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We propose a novel closed curve blending technique that describes continuous and gradual transitions between a pair of closed curves in a plane. The main contribution of this study is to formulate the closed curve blending problem in terms of curvature flow. In this formulation, closed curve blending can be represented as a simple linear flow under linear constraints. This approach enables rapid blending, which is preferred in the production of 2D animation. In the case of curves with different turning numbers, our technique also offers smooth blending by suppressing sharp changes in the tangent vector by incorporating the curvature smoothing flow. This has not been realized with previous methods. We conducted experiments and confirmed that our technique achieved aesthetic blending in under a tenth of a second, two orders of magnitude reduction in computation time compared to state-of-the-art methods. [ABSTRACT FROM AUTHOR]

The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the integrability, increases the topological entropy, splits the separatrices in a transverse way, and breaks all resonant convex caustics. [ABSTRACT FROM AUTHOR]

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form ( [ R 0 , ∞ ) × S n , g ¯ ) with metric g ¯ = d r 2 + ϑ 2 ( r ) σ and non-positive radial sectional curvature. We prove, that for initial mean-convex graphs over S n the flow exists for all times and remains a graph over S n . Under weak further assumptions on the ambient manifold, we prove optimal decay of the gradient and that the flow leaves become umbilic exponentially fast. We prove optimal C 2 -estimates in case that the ambient pinching improves. [ABSTRACT FROM AUTHOR]

We present an interface and geometry preserving (IGP) method for modeling fully Eulerian fluid-structure interaction via phase-field formulation. While the time-dependent mobility model preserves the hyperbolic tangent interface profile, the proposed method maintains the geometry of the solid-fluid interface by minimizing the volume-conserved mean curvature flow. To reduce the curvature flow, we construct a gradient-minimizing velocity field (GMV) for the convection of the order parameter. The constructed velocity field retains the solid velocity in the solid domain while extending the velocity in the normal direction throughout the diffuse interface region. With this treatment, the GMV reduces the normal velocity difference of the isosurfaces of the order parameter, alleviating the undesired thickening or thinning of the diffuse interface region due to the convection. As a result, the time-dependent mobility coefficient is substantially reduced and there is a lesser volume-conserved mean curvature flow. Furthermore, the GMV ensures that the diffuse interface region moves with the solid bulk despite the fluid flow, such that the fluid-solid interface conforms to the geometry of the solid. Using the unified momentum and mass conservation equations via the phase-dependent interpolation, we integrate the IGP method into a fully Eulerian fluid-structure interaction solver based on the incompressible viscous fluid and the neo-Hookean solid models. The kinematics of the solid in an Eulerian frame of reference is resolved by evolving the left Cauchy-Green tensor. We first demonstrate the ability of the phase-field-based IGP method for the convection of circular and square interfaces in a prescribed velocity field. The variational fluid-structure interaction framework with the IGP method is then examined on the deformation of a solid block under cavity flow. We next explore the geometry-preserving effect of the proposed framework when the fluid flow passes over a fixed deformable block in a channel. Finally, the vibration of a flexible plate attached behind a stationary cylinder is considered to demonstrate the ability of the framework in solving unsteady fluid-structure interaction problems. • Novel interface and geometry preserving phase-field method for fully Eulerian FSI. • Gradient minimizing velocity field for accurate evolution of diffuse interface. • Reduced mobility coefficient from suppressed convective distortion and curvature flow. • Robust and accurate integration of the IGP method with fully Eulerian FSI solver. • Systematic spatial and temporal convergence studies for various test cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the curvature flows of convex curves and surfaces in the hyperbolic space. In the first part, we consider the area preserving and length preserving κ α -type curvature flows of smooth closed convex curves in the hyperbolic plane H 2 and show that these two types of flows evolve convex curve to a geodesic circle exponentially in C ∞ topology. In the second part, we study the volume preserving Gauss curvature flow of smooth closed convex surfaces in H 3 and show that the solution remains convex and converges to a geodesic sphere exponentially in C ∞ topology. [ABSTRACT FROM AUTHOR]

We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain Ω The two curves meetonly at one point, forming angles of 120 degrees. The non-closed curve has a fixed end-point on ∂Ω. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on thearea enclosed in the initial loop, if the length of the non-closed curve staysbounded from below during the evolution. Moreover, the closed curve shrinksto a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon. [ABSTRACT FROM AUTHOR]

The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two-dimensional fractional in time partial differential equation. A closed equation governing a slow motion of a small-curvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and non-axisymmetric solutions are obtained. [ABSTRACT FROM AUTHOR]

We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of solutions to an area-preserving curvature flow of planar curves in the upper half plane is investigated. Two endpoints of the curve slide along the horizontal axis with prescribed fixed contact angles. First, by establishing an isoperimetric inequality, we prove the global existence of the solution. We then study the asymptotic behavior of solutions with concave initial data near a traveling wave. [ABSTRACT FROM AUTHOR]

We establish the interior gradient estimate for general 1-D anisotropic curvature flow. The estimate depends only on the height of the graph and not on the gradient at initial time. The proof relies on the monotonicity property of the number of zeros for the parabolic equation.

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (3) an explicit parametrisation of a shrinking figure eight curve. [ABSTRACT FROM AUTHOR]

Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds (J. A. McCoy et al., Annali di Matematica Pura ed Applicata 1-13, ). In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (B. H. Andrews, Invent Math 138(1):151-161, ; Calc Var Partial Differ Equ 39(3-4):649-657, ); these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3. [ABSTRACT FROM AUTHOR]

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann-Dirichlet boundary conditions and more general curvature-dependent speeds. [ABSTRACT FROM AUTHOR]

We study the evolution of convex hypersurfaces [InlineMediaObject not available: see fulltext.] with initial [InlineMediaObject not available: see fulltext.] at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of [InlineMediaObject not available: see fulltext.] is a smooth, closed, and uniformly convex hypersurface. We find a θ* > 0 and a sufficient condition about the anisotropic function f, such that if θ > θ*, then [InlineMediaObject not available: see fulltext.] remains uniformly convex and expands to infinity as t → + ∞ and its scaling, [InlineMediaObject not available: see fulltext.], converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H-log f instead of H-f. [ABSTRACT FROM AUTHOR]

Abstract: The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow. [Copyright &y& Elsevier]

The length-preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be non-simple. It turns out that for certain classes of symmetric curves, the flows converge to m-fold circles. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We investigate the existence, convergence, and uniqueness of modified general curvature flow (MGCF) of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary. [ABSTRACT FROM AUTHOR]

The geometrical properties of metallic nanoparticles such as the size and morphology have significant impacts on the structure and stability of the adsorbed biological entities as well as the nanoscale structural performances. To identify the nanoscale intrinsic geometry from the height images by atomic force microscopy (AFM), we developed a curvature-dependent evolution scheme that can eliminate the noise and smoothen the surfaces. The principal curvatures are computed directly from the first and second derivatives of the discrete AFM height data. The principal curvatures and directions correspond to the eigenvalues and eigenvectors of shape operator matrix, respectively. The evolution equation using the principal curvature flows smoothens the images in the corresponding principal directions. For an idealized model, κ flow successfully identifies the major valley lines to represent the boundary of nanoparticles without referring to the phase information, whereas the mean curvature flow eliminates all the minor ones leaving only the major feature of the boundary. To demonstrate the capability of noise removal, smoothing surfaces, the identification of ridge and valley lines, and the extraction of intrinsic geometry, the developed numerical scheme is applied to real AFM data that include the silver nanoparticles of 24 nm diameter and the gold nanoparticles of 33-56 nm diameters. [ABSTRACT FROM AUTHOR]

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C estimates. [ABSTRACT FROM AUTHOR]

Here a semi-implicit formulation of the gradient augmented level set method is presented. The method is a hybrid Lagrangian--Eulerian method that may be easily applied in two or three dimensions. By tracking both the level set function and the gradient of the level set function, highly accurate descriptions of a moving interface can be formed. Stability is enhanced by the addition of a smoothing term to the gradient augmented level set equations. The new approach allows for the investigation of interfaces evolving by mean curvature and by the intrinsic Laplacian of the curvature. Sample results presented in both two and three dimensions demonstrate the applicability of the scheme. The influence of the smoothing term on stability and accuracy is also investigated. [ABSTRACT FROM AUTHOR]

Abstract: This paper tackles two problems: (1) the reconstruction of 3-D volumes from 2-D post-mortem slices (e.g., histology, autoradiography, immunohistochemistry) in the absence of external reference, and (2) the quantitative evaluation of the 3-D reconstruction. We note that the quality of a reconstructed volume is usually assessed by considering the smoothness of some reconstructed structures of interest (e.g., the gray–white matter surfaces in brain images). Here we propose to use smoothness as a means to drive the reconstruction process itself. From a pair-wise rigid reconstruction of the 2-D slices, we first extract the boundaries of structures of interest. Those are then smoothed with a min–max curvature flow confined to the 2-D planes in which the slices lie. Finally, for each slice, we estimate a linear or flexible transformation from the sparse displacement field computed from the flow, which we apply to the original 2-D slices to obtain a smooth volume. In addition, we present a co-occurrence matrix-based technique to quantify the smoothness of reconstructed volumes. We discuss and validate the application of both our reconstruction approach and the smoothness measure on synthetic examples as well as real histological data. [Copyright &y& Elsevier]

The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper. [ABSTRACT FROM AUTHOR]

Abstract: We give a proof of the isoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a result of Gerhardt [C. Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Differential Geometry 32 (1990) 299–314] and Urbas [J. Urbas, On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z. 205 (1990) 355–372] on an expanding geometric curvature flow. [Copyright &y& Elsevier]

Abstract: We consider closed convex hypersurfaces moving in Euclidean spaces with normal velocity equal to , where is a nonnegative continuous function of t and F is evaluated at the principal curvatures and satisfies the standard conditions. We study long time existence and convergence of the evolving hypersurfaces in three different cases, which include Andrews'' contractive case, McCoy''s mixed volume preserving case and the additional expanding case. [Copyright &y& Elsevier]

[Display omitted] The movement of grain boundaries in pure metals and alloys with a low concentration of dislocations has been historically proved to follow curvature flow behaviour. This mechanism is typically known as grain growth (GG). However, recent 3D in-situ experimental results tend to question this global picture concerning the influence of the curvature on the kinetics of interface migration. This article explains, thanks to 2D anisotropic full-field simulations, how the torque effects can complicate these discussions. It is then illustrated that neglecting torque effects in full-field formulations can have a significant effect on the overall rate of GG, the shape of grains and grain boundaries as well as their local kinetics. The apparent reduced mobility can be much more complex than expected without necessarily questioning the influence of the curvature on the local kinetic equation. [ABSTRACT FROM AUTHOR]

Every surface in the Euclidean space ℝ3 admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3-manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineering fields such as shape classification. In this paper we present an algorithm that is based on a discrete curvature flow to compute constant curvature metrics on 3-manifolds that are hyperbolic and have boundaries of a certain type. We also provide an approach to visualize such a metric by embedding the fundamental domain and universal covering in the hyperbolic space ℍ3. Some experimental results are given for both algorithms. Furthermore, we propose an algorithm to automatically construct truncated tetrahedral meshes for 3-manifolds with boundaries. It can not only generate inputs to the curvature flow algorithm, but could also serve as an automatic tool for geometers and topologists to build simple models for complicated 3-manifolds, and therefore facilitate their research that requires such models. [ABSTRACT FROM AUTHOR]

Abstract: The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane. [Copyright &y& Elsevier]