Related papers: Sharp pinching estimates for mean curvature flow i…

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…

We proved a Bernstein theorem for ancient solution to symplectic mean curvature flow via the complex phase map .

In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the…

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $\phi:M\to N\times\mathbb{R}^{\ell}$, $n+\ell\leq 2m-1$ provided that either $\phi$ is proper with the second fundamental form with…

In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and…

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore…

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $\sigma \in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math.…

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $\psi$ given a strictly convex initial hypersurface in Euclidean space suitably pinched.…

We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.

We consider the effect of surface roughness on solid-solid contact in a Stokes flow. Various models for the roughness are considered, and a unified methodology is given to derive the corresponding asymptotics of the drag force. In this way,…

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty…

We show sharp square function estimates for curves in the plane whose curvature degenerates at a point and estimates sharp up to endpoints for cones over these curves. To this end, for curves of finite type we extend the classical…

We prove the sharp interior gradient estimate for area decreasing graphical mean curvature flow in arbitrary codimension, which generalizes the result in \cite{CM}.

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…

We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is $\sigma_k$-convex.

We prove that smooth convex $\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\leq CH(x,t)(H(x,t)|x-y|+1)^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most…

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary,…