Robert Gulliver

On the variety of manifolds without conjugate points

The longest geodesic segment in a convex ball of a riemannian manifold, where the convexity is ensured by an upper bound on sectional curvatures, is the diameter. This and related results are demonstrated and applied to show that there exist manifolds with sectional curvatures of both signs but without conjugate points.

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofSn orHn, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU kk ωkM (D)k-1 ≤Vol(∂D)k, where ωk is the volume of the unit ball ofRk. Also, we prove that ifD is any domain on a minimal surface inS+n (orHn, respectively), thenD satisfies an isoperimetric inequality2π A≤L2+A2 (2π A≤L2−A2 respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofHn, then(k−1) Vol(U)≤Vol(∂U).

The Case for Differential Geometry in the Control of Single and Coupled PDEs: The Structural Acoustic Chamber

In line with the title of the IMA Summer Program—Geometric Methods in Inverse Problems and PDE Control—the aim of the present article may be summarized as follows: we intend to provide a relatively updated survey (subject to space limitations) of results on exact boundary controllability and uniform boundary stabilization of certain general classes of single Partial Differential Equations as well as of classes of systems of coupled PDEs (in dimension strictly greater than one), that have become available in recent years through novel approaches based on differential (Riemannian) geometric methods.

Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball

We consider the (degenerate) parabolic equationut=G(▽▽u + ug, t) on then-sphereSn. This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦▽u(·,t)¦ <C, whereC depends on the initial conditionu(·, 0) but not ont, nor on the nonlinear functionG. We also prove analogous results for the equationut=G(Δu +cu, ¦x¦,t) on then-ballBn, wherec ≤ λ2(Bn).

Harmonic maps which solve a free-boundary problem

The study of harmonic mappings has seen considerable progress in recent years. Most of the existence and regularity results for boundary-value problems have concerned themselves with the Dirichlet problem, in which the mapping is required to agree with a given mapping on the boundary of its domain. In the present paper, we shall address a free-boundary problem, which is a natural extension of the Plateau boundary conditions and the free boundary conditions for surfaces.

Analysis of a Variational Approach to Progressive Lens Design

We consider a variational approach to the progressive lens design problem. The corresponding Euler--Lagrange equation is a fourth-order nonlinear elliptic partial differential equation. We analyze two linearizations of the equation and show the existence and uniqueness as well as the regularity of the solutions for various boundary conditions. We end with an example of a progressive lens designed by solving the elliptic partial differential equation.

Least area surfaces can have excess triple points

LET F be a closed surface and ,tl a Riemannian 3-manifold. A map

On false branch points of incompressible branched immersions

F-+M is incompressible if 1,: rl(F)-+nl(_lI) is injective. A smooth incompressible immersionfi F-+M is said to be a least clren map if the area offis less than the area of any map from F to M which is freely homotopic to& If 41 is a closed 3-manifold, with rrZ(M) zero, and if F is not S’ or P’, then a theorem of Schoen and Yau [12] asserts that any incompressible map from F to M is homotopic to a least area map. In [2], Freedman et nl. proved some results on the intersections and selfintersections of two-sided least area surfaces, which they summarized by the slogan “least area surfaces intersect least”. (Two-sided means that the map has trivial normal bundle.) In this article we give some examples to show that least area surfaces need not have the minimal possible number of triple points. This is the result of the title. These examples suggest that the results of [2] are the best that can be obtained in general. Iffi F-+Jl is a general position immersion, the only self-intersections will be double curves and triple points. and the usual invariant associated to this self-intersection is the complesity. which is the pair (t, d) where t is the number of triple points and n is the number of double curves. Sow a least area immersion need not be in general position, so Freedman et al. defined a new inv-ariant D(f) of any incompressible immersionfi F+M which, whenfis in general position. is closely related to the number of double curves d(J). They showed that a two-sided least area immersionfminimizes the invariant D(f) among all homotopic maps. They also gave an example to show that, in general, a least area immersion need not minimize the number of double curves d(f), which was another reason for defining a new invariant. However, in the case when the surface involved is the torus T, they showed that a least area mapfi T-+,LI was always self-transverse and that D(J) equals d(J). Thus f has the least number of double curves achievable by any map homotopic to& The main example in this paper is of a Riemannian 3-manifold M and a least area map f: T-+-.11 which has triple points but is homotopic to a general position immersion without triple points. Thusfdoes not minimize the complexity (t, d). At this point, we should mention an important unsolved problem. We stated above that a least area mapfi T+M must be selftransverse. This means that any two sheets off(T) cross transversely, but it does not mean thatf‘is in general position. The mapfcould have curves of triple points (or of even higher multiplicity). X more worrying possibility is thatfcould have a countably infinite set of triple points. An example of a local picture with this property is the intersection in ?z~ of the planes -_=O and y=r with the surface ~=e1’XL sin (l/.x). This last surface is not minimal, but Guliiver has constructed an example of three minimal surfaces which intersect with a countable set of triple points. If M has an analytic metric, thenfis also analytic [6-83 and it

On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary

We prove that a branched immersion of a surface with boundary into a differentiable manifold has no false branch points (in fact, no ramified points) if the immersion induces an isomorphism of fundamental groups and some other natural hypotheses are satisfied. This result has immediate applications to Plateaus problem.

THE USE OF GEOMETRIC TOOLS IN THE BOUNDARY CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS

A general method is developed for finding necessary conditions for a given codimension-two submanifold Γ of a riemannian manifold to be the boundary of an immersed hypersurface of prescribed mean curvature. In the simplest case the condition is a comparison of the magnitude of the mean curvature with the ratio of the volume of the projection of Γ into a hyperplane to the volume of the interior of that projection. The method is applied to show that certain recent existence results for surfaces of prescribed mean curvature may not be quantitatively improved.