Related papers: Multiple Solutions to the Planar Plateau Problem
These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of the evolution of a single closed hypersurface…
Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition…
Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of…
In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there…
The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian…
We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.
We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.
This paper presents results on the extent to which mean curvature data can be used to determine a surface in space or its shape. The emphasis is on Bonnet's problem: classify and study the surface immersions in $\R^3$ whose shape is not…
We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ -\Delta\tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon \kappa(x)…
In this paper, we study planar polygonal curves from the variational methods. We show an unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first…
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a…
We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak…
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the…
We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the…
We consider surfaces of constant Gaussian curvature immersed in 3-dimensional manifolds, and we strengthen the compactness result of Labourie in the case where the ambient manifold is 3-dimensional hyperbolic space. This allows us to prove…
The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed…
We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances…