Related papers: A Note on Inextensible Flows of Curves on Oriented…

A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the…

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses…

We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference…

We prove short-time existence of {\phi}-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.

Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph…

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 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.

Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of…

Models and simulations of the flow of thin films of fluids have many important applications in industrial and natural processes. We consider the motion of a thin layer of an incompressible, Newtonian fluid over an arbitrary solid,…

In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we…

Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of…

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely…

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

We study a geometric flow on curves, immersed in $\mathbb{R}^3$, that have strictly positive torsion. The evolution equation is given by $$X_{t}=\frac{1}{\sqrt{\tau}} \textbf{B}$$ where $\tau$ is the torsion and $\textbf{B}$ is the unit…

In this article we partially classify the space of eternal mean convex flows in $\mathbb{R}^3$ of finite total curvature type, a condition implied by finite total curvature. In particular we show that topologically nonplanar ones must flow…

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.

A recent article by the first two authors together with B Andrews and V-M Wheeler considered the so-called `ideal curve flow', a sixth order curvature flow that seeks to deform closed planar curves to curves with least variation of total…

Surface incompressibility, also called inextensibility, imposes a zero-surface-divergence constraint on the velocity of a closed deformable material surface. The well-posedness of the mechanical problem under such constraint depends on an…

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that…