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A directionally‐split volume‐of‐fluid (VOF) methodology for evolving interfaces under curvature‐dependent speed is devised. The interface is reconstructed geometrically and the volume fraction is advected with a technique to incorporate a topological volume conservation constraint. The proposed approach uses the idea that the role of curvature in a speed function V$$ \mathbf{V} $$ is analogous to the role of viscosity in the corresponding hyperbolic conservation law to propagate complex interfaces where singularities may exist. The approach has the advantage of simple implementation and straightforward extension to more complex multiphase systems by formulating the interface evolution problem using energy functionals to derive an expression for the interface‐advecting velocity. The numerical details of the volume‐of‐fluid based formulation are discussed with emphasis on the importance of curvature estimation. Finally, canonical curves and surfaces traditionally investigated by the level set (LS) method are tested with the devised approach and the results are compared with existing work in LS. [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative proof for the longtime behavior of classical curve shortening flow by using support function to estimate the curvature of evolving curves after an area-preserving rescaling, and then showing the convergence of the rescaled curvature via analyzing the structure of its $ \omega $-limit set. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of prescribing the Q ¯ ′ -curvature on three-dimensional pseudo-Einstein CR manifolds. We study the gradient flow generated by the related functional, and we will prove its convergence to a limit function under suitable assumptions. [ABSTRACT FROM AUTHOR]

In [19] , Smoczyk showed that parabolic expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we obtain representation formula for the corresponding linear hyperbolic evolution equation for the support function of locally convex curves of general nonzero rotation number, finding some natural similarities and differences in behaviour of solutions as compared with the parabolic case. As applications, we consider linear hyperbolic flows that preserve length or generalised enclosed area of the evolving curve, flow of curves with Neumann boundary conditions inside cones and hyperbolic approaches to the Yau problem of flowing one curve to another. We also consider radial graphs evolving by similar linear equations and linear higher order hyperbolic equations. Our results may be compared with the corresponding parabolic cases consider by the first author, Schrader and Wheeler, in [17]. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

For the quermassintegral inequalities of horospherically convex hypersurfaces in the (n + 1) -dimensional hyperbolic space, where n ≥ 2 , we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value-independent curvature estimates for locally constrained flows of inverse type. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution of the gradient flow minimizing the anisoperimetric ratio of convex plane curves. It is shown that for any closed embedded initial curve, which is smooth and convex, the flow exists for all time and the evolving curves converge smoothly to a homothety of boundary of the Wulff shape, which is determined by anisotropic function. [ABSTRACT FROM AUTHOR]

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature, and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs. [ABSTRACT FROM AUTHOR]

In this article we study Chen's flow of curves from theoretical and numerical perspectives. We investigate two settings: that of closed immersed ω -circles, and immersed lines satisfying a cocompactness condition. In each of the settings our goal is to find geometric conditions that allow us to understand the global behaviour of the flow: for the cocompact case, the condition is straightforward and the argument is largely standard. For the closed case however, the argument is quite complex. The flow shrinks every initial curve to a point if it does not become singular beforehand, and we must identify a condition to ensure this behaviour as well as identify the point in order to perform the requisite rescaling. We are able to successfully conduct a full analysis of the rescaling under a curvature condition. The analysis resembles the case of the mean curvature flow more than other fourth-order curvature flow such as the elastic flow or the curve diffusion flow, despite the lack of maximum and comparison principles. Our work is informed by a numerical study of the flow, and we include a section that explains the algorithms used. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments. [ABSTRACT FROM AUTHOR]

We give a flow approach to the prescribed Gaussian curvature problem on compact surfaces with negative Euler characteristic. We identify an explicit new condition on the prescribed function f which guaranties the convergence of the flow to a conformal metric whose Gaussian curvature is f. Blowing up of the flow at infinity is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider the L 2 -gradient flow for the modified p -elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with p ≥ 2 for initial curves in the energy space via minimizing movements. Moreover, we prove the existence of unique global-in-time solutions to the flow with p = 2 and obtain their subconvergence to an elastica as t → ∞. [ABSTRACT FROM AUTHOR]

This article concerns the Lp{L}_{p} Minkowski problem for q-capacity. We consider the case p≥1p\ge 1 and 1

In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For −1≤α1\alpha \gt 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ\psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

We provide a curvature flow approach to the regular Christoffel–Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term. [ABSTRACT FROM AUTHOR]

We consider a novel curvature flow for curves on the 2-dimensional light cone, contained in the 3-dimensional Minkowski space. We show that the solutions to the curvature flow (CF) for such curves are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperbolas are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when c = 0 . We prove that for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Considering non-trivial solutions to the CF, we prove that the corresponding solutions to the ICF may have at most three connected components. Moreover, at each end of such a curve the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone are visualized. [ABSTRACT FROM AUTHOR]

In [25] , Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L 2 (d θ) -gradient flows for a class of higher-order curvature functionals. These give rise to higher order linear parabolic equations for which we derive similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We consider both closed, embedded convex curves and closed, convex curves of higher rotation number. We give some corresponding remarks where relevant on open convex curves. We also consider corresponding 'globally constrained' flows which preserve the length or enclosed area of the evolving curve and a higher order approach to the Yau problem of evolving one convex planar curve to another. In an Appendix, we give some related second order results, including a version of the Yau problem for star-shaped curves. [ABSTRACT FROM AUTHOR]

In this paper, we realize a rapid, smooth iterative morphing algorithm between two immersed and closed spherical curves of arbitrary turning numbers. By compensating in turning numbers, we linearly interpolate between geodesic curvatures and the side lengths of their accurate spherical polygons approximation. Our algorithm is essentially based on a geometrical closure condition for a flow of spherical polygons involving the variation of both sides and exterior angles. A curvature smoothing flow using discrete geodesic curvature is adopted for smooth curve generation. To demonstrate the good properties of the method, a morphing of spherical curves and ruled surfaces is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate so-called slowly traveling wave solutions for a quasilinear parabolic equation in detail. Over the past three decades, the motion of the plane curve by the power of its curvature with positive exponent α has been intensively investigated. For this motion, blow-up phenomena of curvature on cusp singularity in the plane curve with self-crossing points have been studied by several authors. In their analysis, particularly in estimating the blow-up rate, the slowly traveling wave solutions played a significantly important role. In this paper, aiming to clarify the blow-up phenomena, we derive an asymptotic expansion of the slowly traveling wave solutions with respect to the parameter κ , which is proportional to the maximum of the curvature of the curve, as κ goes to infinity. We discovered that the result depends discontinuously on the parameter δ = 1 + 1 / α . It suggests that the blow-up phenomenon may also drastically change according to parameter δ . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

We consider the parabolic polyharmonic diffusion and the L 2 -gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 , then the evolving curve converges exponentially in the C ∞ topology to a straight horizontal line segment. The same behaviour is shown for the L 2 -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m. [ABSTRACT FROM AUTHOR]

In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on manifolds. [ABSTRACT FROM AUTHOR]

This paper concerns the evolution of a closed hypersurface of dimension n (≥ 2) in the Euclidean space ℝ n + 1 under a mixed volume preserving flow. The speed equals a power β (≥ 1) of homogeneous curvature functions of degree one and either convex or concave plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. [ABSTRACT FROM AUTHOR]

A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres. [ABSTRACT FROM AUTHOR]

A recent article Li and Lv (J. Geom. Anal. 30: 417–447, 2020) considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article, we extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptotically round and convergence is exponential in the C ∞ -topology. [ABSTRACT FROM AUTHOR]

This article concerns the Orlicz-Minkowski problem for p -capacity for 1 < p < n. We use the flow method to obtain a new existence result of solutions to this problem by an approximation argument for general measures. [ABSTRACT FROM AUTHOR]

We analyze a new mathematical and numerical framework, the " Voronoi Implicit Interface Method" (" VIIM" ), first introduced in Saye and Sethian (2011) [R.I. Saye, J.A. Sethian, The Voronoi Implicit Interface Method for computing multiphase physics, PNAS 108 (49) (2011) 19498-19503] for tracking multiple interacting and evolving regions (" phases" ) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.). From a mathematical point of view, the method provides a theoretical framework for moving interface problems that involve multiple junctions, defining the motion as the formal limit of a sequence of related problems. Discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions, triple points and quadruple points in two dimensions, as well as triple lines, etc. in higher dimensions. Topological changes in the system occur naturally, with no surgery required. In this paper, we present the method in detail, and demonstrate several new extensions of the method to different physical phenomena, including curvature flow with surface energy densities defined on a per-phase basis, as well as multiphase fluid flow in which density, viscosity and surface tension can be defined on a per-phase basis.We test this method in a variety of ways. We perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems, including verification of von Neumann-Mullins' law in two dimensions (and its analog in three dimensions), as well as normal driven flow and curvature flow with and without constraints, demonstrating topological change and the effects of different boundary conditions. We couple the method to a second order projection method solver for incompressible fluid flow, and study the effects of membrane permeability and impermeability, large shearing torsional forces, and the effects of varying density, viscosity and surface tension on a per-phase basis. Finally, we demonstrate convergence in both space and time of a topological change in a multiphase foam. © 2012 Elsevier Inc..

Space curve motion describes dynamics of material defects or interfaces and can be found in image processing or vortex dynamics. This article analyzes some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop noncircular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. In the second part, the Avoidance principle for spherical curves under the curve shortening flow in R3 is shown by generalizing the arguments developed by Hamilton and Gage. [ABSTRACT FROM AUTHOR]

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space R 2 must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term "affine geometry" refers to "equi-affine geometry".) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in R 2 ? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in R 2 , and show the general-affine maximal curves in R 2 are much more abundant and include the explicit curves y = x α (α is a constant and α ∉ { 0 , 1 , 1 2 , 2 }) and y = x log ⁡ x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA (n) = GL (n) ⋉ R n. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved. [ABSTRACT FROM AUTHOR]

Level set methods are a powerful tool for implicitly representing deformable surfaces. Since their inception, these techniques have been used to solve problems in fields as varied as computer vision, scientific visualization, computer graphics and computational physics. With the power and flexibility of this approach; however, comes a large computational burden. In the level set approach, surface motion is computed via a partial differential equation (PDE) framework. One possibility for accelerating level-set based applications is to map the solver kernel onto a commodity graphics processing unit (GPU). GPUs are parallel, vector computers whose power is currently increasing at a faster rate than that of CPUs. in this work, we demonstrate a GPU-based, three-dimensional level set solver that is capable of computing curvature flow as well as other speed terms. Results are shown for this solver segmenting the brain surface from an MRI data set.

This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in RmRm by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence. [ABSTRACT FROM AUTHOR]

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles. [ABSTRACT FROM AUTHOR]

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalizes the previous result [S. Guo, Convex hypersurfaces evolving by functions of the mean curvature, preprint (2016), arXiv:1610.08214 ] for convex hypersurfaces in the Euclidean space by the author to the setting in the hyperbolic space for the same class of flows. [ABSTRACT FROM AUTHOR]

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y) , where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. ['Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit', Netw. Heterog. Media 1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$. [ABSTRACT FROM AUTHOR]

In this paper, we consider the curvature flow with driving force on fixed boundary points in the plane. We give a general local existence and uniqueness result of this problem with C2 initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given. [ABSTRACT FROM AUTHOR]