Frank Morgan

Proof of the Double Bubble Conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 .

Regularity of isoperimetric hypersurfaces in Riemannian manifolds

We add to the literature the well-known fact that an isoperimetric hypersurface S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold. If the metric is merely Lipschitz, then S is still Holder differentiable.

Induced-polarization detection and mapping of contaminant plumes

Several laboratory and scaled model investigations suggest that organic contaminants affect the surface electrical properties of exposed soils/rocks and therefore produce measurable induced polarization IP signatures. However, there is little field evidence of an IP methodology for contaminant mapping. A2D time-domain IP method is developed for mapping the FS12 contaminant plume at the Massachusetts Military Reservation MMR located in Cape Cod, Massachusetts. The FS-12 plume consists of approximately 265 m 3 of fuel that erupted from a broken underground pipeline in the early 1970s. Benzene and ethylene dibromide EDB are the primary contaminants at FS-12, with concentrations exceeding the allowed maximum concentration levels MCL, while other constituents of the plume did not exceed their MCL. Therefore, the contaminants of interest are benzene and EDB, partly because of their health risk and partly because they present the highest concentrations 2400 and 1000 g/L, respectively among the plume constituents and are therefore more likely to be related to the polarization source. IP data were acquired along a survey line that partially transects the plume extending over contaminated and uncontaminated zones and were inverted to give 2D resistivity and chargeability plots to 100 m depth and a horizontal extent of 400 m. By separately inverting IP data derived from time windows located at short and long decay times, a timedomain gross spectral chargeability difference is produced. Both the chargeability and gross spectral chargeability difference show good agreement with the known location of the plume from monitoring wells, with the IP chargeability section suggesting contaminant distribution detail that cannot otherwise be inferred from the sparse borehole distribution.

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

Clusters minimizing area plus length of singular curves

Soap bubble clusters, which seek the least-area way to enclose and separate given volumes of air, often have been used as models for metallic structures, which likewise seek to minimize interfacial areas between regions. For metals, however, there may well be additional energy associated with the singular curves, affecting both the global and local structure (cf. [MT, Bo]). See Fig. 1. We give the first existence and regularity results for clusters of prescribed volumes in R 3 minimizing area plus length of singular curves. Almgrens general theory of soap bubble clusters [A2] does not apply, because it provides no control over the singular curves, although some of his ideas and lemmas prove useful.

Isoperimetric regions in cones

We consider cones C = 0? M n and prove that if the Ricci curvature of C is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.

An isoperimetric comparison theorem for Schwarzschild space and other manifolds

We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric (n - 1)-spheres of a spherically symmetric n-manifold are isoperimetric hypersurfaces, meaning that they minimize (n - 1)-dimensional area among hypersurfaces enclosing the same n-volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual (3 + 1)-dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.

OPEN PROBLEMS IN SOAP BUBBLE GEOMETRY

The Burlington Mathfest in August 1995 included an AMS Special Session on Soap Bubble Geometry, organized by Frank Morgan. At the end of the session, participants were asked to pose open problems related to bubble geometry. We have collected those problems here, adding a few introductory comments. Participants in the special session included the following: Fred Almgren, Princeton U. Megan Barber, Williams C. Ken Brakke, Susquehanna U. John Cahn, NIST Joel Foisy, Duke U. Christopher French, U.Chicago Scott Greenleaf, SUNY Stony Brook Karsten Groes-Brauckmann, Bonn Joel Hass, UC Davis Aladar Heppes, Budapest Michael Hutchings, Harvard U. Jenny Kelley, Rutgers U. Andy Kraynik, Sandia Rob Kusner, U.Massachusetts Rafael Lopez, Granada Joe Masters, U.Texas Helen Moore, Bowdoin C. Frank Morgan, Williams C. Ivars Peterson, Science News Robert Phelan, Dublin Joel Shore, McGill U. John Sullivan, U.Minnesota Italo Tamanini, Trento Jean Taylor, Rutgers U. Jennifer Tice, Williams C. Brian Wecht, Williams C. Henry Wente, U.Toledo Brian White, Stanford U.

Hydrogeophysical methods for analyzing aquifer storage and recovery systems

Hydrogeophysical methods are presented that support the siting and monitoring of aquifer storage and recovery (ASR) systems. These methods are presented as numerical simulations in the context of a proposed ASR experiment in Kuwait, although the techniques are applicable to numerous ASR projects. Bulk geophysical properties are calculated directly from ASR flow and solute transport simulations using standard petrophysical relationships and are used to simulate the dynamic geophysical response to ASR. This strategy provides a quantitative framework for determining site-specific geophysical methods and data acquisition geometries that can provide the most useful information about the ASR implementation. An axisymmetric, coupled fluid flow and solute transport model simulates injection, storage, and withdrawal of fresh water (salinity ∼500 ppm) into the Dammam aquifer, a tertiary carbonate formation with native salinity approximately 6000 ppm. Sensitivity of the flow simulations to the correlation length of aquifer heterogeneity, aquifer dispersivity, and hydraulic permeability of the confining layer are investigated. The geophysical response using electrical resistivity, time-domain electromagnetic (TEM), and seismic methods is computed at regular intervals during the ASR simulation to investigate the sensitivity of these different techniques to changes in subsurface properties. For the electrical and electromagnetic methods, fluid electric conductivity is derived from the modeled salinity and is combined with an assumed porosity model to compute a bulk electrical resistivity structure. The seismic response is computed from the porosity model and changes in effective stress due to fluid pressure variations during injection/recovery, while changes in fluid properties are introduced through Gassmann fluid substitution.

Area-minimizing surfaces, faces of Grassmannians, and calibrations

(1988). Area-minimizing Surfaces, Faces of Grassmannians, and Calibrations. The American Mathematical Monthly: Vol. 95, No. 9, pp. 813-822.