Second-Order Elliptic Equation with Singularities

On the compact Riemannian manifold of dimension n≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.

Classical Neumann Problems for Hessian Equations and Alexandrov–Fenchel’s Inequalities

Recently, the 1st named author together, with Xinan Ma [12], has proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems for uniformly convex domains in $\mathbb {R}^{n}$. As an application, we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov–Fenchel inequalities arising from convex geometry. This geometric application is motivated by Reilly [18].

Products of pairs of Dehn twists and maximal real Lefschetz fibrations

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

Products of pairs of Dehn twists and maximal real Lefschetz fibrations

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

ECONOMICAL COVERS WITH GEOMETRIC APPLICATIONS

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let $H = (V, E)$ be a $k$-uniform, $D$-regular hypergraph on $n$ vertices, in which no two vertices are contained in more than $o(D / e^{2k} \log D)$ edges as $D$ tends to infinity. Our results include the fact that if $k = o(\log D)$, then there is a cover of $(1 + o(1)) n / k$ edges, extending the known result that this holds for fixed $k$. On the other hand, if $k \geq 4 \log D $ then there are $k$-uniform, $D$-regular hypergraphs on $n$ vertices in which no two vertices are contained in more than one edge, and yet the smallest cover has at least $\Omega (\frac {n}{k} \log (\frac {k}{\log D} ))$ edges. Several extensions and variants are also obtained, as well as the following geometric application. The minimum number of lines required to separate $n$ random points in the unit square is, almost surely, $\Theta (n^{2/3} / (\log n)^{1/3}).$2000 Mathematical Subject Classification: 05C65, 05D15, 60D05.