Related papers: An $\varepsilon$-regularity theorem for line bundl…
A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved,…
In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(|y|))$ over $\mathbb{R}^n$, $n\geq 2$, satisfying there exist positive constants $\epsilon$, $c$ and $N$ such that $ u'(r)\geq c…
We investigate the effect of small suction Reynolds number and permeability parameter on the stability of Poiseuille fluid flow in a porous medium between two parallel horizontal stationary porous plates . We have shown that the perturbed…
A streamwise-constant model is presented to investigate the basic mechanisms responsible for the change in mean flow occuring during pipe flow transition. Using a single forced momentum balance equation, we show that the shape of the…
We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing…
In this paper, we prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves Baker's convergence theorem. In particular, we obtain a new differentiable sphere theorem for submanifolds in…
We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize…
Our interest is a regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of continuous Hermitian metrics with semi-positive curvatures on the so-called Zariski's…
We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is…
In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable…
A turbulent-laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow,…
We report a theory deriving bulk flow scaling for canonical wall-bounded flows. The theory accounts for the symmetries of boundary geometry (flat plate channel versus circular pipe) by a variational calculation for a large-scale energy…
The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows.…
In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent…
In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.
We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining…
Let $E$ be a hermitian complex vector bundle over a compact K\"ahler surface $X$ with K\"ahler form $\omega$, and let $D$ be an integrable unitary connection on $E$ defining a holomorphic structure $D^{\prime\prime}$ on $E$. We prove that…
The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity,…
A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean…
In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems…