SGS XI: Sunday May 6, 2007
Georgia Institute of Technology

It is shown that a family of compact submanifolds with boundary has a non-empty intersection in Rⁿ provided a certain geometric estimate holds. The inequality in question involves three features: the intrininsic sizes of the submanifolds, a weighed measure of the effect of translations, and the distortion of the configurations of normal spaces. An application is the following global result. Let M₁,... M_n be properly embedded connected smooth hypersurfaces without boundary. Consider the (unoriented) Gauss map given by G_j:M_j→ G(1,n)≅ RP^n-1, G_j(p) = [T_pM_j]^⊥. If every hyperplane in RP^n-1 intersects at most n-1 of the sets Ḡ₁(M₁),... Ḡ_n(M_n), then M₁ ∩... M_n consists of a single point. If time allows, we will also discuss some recent applications of related ideas to several complex variables and algebraic geometry.

We give a brief introduction to geometric knot theory and follow with a discussion of the distortion of a curve. The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that any closed curve has distortion at least π/2 and asked about the distortion of knots. Here we use the existence of an essential secant to show that any nontrivial knot of finite total curvature in space has distortion at least 5π/3.

We introduce a number of questions (five?) of sharp isoperimetric type for Riemannian manifolds. By "isoperimetric type" we mean inequalities between global geometric quantities such as volume, diameter, lengths of closed geodesics, or eigenvalues of the Laplace operator. The sharp inequalities come with rigidity statements that are sometimes more interesting than the inequalities. Each topic has a recent result or counter example (mostly two dimensional) as well as interesting open questions. Some example questions: "What is the optimal relationship between the diameter and the length of the shortest closed geodesic for a metric on S²?", "What is the relationship between the volumes and the marked length spectra of metrics of negative curvature on a given manifold?", "How big is the volume of a ball of radius r inside the injectivity radius?"

We study the rigidity of complete embedded constant mean curvature surfaces in space. One of the results that we prove is the following: Let M be a complete embedded constant mean curvature surface of finite genus. If M has bounded norm of the second fundamental form and M is not the helicoid then M is rigid. Here rigid means that the inclusion map of M into space represents the unique isometric immersion of M into space with the same constant mean curvature up to ambient isometries. This is joint work with Bill Meeks.

Given a Kähler metric, one may attempt to impose requirements on the curvature of another metric in its conformal class. A typical such requirement is the almost Einstein condition. This leads to an equation involving the Kähler metric, its Ricci curvature and a Hessian of an auxiliary function. For a Kähler metric satisfying a more general form of this equation, we deduce properties that lead to a classification, both local, and of all compact manifolds on which such a metric exists. Within this more general class, the manifolds that are in fact almost Einstein, are determined with the aid of a discrete set of points on an auxiliary real plane "moduli" curve. We also describe how this construction can be applied to other curvature requirements within the conformal class, such as the Ricci soliton and the Cotton space conditions. This work is joint, in part with Derdzinski, in part with Gover and Nagy.

A theorem due to Douglas and Rado guarantees the existence of a minimal surface spanning a fixed Jordan curve in R³. If the boundary is not fixed, then various applications of the Weierstrass representation theorem can often be used to establish existence. A recent application involves searching for meromorphic 1-forms on the alleged surface so that the boundary curves are geodesics with respect to the metric induced by each 1-form. In particular, if the desired minimal surface is bounded by Euclidean line segments or planar, principal curves, then a connection between the Weierstrass representation theorem and the second fundamental form can often be used to prove existence. We will discuss applications to graphs over annuli, capillary graphs, soap films and partitions of space.