Simply connected translating solitons contained in slabs

In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.

A dynamical approach to the variational inequality on modified elastic graphs

We consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

On the Convergence of the Elastic Flow in the Hyperbolic Plane

We examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.

Rigidity of positively curved shrinking Ricci solitons in dimension four

We classify four-dimensional shrinking Ricci solitons satisfying $Sec\, \ge \,{1 \over {24}}R$ , where Sec and R denote the sectional and the scalar curvature, respectively. They are isometric to either ℝ4 (and quotients), 𝕊 4, ℝℙ4 or ℂ ℙ2 with their standard metrics.

Elastic flow of networks: long-time existence result

We provide a long-time existence and sub-convergence result for the elastic flow of a three-network in ℝn under some mild topological assumptions. The evolution is such that the sum of the elastic energies of the three curves plus their weighted lengths decrease in time. Natural boundary conditions are considered at the boundary of the curves and at the triple junction.

Prescribing scalar curvatures: non compactness versus critical points at infinity

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

We define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

On obstacle problem for mean curvature flow with driving force

In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

Minimal cluster computation for four planar regions with the same area

The topology of a minimal cluster of four planar regions with equal areas and smallest possible perimeter was found in [9]. Here we describe the computation used to check that the symmetric cluster with the given topology is indeed the unique minimal cluster.