Mean Convex Research Articles

Qualitative properties for a system coupling scaled mean curvature flow and diffusion

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but gives an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.

Curvature measures and soap bubbles beyond convexity

Extending the celebrated results of Alexandrov (1958) and Korevaar–Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on a natural geometric condition involving the curvature measures. More specifically we show, under a weak mean convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach A⊆Rn+1 with finite and positive volume such that, for some k∈{0,…,n−1}, the k-th generalized curvature measure Θkϕ(A,⋅), which is defined on the unit normal bundle of A with respect to the relative geometry induced by ϕ, is proportional to the top order generalized curvature measure Θnϕ(A,⋅). If k=n−1 the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach.These findings are new even in the Euclidean space.Several auxiliary and related results are proved, which are of independent interest. They include the extension of the classical Steiner–Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space, a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle–Lussardi–Villa (2021), and general versions of the Heintze–Karcher inequality.

Nonhomogeneous expanding flows in hyperbolic spaces

In the present paper, we consider star-shaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the long-time existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

Mean curvature flow of graphs in generalized Robertson–Walker spacetimes with perpendicular Neumann boundary condition

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a generalized Robertson–Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if the initial hypersurface is mean convex, then the evolving hypersurfaces remain mean convex during the flow.

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces

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, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the H^1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

Mean convex properly embedded $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$

We establish curvature estimates and a convexity result for mean convex properly embedded [\varphi,\vec{e}_{3}] -minimal surfaces in \mathbb{R}^3 , i.e., \varphi -minimal surfaces when \varphi depends only on the third coordinate of \mathbb{R}^3 . Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in \mathbb{R}^3 , we use a compactness argument to provide curvature estimates for a family of mean convex [\varphi,\vec{e}_{3}] -minimal surfaces in \mathbb{R}^{3} . We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded [\varphi,\vec{e}_{3}] -minimal surface in \mathbb{R}^{3} with non-positive mean curvature when the growth at infinity of \varphi is at most quadratic.

SO(2) Symmetry of the Translating Solitons of the Mean Curvature Flow in $$\mathbb {R}^4$$

In this paper, we prove that the translating solitons of the mean curvature flow in \(\mathbb {R}^4\) which arise as blow-up limit of embedded, mean convex mean curvature flow must have SO(2) symmetry.

Differential algebraic observer-based trajectory tracking for parallel robots via linear matrix inequalities

This paper develops a novel observer-based trajectory tracking technique for parallel robots, modelled as differential algebraic equations, which assumes that only positions are available for control purposes while joint velocities should be estimated. Based on the direct Lyapunov method and a recently appeared factorisation for expressions in the differential mean value theorem, convex modelling and Finsler's lemma are combined to incorporate restrictions into the analysis. Two generalisations are thus achieved: the inner-loop feedback is allowed to use velocity estimates instead of the real values and the outer-loop feedback becomes fully nonlinear while taking into account the parallel characteristics of mechanisms. Moreover, both the observer and the controller design conditions are linear matrix inequalities, which can be efficiently solved via commercially available software. Illustrative examples are provided that show the advantages of the proposal against former works on the subject.

Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2\pi/e}+\delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.

$K$-mean convex and $K$-outward minimizing sets

We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow.

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, in this case the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. For the first mode, the relevant bound only depends on the space dimension, as well. Moreover, it largely improves on an earlier estimate obtained for convex domains by the first author and co-authors. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.

Discs area-minimizing in mean convex Riemanniann-manifolds

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to$\mathbb {D}^{2}\times T^{n}$with non-zero degree, where$\mathbb {D}^{2}$is a disc and$T^{n}$is an$n$-dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015,J. Geom. Anal.,25, 1001–1017) to higher dimensions.

Mean Convex Mean Curvature Flow with Free Boundary

AbstractIn this paper, we generalize White's regularity and structure theory for mean‐convex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple‐approximation scheme, which combines ideas from Edelen [9], Haslhofer‐Hershkovits [16], and Volkmann [43]. Other important new ingredients are a Bernstein‐type theorem and a sheeting theorem for low‐entropy free boundary flows in a half‐slab, which allow us to rule out multiplicity 2 (half‐)planes as possible tangent flows and, for mean‐convex domains, as possible limit flows. © 2021 Wiley Periodicals LLC.

Closed [formula omitted]-structures on nilmanifolds

In this paper we consider closed SL(3,C)-structures which are either mean convex or tamed by a symplectic form. These notions were introduced by Donaldson in relation to G2-manifolds with boundary. In particular, we classify nilmanifolds which carry an invariant mean convex closed SL(3,C)-structure and those which admit an invariant mean convex half-flat SU(3)-structure. We also prove that, if a solvmanifold admits an invariant tamed closed SL(3,C)-structure, then it also has an invariant symplectic half-flat SU(3)-structure.

Low entropy and the mean curvature flow with surgery

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $$\varLambda _{n-2}$$ . In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $$H - \frac{\langle x, \nu \rangle }{2} > 0$$ is possible and as an application we use the surgery flow to show that smooth n-dimensional closed self shrinkers with entropy less than $$\varLambda _{n-2}$$ are isotopic to the round n-sphere.

On the Mean Convexity of a Space-and-Time Neighborhood of Generic Singularities Formed by Mean Curvature flow

We consider a nondegenerate case of generic singularity when the limit cylinder of the rescaled mean curvature flow is \({\mathbb {R}}^3 \times {\mathbb {S}}^1\). We obtain a detailed description of a possibly small, but fixed, neighborhood of the singularity, up to (and including) the blowup time, and find that it is mean convex and the singularity is isolated. In the considered nondegenerate cases, we confirm the conjectures of mean convexity and isolation of singularities.

A note on mean convex 𝜆-surfaces in ℝ³

Inspired by the work of Spruck and Xiao on mean convex translators, in this note, we show that any closed and mean convex λ \lambda -surface in R 3 \mathbb {R}^3 with λ ≤ 0 \lambda \leq 0 must be convex. We also give a curvature estimate for mean convex λ \lambda -surfaces in R 3 \mathbb {R}^3 .

Ancient and eternal solutions to mean curvature flow from minimal surfaces

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in \({\mathbb {R}}^{n+1}\) for \(n \ge 2\). These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in \({\mathbb {R}}^{n+1}\) for \(n \ge 2\). These eternal solutions are not solitons, are \(O(n)\times O(1)\)-invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As \(t \rightarrow \infty \), the profile curves converge to a grim reaper for \(n \ge 3\) and become flat for \(n=2\). Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded \(O(n)\times O(1)\)-invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.

Self-Expanders of the Mean Curvature Flow

We study self-expanding solutions M^{m}subset mathbb {R}^{n} of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function |A|2/|H|2 attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/|H| is parallel in the normal bundle, then a similar result holds in higher codimension for the function |Aξ|2/|H|2, where Aξ is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex self-expanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of non-negative scalar curvature. In particular, in dimension 2 any mean convex self-expander that is asymptotic to a cone must be strictly convex.

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.