneous parabolic equations in non-divergence form initiated by the first author in [1]. We establish a forward-in-time intrinsic Harnack inequality, which in…

The main goal of this paper is to prove $L^1$-comparison and contraction principles for weak solutions (in the sense of distributions) of Hele-Shaw flow with a linear Drift. The flow is considered with a general reaction term including the…

We construct a topological invariant for a Morse-Smale flow on a 3-manifold and prove that the flows are topologically equivalent iff their invariants are same.

In this paper, we establish a scalar-mean curvature comparison theorem for the long neck problem on odd-dimensional spin manifolds. This extends previous work of Cecchini and Zeidler, and gives a complete answer to Gromov's long neck…

We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an…

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian…

This paper is the second part of a series of papers about a new notion of T-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not allow the identification of the directed segment with the 3-dimensional…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th…

In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a…

We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self…

In this paper, we prove a pseudolocality-type theorem for $\mathcal L$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In…

We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp $C^0$-estimate and a quadratic concavity estimate for the…

In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling…

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the…