Related papers: Willmore flow of surfaces in Riemannian spaces I: …

Instead of investigating the Willmore flow for two-dimensional, closed immersed surfaces directly we turn to its inversion. We give a lower bound on the lifespan of this inverse Willmore flow, depending on the concentration of curvature in…

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…

This paper studies the regularity of constrained Willmore immersions into $\R^{m\ge3}$ locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions…

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for…

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result by Kuwert and Sch\"atzle in 2002 and prove short time existence and uniqueness of the…

We consider a closed surface in $\mathbb{R}^3$ evolving by the volume-preserving Willmore flow and prove a lower bound for the existence time of smooth solutions. For spherical initial surfaces with Willmore energy below $8\pi$ we show long…

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The…

We consider the $L^2$ gradient flow for the Willmore functional in Riemannian manifolds of bounded geometry. In the euclidean case E.\;Kuwert and R.\;Sch\"atzle [\textsl{Gradient flow for the Willmore functional,} Comm. Anal. Geom., 10:…

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow…

Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy $W=\int H^2$ under compactly supported infinitesimal conformal variations. Examples include all constant mean…

We study the mean curvature flow of smooth $n$-dimensional compact submanifolds with quadratic pinching in a Riemannian manifold $\mathcal{N}^{n+m}$. Our main focus is on the case of high codimension, $m\geq 2$. We establish a codimension…

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows we prove long time existence and smooth convergence to a radial coordinate slice. In the case…

We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a…

In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on…

In this paper, we systemally study the long time behavior of the curve shortening flow in a closed or non-compact complete locally Riemannian symmetric manifold. Assume that we have a global flow. Then we can exhibit a a limit for the…

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic,…

In this paper, we study the $\sigma_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show…

Here we continue the investigation of the M\"obius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $\Sigma$ into $\mathbb{R}^n$ respectively…

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on…

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{\lambda_1,\lambda_2}$, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted enclosed volume, for…