[From the introduction] The present work was motivated by a reexamination of some fundamental constrained variational problems that arise in equilibrium theory in fluid dynamics and Magnetohydrodynamics. These problems lead formally to non-linear eigenvalue problems as variational equations. Indeed, it is traditional that many fundamental problems of mathematical physics can b e formulated as nonlinear eigenvalue problems of the form \[-\Delta u = \Lambda(u), u\in H_0^1(\Omega)\] The profile function $\Lambda(u) is usually expressed in the form \$\Lambda(u)=\lambda f(u)$ where $f(u)$ is a known function and $\lambda$ is a Lagrange multiplier.

Using the method of Minkowski or Alexandrov one finds simple discretizations of elliptic Monge-Amp\022ere equations, including the equation of graphs with prescribed positive Gaussian curvature. It is shown how these discrete problems can b e solved numerically, and computer graphics of the piecewise linear, convex solutions are presented.

[Excerpt] I would like to describe some recent research, concerning properly embedded minimal surfaces with periodicity that I have been doing with Michael Callahan and Bill Meeks III. The work includes the construction of new examples and the characterization of the qualitative behavior of all examples in an important class. It is based, in turn, on an analysis of the geometric behavior of such surfaces at infinity. This talk is divided into two parts; the first discusses the examples and qualitative results; and the second gives a feeling for the analytical background.

Working primarily within the conformal category, we develop complementary existence and rigidity theorems for periodic minimal surfaces in $R^n$ . Applying this theory, we prove: 1. Every flat three-torus contains an infinite number of genus 3 embedded minimal surfaces; 2. Necessary and sufficient conformal conditions for a closed Riemann surface of genus g to conformally minimally immerse in a flat $3$- or $(2g-1)$-torus; 3. The existence of distinct isometric minimal surfaces in flat tori; 4. Special results on the geometry of minimal surfaces of genus 3 and of classical examples of minimal surfaces in flat three-tori; 5. The determination of the group of symmetries of certain minimal surfaces in R³.

[From the introduction] We set forth in this paper rigorous justification of a new approach for defining and then investigating the evolution of a hypersurface in R n moving according to its mean curvature. This problem has been long studied using parametric methods of differential geometry: see, for instance, Gage, Gage-Hamilton, Grayson, Huisken, etc., etc. In this classical setup, we are given at time 0 a smooth hypersurface $\Gamma_0$ which is, say, the connected boundary of a bounded open subset of $R^n$. As time progresses we allow the surface to evolve, by moving each point in the opposite direction to the mean curvature vector, at a velocity equal to $(n-1)$ times the absolute value of the mean curvature at that point. Assuming this evolution is smooth, we define thereby for each $t > 0$ a new hypersurface $\Gamma_t$. The primary problem is then to study geometric properties of $\{\Gamma_t\}_{t>0} in terms of $\Gamma_0$.

[From the introduction] We present in this paper a new, elementary, and fairly concise proof of short time existence for the classical motion of a smooth hypersurface evolving according to its mean curvature. In this problem we are given initially a smooth connected hypersurface $\Gamma_0$ which is the boundary of a bounded open set $U\subset R^n$. We then allow $\Gamma_0$ to evolve in time into a family of surfaces $\{\Gamma_t\}_{t>0}$ by moving each point on $\Gamma_t (t\ge 0)$ in the opposite direction to its mean curvature vector, at a velocity equal to $(n-1)$ times the absolute value of the mean curvature. Our intent is to verify that for small times at least, the classical motion as envisioned in fact exists and is unique. This assertion was first proved by R. Hamilton [4], and we discuss below the relation of our work to his.

[From the introduction] In this paper, we consider the Dirichlet problem for the conformally invariant model problem of critical Sobolev growth: \[\] Problem (1.1) is formally the Euler-Lagrange equations for the variational problem \[\] for $u$ in the admissible class \[\] where $h$ is the harmonic extension of $\phi\ge 0$.