Related papers: New area-minimizing Lawson-Osserman cones

We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve…

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson…

Let $ l : \Sigma \to X$ be a weakly Lagrangian map of a compact orientable surface $ \Sigma$ in a K\"ahler surface $ X$ which is area minimizing in its homotopy class of maps in $ W^{1,2}(\Sigma, X)$, the Sobolev space of maps of square…

In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane. In this paper we resolve one of the open problems stated in that paper, by showing that finding a MaxMin…

In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms.

We compute the minimal exponent of the affine cone over a complete intersection of smooth projective hypersurfaces intersecting transversely. The upper bound for the minimal exponent is proved, more generally, in the weighted homogeneous…

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main…

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it…

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges.…

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for…

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in $\mathbb{R}^{2n}$, with $n \ge 2$. If $p \ge 2n-1$, then among hypersurfaces in $\mathbb{R}^{2n}$ which are $O(n) \times…

Some foundational results on the geometry of Lorentz-Minkowski spaces and Finsler spacetimes are obtained. We prove that the local light cone structure of a reversible Finsler spacetime with more than two dimensions is topologically the…

In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at…

The smallest enclosing circle problem asks for the circle of smallest radius enclosing a given set of finite points on the plane. This problem was introduced in the 19th century by Sylvester [17]. After more than a century, the problem…

We prove that the area of cross-sections of light-cones, in space-times satisfying suitable energy conditions, is smaller than or equal to that of the corresponding cross-sections in Minkowski, or de Sitter, or anti-de Sitter space-time.…

We show the uniqueness of the cylindrical tangent cone $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ for area-minimizing hypersurfaces in $\mathbb{R}^9$, completing the uniqueness of all tangent cones of the form $C_{p,q} \times…

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of…

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical…

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|\lambda_n|\leq d^{\gamma}$ contains a clique of size $d^{1-O(\gamma)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine…

We prove the regularity of cohomogeneity two equivariant isotopy minimization problems. Based on this, we develop cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in 1988. As an…