Related papers: Stable approximations for axisymmetric Willmore fl…
In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level…
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially…
We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the…
We use the minimizing movement theory to study the gradient flow associated with a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, one can define a Willmore energy on regular…
We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small…
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…
In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both…
We show that the concept of $H^2$-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class $W^{2,p}$ from a…
We study the Willmore problem with free boundary by means of a new {\L}ojasiewicz-Simon gradient inequality for functionals on infinite dimensional manifolds. In contrast to previous works, we do not rely on a gradient-like representation…
An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface,…
In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of…
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…
In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha\sigma_2^{1/2},$ where $\sigma_2^{1/2}$ is the normalized square root of the scalar curvature, $\alpha\geq 2,$ and $r$ is the…
We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order…
The Willmore Problem seeks closed surfaces in $\mathbb{S}^3\subset\mathbb{R}^4$ of a given topological type minimizing the squared-mean-curvature energy $W = \int |H_{\mathbb{R}^4}|^2 = area + \int |H_{\mathbb{S}^3}|^2$. The longstanding…
We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing…
The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the…
In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We…
We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of finite codimensions in a suitable…
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…