Stability of translating solutions to mean curvature flow

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.     

Mean Curvature Ow of Graphs in Σ1 × Σ2

Let Σ_1 and Σ_2 be m and n dimensional Riemannian manifolds of constant curvature respectively. We assume that w is a unit constant m-form in Σ_1 with respect to which Σ_0 is a graph. We set v = 〈e_1 ∧ … ∧ e_m, 〉), where {e_1, …, e_m} is a normal frame on Σ_t. Suppose that Σ_0 has bounded curvature. If v(x, 0) ≥ v0 > frac{sqrt{p}}{2} for all x, then the mean curvature flow has a global solution F under some suitable conditions on the curvatrue of Σ_1 and Σ_2.     

Mean curvature flow with a forcing term in minkowski space

We study the forced mean curvature flow of graphs in Minkowski space and prove longtime existence of solutions. When the forcing term is a constant, we prove convergence to either a constant mean curvature hypersurface or a translating soliton – depending on the boundary conditions at infinity. It is a pleasure to thank my PhD advisors Klaus Ecker and Gerhard Huisken for their assistance and encouragement. I also thank Maria Athanassenas, Oliver Schnürrer and Marty Ross for their interest and useful comments, and the Max Planck Gesellschaft for financial support.     

Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

A Geometric Problem and the Hopf Lemma. Ⅱ

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1 =constant in case M satisfies: for any two points (X′, Xn+1), (X′, ^Xn+1)on M, with Xn+1 ＞ ^Hn+1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part Ⅰ dealt with corresponding one dimensional problems.     

A Geometric Problem and the Hopf Lemma. Ⅱ

Abstract A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X_n+1 =constant in case M satisfies: for any two points (X′,X_n+1), on M, with , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0401118.     

Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

We discuss the motion of noncompact axisymmetric hypersurfaces Γ _t evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.     

The size of the singular set in mean curvature flow of mean-convex sets

We prove that when a compact mean-convex subset of (or of an -dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most . Examples show that this is optimal. We also show that, as , the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most . If , the convergence is everywhere smooth and hence after some time , the moving surface has no singularities

     

Smectic Liquid Crystals: Materials with One-dimensional, Periodic Order

Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Although smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers becomes nontrivial. We discuss various topological and geometrical aspects of these materials and present recent progress on finding some exact layer configurations. We also exhibit connections to the study of certain embedded minimal surfaces and briefly summarize some important open problems.     

The nature of singularities in mean curvature flow of mean-convex sets

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of or of an -dimensional riemannian manifold. If , the moving boundary is shown to be very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If , the same results are shown up to the first time that singularities occur.

     

Construction of Lagrangian Self-similar Solutions to the Mean Curvature Flow in \mathbb{C}^n

We give new examples of self-shrinking and self-expanding Lagrangian solutions to the Mean Curvature Flow (MCF). These are Lagrangian submanifolds in , which are foliated by (n − 1)-spheres (or more generally by minimal (n − 1)-Legendrian submanifolds of ), and for which the study of the self-similar equation reduces to solving a non-linear Ordinary Differential Equation (ODE). In the self-shrinking case, we get a family of submanifolds generalising in some sense the self-shrinking curves found by Abresch and Langer.     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].     

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4 Chen , Y. , Giga , Y. , Goto , S. ( 1991 ). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Diff. Geom. 33 : 749 – 786 .[Crossref], [Web of Science ®] , [Google Scholar]] and [12 Evans , L. C. , Spruck , J. ( 1991 ). Motion of level sets by mean curvature. I . J. Diff. Geom. 33 ( 3 ): 635 – 681 . [Google Scholar]]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.