Related papers: Fourth-order flows in surface modelling
We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics.
This article deals with flow of plane curves driven by the curvature and external force. We make use of such a geometric flow for the purpose of image segmentation. A parametric model for evolving curves with uniform and curvature adjusted…
Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher…
Studying various functionals and associated gradient ows are known problems in differential geometry. The perpose of this article is to provide a general overview of curvature functionals in Finsler geometry and use their information for…
This is an expository article describing the conformalized mean curvature flow, originally introduced by Kazhdan, Solomon, and Ben-Chen. We are interested in applying mean curvature flow to surface parametrizations. We discuss our own…
Modeling of phenomena such as anomalous transport via fractional-order differential equations has been established as an effective alternative to partial differential equations, due to the inherent ability to describe large-scale behavior…
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time…
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential…
We introduce fourth fundamental form $IV,$ and $i$-th curvature formulas of hypersurfaces in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}$. Defining fourth fundamental form and $i$-th curvatures for hypersurfaces, we calculate…
This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…
In this text we outline the major techniques, concepts and results in mean curvature flow with a focus on higher codimension. In addition we include a few novel results and some material that cannot be found elsewhere.
We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow…
This work thoroughly examines several analytical tools, each possessing a different level of mathematical intricacy, for the purpose of characterizing the orientation distribution function of elongated objects under flow. Our investigation…
The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity,…
This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given…
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 present the necessary and sufficient conditions of the well-posedness of the initial value problem for certain fourth-order linear dispersive systems on the one-dimensional torus. This system is related with a dispersive flow for closed…
This article is a survey article that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their…
This article describes the mean curvature flow, some of the discoveries that have been made about it, and some unresolved questions.
Incompressible fluids on curved surfaces are considered with respect to the interplay between topology, geometry and fluid properties using a surface vorticity-stream function formulation, which is solved using parametric finite elements.…