Related papers: Revisiting ancient noncollapsed flows in $\mathbb{…
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this…
We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as…
In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the…
In this paper we prove a new matrix Li-Yau-Hamilton estimate for K\"ahler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be…
We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate…
The famous flow decomposition theorem of Gallai (1985) states that any static edge $s$,$d$-flow in a directed graph can be decomposed into a nonnegative linear combination of incidence vectors of paths and cycles. In this paper, we study…
We establish a weak compactness theorem for the moduli space of closed Ricci flows with uniformly bounded entropy, each equipped with a natural spacetime distance, under pointed Gromov-Hausdorff convergence. Furthermore, we develop a…
It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\kappa$-solution. We prove that the every noncompact ancient $\kappa$-solution in dimension $3$ is…
We give a new proof of the existence of compact surfaces embedded in $R^3$ with Anosov geodesic flows. This proof starts with a non-compact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone…
We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become…
In this paper, we study the classification of $\kappa$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of…
We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.
We consider a subset $S$ of the complex Lie algebra $\so(n,\C)$ and the cone $C(S)$ of curvature operators which are nonnegative on $S$. We show that $C(S)$ defines a Ricci flow invariant curvature condition if $S$ is invariant under…
Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ($\mathcal{K} \geq 0$), and the proof relies on the Alexandrov-Bakelman-Pucci method.…
We present a new non perturbative renormalization group for classical simple fluids. The theory is built in the Grand Canonical ensemble and in the framework of two equivalent scalar field theories as well. The exact mapping between the…
We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we…
We prove the nonexistence of finite time self-similar singularities in an ideal viscoelastic flow in $R^3$. We exclude the occurrence of Leray-type self-similar singularities under suitable integrability conditions on velocity and…
In this paper, we prove that any non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional…
In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through…
We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous…