Related papers: A sharp inscribed radius estimate for fully nonlin…

In this paper we consider the local $L^p$ estimate of Riemannian curvature for the Ricci-harmonic flow or List's flow introduced by List \cite{List2005} on complete noncompact manifolds. As an application, under the assumption that the flow…

We prove a sharp (up to $C_\epsilon R^\epsilon$) $L^7$ square function estimate for the moment curve in $\mathbb{R}^3$.

In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.

We consider the motion of incompressible viscous fluid in a rectangle, imposing the periodicity condition in one direction and the no-slip boundary condition in the other. Assuming that the flow is subject to an external random force, white…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse…

In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow…

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature…

Let $S$ be a surface of nonpositive curvature of genus bigger than 1 (i.e. not the torus). We prove that any flat strip in the surface is in fact a flat cylinder. Moreover we prove that the number of homotopy classes of such flat cylinders…

We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder.

In this paper, we establish uniform and sharp volume estimates for the singular set and the quantitative singular strata of mean curvature flows starting from a smooth, closed, mean-convex hypersurface in $\mathbb R^{n+1}$.

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a…

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.

We prove sharp spectral gap estimates on compact manifolds with integral curvature bounds. We generalize the results of Kr\"oger (Kr\"oger '92) as well as of Bakry and Qian (Bakry-Qian '00) to the case of integral curvature and confirm the…

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under…

In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is…

We use high-low frequency methods developed in the context of decoupling to prove sharp (up to $C_\epsilon R^\epsilon$) square function estimates for the moment curve $(t,t^2,\ldots,t^n)$ in $\mathbb{R}^n$. Our inductive scheme incorporates…

We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An…

We have an idea on the influence of a nonlinear term (tending to 0) on the prescribed scalar curvature equation to have an uniform estimate.