Related papers: Annular and boundary reducing Dehn fillings

This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings $M(\alpha)$ and $M(\beta)$ on a given 3-manifold $M$,…

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open…

We consider two systems of curves $(\alpha_1,...,\alpha_m)$ and $(\beta_1,...,\beta_n)$ drawn on a compact two-dimensional surface $M$ with boundary. Each $\alpha_i$ and each $\beta_j$ is either an arc meeting the boundary of $M$ at its two…

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not…

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty…

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In…

Let $M_{\langle u,v,w\rangle}\in C^{uv}\otimes C^{vw}\otimes C^{wu}$ denote the matrix multiplication tensor (and write $M_n=M_{\langle n,n,n\rangle}$) and let $det_3\in ( C^9)^{\otimes 3}$ denote the determinant polynomial considered as a…

In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension at least 3 and at most 7, there exist infinitely many almost properly embedded free boundary minimal hypersurfaces. This settles the free…

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by…

We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension $\ge 2$, is…

The length function $\ell_2(r,R)$ is the smallest length of a binary linear code with codimension (redundancy) $r$ and covering radius $R$. We obtain the following new upper bounds on $\ell_2(r,R)$, which yield a decrease $\Delta(r,R)$…

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at…

We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…

Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic…

There is a sequence of positive numbers $\delta_{2n}$, such that for any connected $2n$-dimensional Riemannian manifold $M$, there are two mutually exclusive possibilities: $1)$ There is a complex structure on $M$ making it into a K\"ahler…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly…

For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold…

In this article we prove that all boundary points of a minimal oriented hypersurface in a Riemannian manifold are regular, that is, in a neighborhood of any boundary point, the minimal surface is a $\mathcal{C}^{1, \frac14}$ submanifold…