Related papers: Critical behavior of collapsing surfaces

We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with…

In this paper, we study the properties of nondegenerate cylindrical singularities of mean curvature flow. We prove they are isolated in spacetime and provide a complete description of the geometry and topology change of the flow passing…

Correlation functions are sensitive to the presence of a boundary. Surface modulations give rise to modified near surface correlations, which can be measured by scattering probes. To determine these correlations, we develop a perturbative…

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a…

In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a K\"ahler-Einstein surface. We prove that the blow-up flow $\Sigma_s^\infty$ at a singular point $(X_0, T_0)$ of…

The sliding motion of objects is typically governed by their friction with the underlying surface. Compared to translational friction, however, rotational friction has received much less attention. Here, we experimentally and theoretically…

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the…

We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial…

In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities…

In spherical symmetry compelling numerical evidence suggests that in general relativity solutions near the threshold of black hole formation exhibit critical behavior. One aspect of this is that threshold solutions themselves are…

Wetting phenomena, molecular protrusions of lipid bilayers and membrane stacks under lateral tension provide physical examples for interacting surfaces with tension. Such surfaces are studied theoretically using functional renormalization…

We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces…

We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface, we derive optimal error bounds with respect to the $L^2$-- and $H^1$--norms for a…

Recently a new no-global-recollapse argument was given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this paper we discuss important properties of the mean curvature flow…

The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary…

Error estimates are proved for an evolving surface finite element semi-discretization for anisotropic mean curvature flow of closed surfaces. For the geometric surface flow, a system coupling the anisotropic evolution law to parabolic…

In the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour. On the other hand surface defects, whether they are of natural origin or created artificially, are known to…

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean…

In this paper, we give an example of a compact mean-convex hypersurface with a single singular point moved by mean curvature having a sequence of singular epochs (times) converging to zero.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$…