## Research Papers **Google Citations**

[M1] *A s**mooth curve in ***R**^{4}* bounding a continuum of area minimizing surfaces.* Duke Math. J. 43 (1976), 867-870.

The first example is given of a smooth curve in **R*** ^{n}* which bounds infinitely many minimal surfaces. Symmetries of the curve give rise to a continuum of nonorientable surfaces, which are actually area-minimizing.

[M2] *Almost every curve in*** R**^{3}* bounds a unique area minimizing surface.* Inventiones Math. 45 (1978), 253-297. (Princeton University dissertation, 1977, supervised by Professor Frederick J. Almgren, Jr., in geometric measure theory.)

A geometrically natural, probabilistic measure µ, akin to Brownian motion, is defined on the space of smooth Jordan curves in **R**^{3}. It is proved that for µ-almost every curve *B*, there is a *unique *surface of least area bounded by *B*. (Actually shows bad set σ-porous.)

[M3] *Measures on spaces of surfaces.* Arch. Rat. Mech. Anal. 78 (1982), 335-359.

Our probabilistic measure is generalized from curves in **R**^{3} to the space of smooth, compact, k-dimensional submanifolds of **R*** ^{n}*. Applications include generic results on uniqueness for k-dimensional area-minimizing surfaces in

**R**

*and on transversality, immersions, and embeddings. The methods combine geometric measure theory, probability theory, partial differential equations, and pseudodifferential operators.*

^{n}[M4] *A smooth curve in*** R**

^{3}bounding a continuum of minimal manifolds. Arch. Rat. Mech. Anal. 75 (1981), 193-197.

The first example in **R**^{3} is given of a smooth curve which bounds infinitely many minimal surfaces. The curve, consisting of four circles, actually bounds continua of unstable minimal surfaces of arbitrarily large genus.

[M5] *Generic uniqueness results for hypersurfaces minimizing the integral of an elliptic integrand with constant coefficients. *Indiana U. Math. J. 30 (1981), 29-45.

Generic uniqueness results are extended from *area*-minimizing surfaces to hypersurfaces minimizing more general integrals.

[M6] *On the singular structure of two-dimensional area minimizing surfaces in ***R***^{n}*.

*Math. Ann. 261 (1982), 101-110.*

Tangent cones at singularities in 2-dimensional area-minimizing integral currents (oriented surfaces) in **R*** ^{n}* are shown to be sums (i.e., unions) of complex planes (for some orthogonal complex structure on some subspace of

**R***). Singularities in 2-dimensional area-minimizing flat chains modulo two (unoriented surfaces, contrary to misstatement at beginning of Introduction) in*

^{n}

**R***are shown to be isolated points where at most*

^{n}*n*/2 smooth submanifolds intersect orthogonally.

Note. In 2006 Mario Micallef pointed out to me a 1931 article by Douglas in which he essentially proves these results! J. Douglas, The problem of Plateau for two contours, J. Math. Phys. 10 (1931), 315-359, Sect. 9.

[M7] with R. Gulliver, *The symmetry group of a curve preserves a plane.* Trans. AMS 84 (1982), 408-411.

The symmetries of a Jordan curve in Euclidean or hyperbolic n-space are completely characterized.

[M8] *On the singular structure of three-dimensional area-minimizing surfaces in R^{n}*. Trans. AMS 276 (1983), 137-143.

A sufficient condition is given for the sum (union) of two 3-planes in **R*** ^{n}* to be area-minimizing. The results suggest the general stability of singularities.

[M9] *Area minimizing currents bounded by higher multiples of curves. *Rend. Circ. Mat. Pal. 33 (1984), 37-46.

A question on the least area bounded by multiples of a curve, posed by L.C. Young in 1963, is answered by an example.

[M10] *Examples of unoriented area-minimizing surfaces.* Trans. AMS 283 (1984), 225-237.

A comprehensive study is made of constructions of area-minimizing flat chains modulo two. New results show that any area-minimizing submanifold of **R*** ^{n}* occurs as the singular set of some area-minimizing flat chain modulo two in some

**R***.*

^{N}[M11] *The exterior algebra *Λ^{k}**R*** ^{n}* and area minimization. Lin. Alg. and its Appl. 66 (1985), 1-28.

The structure of the exterior algebra Λ^{k}**R*** ^{n}* is studied in low dimensions, and consequences are drawn for k-dimensional area-minimizing surfaces in

**R***. Results include a classification of calibrated geometries of 3-dimensional surfaces in*

^{n}

**R**^{6}, a comass equality with implications on when the Cartesian product of area-minimizing surfaces is area-minimizing, and new examples of area-minimizing surfaces with singularities.

[M12] with R. Harvey, *The faces of the Grassmannian of 3-planes in*** R**

^{7}

*.*Inventiones Math. 83 (1986), 191-228.

A classification is given of the faces of the Grassmannian of oriented 3-planes in **R**^{7}, and hence of the calibrated geometries of 3-dimensional area-minimizing surfaces in **R**^{7}. There are five discrete types of faces and five infinite families of types. New phenomena include faces which are not totally geodesic in the Grassmannian: nonround **S**^{1}‘s, **S**^{2}‘s, and **S**^{3}‘s.

[M13] with R. Harvey, *The comass ball in *Λ^{3}**R**^{6}**. **Indiana U. Math J. 35 (1986), 145-156.

The structure of the set of calibrations in * *Λ^{3}**R**^{6} is studied. (In [M12], these results are applied to identify the calibrated geometries of 3-dimensional area-minimizing surfaces in **R**^{7}.)

[M14] *On finiteness of the number of stable minimal hypersurfaces with a fixed boundary. *Indiana U. Math. J. 35 (1986), 779-833. (Research announcement appeared in Bull. AMS 13 (1985), 133-136.)

The finiteness of the number of area-minimizing or stable minimal hypersurfaces with a fixed boundary is established in several settings, including certain ambient manifolds. In particular, for a uniformly extremal system of * *C^{3} Jordan curves in **R**^{3}, the classical Plateau-Douglas problem has only finitely many least-area solutions of fixed topological type, and only finitely many types occur. Moreover, for a real-analytic Jordan curve in certain noncompact manifolds, there are only finitely many solutions to the classical Plateau problem. Applications include what seems to be the first regularity theorem for area-minimizing normal currents.

[M15] with J. Dadok and R. Harvey, *Calibrations on*** R**

^{8}

*.*Trans. AMS 307 (1988), 1-40.

Results are given on the calibrated geometries of 4-dimensional area-minimizing surfaces in * ***R**^{8}. New phenomena include the first example of a face of the Grassmannian which is not a finite union of embedded manifolds.

[M16] *A regularity theorem for minimizing hypersurfaces modulo *ν. Trans. AMS 297 (1986), 243-253.

It is well known that an (*n*-1)-dimensional area-minimizing flat chain modulo ν *S* in **R*** ^{n}* can have an (

*n*-2)-dimensional singular set. Nevertheless, it is proved that if

*S*has a smooth, extremal boundary of at most

*n*/2 components, then the singular set has Hausdorff dimension at most

*n*-8.

[M17] *Harnack-type mass bounds and Bernstein theorems for area-minimizing flat chains modulo *ν. Comm. P.D.E. 11 (1986), 1257-1295.

For an area-minimizing flat chain modulo ν with no boundary inside the unit ball, an absolute upper bound is given for the amount of area inside a shrunken ball. Such Harnack-type estimates lead to generalizations of Bernstein’s Theorem. For example, for *n* ≤ 5, a 2-dimensional, area-minimizing locally flat chain modulo 2 without boundary in **R*** ^{n}* which has at least one singularity must consist of 2 orthogonal planes.

[M18] *Calibrations modulo* ν. Adv. in Math. 64 (1987), 32-50.

The theory of calibrations is extended to show that certain surfaces are area-minimizing modulo ν. For example, a complex algebraic variety in **C*** ^{n}* of degree

*d*is area-minimizing modulo ν for all ν ≤ 2

*d*.

[M19] with H. Gluck and W. Ziller, *Calibrated geometries in Grassmann manifolds. *Comment. Math. Helvetici 64 (1989), 256-268.

New parallel calibrations on the Grassmannian of oriented *m*-planes in * ***R*** ^{n}* prove certain subGrassmannians to be homologically area-minimizing.

[M20] *Least-volume representatives of homology classes in *G(2,**C**^{4}). Ann. scient. Šc. Norm. Sup. 22 (1989), 127-135.

Least-volume representatives are found for every integer homology class in the Grassmannian G(2,* ***C**^{4}) of complex 2-planes in **C**^{4}. (In degree 4 the homology has rank 2, so that there are lots of classes.)

[M21] *The cone over the Clifford torus in R^{4}*

*is*

*Φ-minimizing.*Math. Ann. 289 (1991), 341-354.

The regularity results for area-minimizing hypersurfaces (integral currents) are shown to fail for hypersurfaces minimizing the integrals of certain other elliptic integrands *Φ*. In particular, the cone over **S ^{1}**x

**S**in

^{1}

**R**^{4}is

*Φ*-minimizing.

[M22] *A sharp counterexample on the regularity of **Φ*-minimizing hypersurfaces. Bull. Amer. Math Soc. 22 (1990), 295-299.

An announcement of [M21].

[M23] *Size-minimizing rectifiable currents.* Invent. math. 96 (1989), 333-348.

Traditionally in computing the area of a surface (rectifiable current) one counts multiplicities. *Size*is an alternative to area which does not count multiplicities. This paper addresses basic questions of the existence, regularity, and behavior of size minimizers. A gap in the proof of Existence Theorem 2.11 was filled by Thierry De Pauw and Robert Hardt, “Size minimization and approximating problems,” Calc. Var. PDE 17 (2003), 405-442, Rmk. 2.3.5.

[M24] *The torus lemma on calibrations, extended. *Proc. Am. Math. Soc. 107 (1989), 675-678.

This extension of the torus lemma characterizes faces of the Grassmannian in terms of their intersections with a maximal torus.

[M25] With Francis C. Larche and John Sullivan,*Some results on the phase behavior in coherent equilibria. *Scripta Metallurgica et Materialia 24 (1990), 491-494.

Alloys such as gold and copper can have two different solid phases. We show under certain hypotheses that the compositions within each phase are *nonincreasing *in the overall composition.

[M26] *Calibrations and the size of Grassmann faces.* Aequationes Math. 43 (1992), 1-13.

The richness of examples and model singularities provided by a (constant-coefficient) calibration depends on the size of the associated face of the Grassmannian. This paper provides upper and lower bounds on the size of Grassmann faces for area and for more general integrands.

[M27] With John Sullivan and Francis C. Larche, *Monotonicity theorems for two-phase solids*. Archive Rat. Mech. Anal. 124 (1993), 329-353.

This paper extends the mathematical theory of [M25], showing how certain quantities vary in given parameters for energy minima. The tools include convex function theory and the calculus of variations.

[M28] With Gary Lawlor, *Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms.* Pacific J. Math. 166 (1994), 55-82.

We prove cones such as the cone over the tetrahedron minimizing in a context that applies to soap films, immiscible fluids, shortest networks, and more general norms than area. The proof is direct, using a new kind of calibration.

[M29] With Gary Lawlor, *Minimizing cones and networks: immiscible fluids, norms, and calibrations,* in Jean Taylor, ed., Computing Optimal Geometries, AMS Selected Lectures in Math., 1991.

[M30] With Jean Taylor, *The tetrahedral point junction is excluded if triple junctions have edge energy. *Scripta Metallurgica et Materialia 25 (1991), 1907-1910.

If singular curves carry any cost, the cone over the tetrahedron is no longer minimizing. Estimates tell on what scale to look for a resolution of this singularity in immiscible fluids or grain boundaries.

[M31] With Herman Gluck and Dana Mackenzie, *Volume-minimizing cycles in Grassmann manifolds,* research announcement. An announcement of [M32].

[M32] With Herman Gluck and Dana Mackenzie, *Volume-minimizing cycles in Grassmann manifolds.* Duke Math. J. 79 (1995), 335-404.

Least-volume representatives are found for example for every real homology class in H^{4} of the Grassmannian G(4,**R**^{8}) of oriented 4-planes in **R**^{8}, which has rank 3. The surfaces are calibrated by quaternionic and Pontryagin forms. For H^{4}G(3,**R**^{7}), in one Pontryagin homology class there are no calibrated surfaces, and therefore infinitely many associated minimal surfaces. The proofs rely on comass estimates for the relevant calibrations.

[M33] With Joel Hass, *Geodesics and soap bubbles in surfaces.* Math. Z. 223 (1996), 185-196.

On a Riemannian surface there is a curve, often of constant curvature, which minimizes length among all curves bounding the same area or curvature. One consequence is a simple proof of an argument, suggested by Poincare, for a simple geodesic on a convex 2-sphere.

[M34] With Z. Furedi and J. Lagarias, *Singularities of minimal surfaces and networks and related extremal problems in Minkowski space. *DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 6, 1991, 95-106.

Bounds on the number of points which are equidistant or have other such properties in Minkowski space. The questions arise from a study of singular minimal surfaces and networks.

[M35] *Soap bubbles in R^{2}* and in surfaces. Pac. J. Math. 165 (1994), 347-361.

Existence and regularity for least-perimeter enclosures of prescribed areas. We prove that planar soap bubbles consist of arcs of circles meeting in threes at 120-degree angles, thus providing a simplified illustration of the rather technical methods of Almgren. Our theory also provides the option of requiring regions to be connected (in which case they might bump up against each other) or more finely prescribing combinatorial type.

[M36] *Surfaces minimizing area plus length of singular curves.* Proc. AMS 122 (1994), 1153-1161.

The first existence and regularity results on surfaces minimizing area plus length of singular curves, as in energy-minimizing interfaces in materials.

[M37] (** M,**ε,δ)-minimal curve regularity. Proc. AMS 120 (1994), 677-686.

(** M,**ε,δ)-minimal curves are proved to be embedded C

^{1,α/2}curves meeting in threes at 120 degree angles.

[M38] With John E. Brothers, *The isoperimetric theorem for general integrands*. Mich. Math. J. 41 (1994), 419-431.

A relatively simple, general proof that the Wulff crystal uniquely minimizes surface energy for given volume.

[M39] *Clusters minimizing area plus length of singular curves.* Math. Ann. 299 (1994), 697-714.

The first existence and regularity results for clusters of prescribed volumes in **R**^{3} minimizing area plus length of singular curves, as in metals. Extension §4.6 to general norms requires a lowersemicontinuity hypothesis; see e.g. [M44].

[M40] With Christopher French and Scott Greenleaf, *Wulff clusters in* **R**^{2}. J. Geom. Anal. 8 (1998), 97-115.

The first existence and regularity results on the cheapest way to enclose and separate planar regions of prescribed areas, where cost is given by a general norm F, thus generalizing the Wulff shape for enclosing a single region.

[M41] *Strict calibrations.* Matemática Contemporânea 9 (1995), 139-152.

Strict calibrations have comass strictly less than one off the calibrated surface S and hence prove S *uniquely* area-minimizing. Ordinary and strict calibrations, with the usual closure condition relaxed, can prove constant-mean-curvature surfaces area-minimizing for fixed volume constraints. Strict calibrations are sufficiently adaptable to prove minimizing properties of certain triple junctions of constant-mean-curvature surfaces.

[M42] With Joel Hass, *Geodesic nets on the 2-sphere.* Proc. AMS 124 (1996), 3843-3850.

We prove the existence of certain nets of geodesics meeting in threes or more in equilibrium on certain Riemannian 2-spheres.

[M43] With Gary Lawlor, *Curvy slicing proves that triple junctions locally minimize area.* J. Diff. Geom. 44 (1996), 514-528.

In soap films three minimal surfaces meet at 120-degree angles. We use a novel curvy slicing argument to prove that small pieces minimize area for given boundary. The argument applies in general dimension and codimension. See also Petrache and Züst.

[M44] *Lowersemicontinuity of energy of clusters.* Proc. Royal Soc. Edinburgh 127A (1997), 819-822.

We discuss existence and lowersemicontinuity for clusters of materials minimizing an energy given by a collection of norms *F _{ij}* on the interfaces between regions

*R*and

_{i}*R*. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

_{j}[M45] *For the minimal surface equation, the set of solvable boundary values need not be convex.* Bull. Austral. Math. Soc. 53 (1996), 369-372.

One might think that if the minimal surface equation had a solution on a smooth domain *D* in **R*** ^{n}* with boundary values

*f*, it would have a solution with boundary values

*tf*for all 0 ≤

*t*≤ 1. We give a counterexample in

**R**

^{2}.

[M46] *An isoperimetric inequality for the thread problem.* Bull. Austral. Math Soc. (1997), 489-495.

Given a fixed curve *C*_{0} in **R**n of length *L*_{0} and a variable curve *C* of fixed length *L* ≤ *L*_{0}, the thread problem seeks a least-area surface bounded by* C*_{0} + *C*. We show that an extreme case is a circular arc and its chord. We provide some counterexamples and generalizations to higher dimensions.

[M47] *The hexagonal honeycomb conjecture.* Trans. AMS 351 (1999), 1753-1763.

The Hexagonal Honeycomb Conjectured, not proved until 1999 by Thomas Hales, says that the planar hexagonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. Hales’s proof depends on a truncation lemma from this paper, which also had proved existence. See Math Chat and Hales.

[M48] With Kenneth Brakke, *Instability of the wet X soap film. *J. Geom. Anal. 8 (1998), 749-767.

We show that adding slight thickness to an soap film shaped like an X leaves it unstable, although adding much thickness makes it stable. Analogous questions about other singularities remain controversial.

[M49] With Colin Adams, *Isoperimetric curves on hyperbolic surfaces. *Proc. AMS 126 (1999), 1347-1356.

Only for a few Riemannian surfaces is known the least-perimeter enclosure of prescribed area. We characterize solutions for hyperbolic surfaces.

[M50] *Immiscible fluid clusters in* **R**^{2} and **R**^{3}. Mich. Math. J. 45 (1998), 441-450.

We prove that an energy-minimizing planar cluster of immiscible fluids consists of finitely many circular arcs meeting at finitely many points, as long as the interfacial energies satisfy strict triangle inequalities. For **R**^{3}, we generalize soap bubble cluster regularity results to clusters of immiscible fluids with interfacial energies near unity.

[M51] *Perimeter-minimizing curves and surfaces in ***R*** ^{n} enclosing prescribed multi-volume*. Asian J. Math. 4 (2000), 373-382.

Planar curves minimizing length for given area are classically characterized as circular arcs. We give a new generalization to **R*** ^{n}* of such area constraints and characterize the minimizing curves. We also consider surfaces satisfying new generalized volume constraints.

[M52] With Michael Hutchings and Hugh Howards. *The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature.* Trans. AMS 352 (2000), 4889-4909.

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of strictly decreasing 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. (After submitting this paper for publication and returning from leave, in cleaning out old files Morgan discovered a preprint of [BeC] which Cao gave him at the Lehigh Geometry/Topology Conference in June, 1994. Morgan got interested in this problem in 1996. The basic idea of Theorem 2.1 came from Hutchings in an email message of December 19, 1996.)

[M53] With David L. Johnson, *Some sharp isoperimetric theorems for Riemannian manifolds.* Indiana U. Math J. 49 (2000), 1017-1041.

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using differential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

In inequality (3.7), P^{2} should be -P′^{2}.

[M54] With Michael Hutchings, Manuel Ritoré, and Antonio Ros, *Proof of the Double Bubble Conjecture. *Ann. Math. 155 (March, 2002), 459-489.

We prove that the standard double bubble provides the least-area way to enclose a separate two regions of prescribed volume in **R**^{3}. See Lawlor’s “Double bubbles for immiscible fluids in **R*** ^{n}*,” arXiv.org (2012).

[M55] With Michael Hutchings, Manuel Ritoré, and Antonio Ros, *Proof of the Double Bubble Conjecture, *ERA AMS 6 (2000), 45-49. http://www.ams.org/journal-getitem?pii=S1079-6762-00-00079-2.

[M56] With Manuel Ritoré, *Isoperimetric regions in cones. *Trans. AMS 354 (2002), 2327-2339. Available on the web at http://www.ugr.es/~ritore/preprints/cone.pdf.

We consider cones *C* over *M ^{n}* and prove that if the Ricci curvature of

*M*is at least

*n*-1, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions. For an alternative approach by the Ros Product Theorem, see [M72, Sect. 3]. Díaz et al. [D] and later and independently Jimmy Petean (email 11/22/10) have answered one of our open questions (Rmk. 3.10) by observing that geodesic spheres about the vertex are not isoperimetric in a cone over a long skinny torus.

[D] Alexander Díaz, Nate Harman, Sean Howe, and David Thompson, Isoperimetric problems in sectors with density, Adv. Math. (2012).

Montiel [M, p. 586, case κ = 0, *f(r) = cr*], following the Euclidean proof of Barbosa-doCarmo, essentially contains our main result Theorem 3.6 except for the admission of singularities. His more general proof is flawed by assuming without justification that φ = f′ is constant (top of p. 596), which holds just for cones.

[M] Sebastián Montiel, Stable constant mean curvature hypersurfaces in some Riemannian manifolds, Comment. Math. Helv. 73 (1998), 584-602.

[M57] With Hubert Bray, *An isoperimetric comparison theorem for Schwarzschild space and other manifolds.* Proc. AMS 130 (2002), 1467-1472.

We give a very general isoperimetric comparison theorem, which implies for example that geodesic spheres in the Schwarzschild space minimize area for given volume, which in turn has applications to the Penrose Inequality in general relativity.

Note: Theorem 2.1 should assume ϕ_{0} nondecreasing for *r* ≥ *r*_{1}, as is automatic in the corollaries and applications.

[M58] *Area-minimizing surfaces in cones.* Comm. Anal. Geom. 10 (2002), 971-983.

We show that a *k*-dimensional area-minimizing surface can pass thrugh an acute conical singularity if and only if *k* ≥ 3. The larger *k*, the more acute the conical singularity can be.

Cited by: Qi Ding, J. Jost, Y. L. Xin, Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature, arXiv.org.

[M59] With Roger Bolton, *Hexagonal economic regions solve the location problem.* Amer. Math. Monthly 109 (February 2002), 165-172.

We show in a certain mathematical sense that congruent regular hexagons solve the location problem, i.e., provide optimal market regions about centers of production.

[M60] *Small double bubbles are standard.* Electronic Proceedings of the 78th annual meeting of the Lousiana/Mississippi Section of the MAA, Univ. of Miss., March 23-34, 2001,

http://www.mc.edu/campus/users/travis/maa/proceedings/spring2001/

We prove that in a smooth, compact, two-dimensional submanifold of **R*** ^{N}*, the least-perimeter way to enclose and separate two regions of small prescribed areas is a standard double bubble, consisting of three constant-curvature curves meeting in threes at 120 degrees. This paper is largely superseded by the next one, which proves that small stable double bubbles are standard.

[M61] With Wacharin Wichiramala, *The standard double bubble is the unique stable double bubble in R*

^{2}

*.*Proc. AMS 130 (2002), 2745-2751.

We prove that the only equilibrium double bubble in **R**^{2} which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.

[M62] *A round ball uniquely minimizes gravitational potential energy,* Proc. AMS 133 (2005) 2733-2735.

We prove that among measurable bodies in **R**^{3} of mass m_{0} and density at most 1, a round ball of unit density uniquely minimizes gravitational potential energy.

[M63] *Regularity of isoperimetric hypersurfaces in Riemannian manifolds. *Trans. AMS 355 (2003) 5041-5052.

We add to the literature the well-known fact that an isoperimetric hypersurfaces *S* of dimension at most six in a smooth Riemannian manifold *M* is a smooth submanifold. If *M* is merely C^{1,1}, then *S* is still C^{1,1/2}.

[M64] *Clusters with Mu**ltiplicities in R*^{2}*.* Pacific J. Math. 221 (2005) 123-146.

Perimeter-minimizing planar double soap bubbles in which regions are allowed to overlap with multiplicities meet in fours, fives, and sixes as well as threes. We further provide certain generalizations to immiscible fluids and higher dimensions, and an associated theory of calibrations. We work in the category of flat chains with coefficients in a normed group.

[M65] With Kenneth A. Brakke, Instabilities of cylindrical bubble clusters. Eur. Phys. J. E 9 (2002) 453-460.

We use the second variation formula to compute instabilities for certain cylindrical bubble clusters and compare to earlier simulations, experiments, and computations of Cox, Weaire, and Fortes.

[M66] With D. Weaire, N. Kern, S. J. Cox, and J. M. Sullivan, Periodicity of pressures in periodic foams. Proc. Roy. Soc. London A 460 (2004), 569-579.

We show that periodic foams in equilibrium have periodic pressures. Also we show that a planar equilibrium foam with congruent bubbles must be a fully periodic arrangement of hexagons.

[M67] Streams of cylindrical water. Math. Intelligencer 26 (2004), 70-72.

Just as isotropic surface energy produces round water droplets and unstable undulating streams, crystalline energy produces cylindrical droplets and crystalline unduloid

[M68] Cylindrical surfaces of Delaunay. Rend. Circ. Mat. Palermo 53 (2004), 469-477.

For the cylindrical norm on **R**^{3}, for which the isoperimetric shape is a cylinder rather than a round ball, there are analogs of the classical Delaunay surfaces of revolution of constant mean curvature.

[M69] Hexagonal surfaces of Kapouleas. Pacific J. Math. 220 (2005), 379-387.

For the “hexagonal” norm on **R**^{3}, for which the isoperimetric shape is a hexagonal prism rather than a round ball, we give analogs of the compact immersed constant-mean-curvature surfaces of Kapouleas.

[M70] Planar Wulff shape is unique equilibrium. Proc. Amer. Math. Soc. 133 (2005), 809-813.

In **R**^{2}, for any norm, an immersed closed rectifiable curve in equilibrium for fixed area must be the Wulff shape (possibly with multiplicity). Question whether higher-dimensional smooth equilibrium is Wulff shape later answered positively by He, Li, Ma, and Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, *Indiana U Math J* 58 (2008); see also Delgadino and Maggi, Alexandrov’s theorem revisited.

[M71] With Colin Adams and John M. Sullivan, When soap bubbles collide. Amer. Math. Monthly 114 (April, 2007), 329-337; arXiv.org.

Can you fill **R*** ^{n}* with a froth of “soap bubbles” that meet at most

*n*at a time? Not if they have bounded diameter, as follows from Lebesgue’s Covering Theorem. We provide some related results and conjectures.

[M72] In polytopes, small balls about some vertex minimize perimeter. J. Geom. Anal. 17 (2007), 97-106; arXiv.org.

In (the surface of) a convex polytope P* ^{n}* in

**R**

^{n+1}, for small prescribed volume, geodesic balls about some vertex minimize perimeter. (Remark 3.11 is proved in [M83].)

[M73] Regularity of area-minimizing surfaces in 3D polytopes and of invariant surfaces in **R*** ^{n}*. J. Geom. Anal. 15 (2005), 321-341; arXiv.org.

In (the surface of) a convex polytope *P*^{3} in** R**

^{4}, an area-minimizing surface avoids the vertices of

*P*and crosses the edges orthogonally. In a smooth Riemannian manifold

*M*with a group of isometries

*G*, an area-minimizing

*G*-invariant oriented hypersurface is smooth (except of a very small singular set in high dimensions). Already in 3D, area-minimizing

*G*-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also discuss flat chains modulo nu and soap films. For details on Remark 4.2, see subsequent paper, In orbifolds, small isoperimetric regions are small balls.

[M74] With Aládar Heppes, Planar clusters. Phil. Mag. 85 (2005), 1333-1345; arXiv.org**.**

We provide upper and lower bounds on the least-perimeter way to enclose and separate *n* regions of equal area in the plane. Along the way, inside the hexagonal honeycomb, we provide minimizers for each *n*.

[M75] A note on cross-profile morphology for glacial valleys, Short Communications, Earth Surface Processes and Landforms 30 (2005) 513-514.

We provide an improvement on the Hirano-Aniya catenary model for the cross-profile morphology of a glacial valley.

[M76] Manifolds with density. Notices Amer. Math. Soc. 52 (2005), 853-858.

We discuss the category of Riemannian manifolds with density and present easy generalizations of the volume estimate of Heintze and Karcher and thence of the isoperimetric inequality of Levy and Gromov. For some minor corrections see Morgan’s *Geometric Measure Theory,* Chapter 18.

[M77] With Jack Cook and Jonathan Lovett, Rotation in a normed plane. Amer. Math. Monthly, 114 (Aug.-Sept. 2007), 628-632.

Given a norm on a plane, we show that if you can isometrically rotate a generic “irrational” unit rhomus along with its diagonals, then the norm is Euclidean (up to linear equivalence).

[M78] Isoperimetric estimates in products. Ann. Global Anal. Geom. 30 (2006), 73-79; http://dx.doi.org/10.1007/s10455-006-9028-6.

In a product M_{1}xM_{2} of Riemannian manifolds, the least perimeter required to enclose given volume among general regions is at least 1/√2 times that among regions of product form, assuming that the isoperimetric profiles of M_{1} and M_{2} are concave.

[M79] Myers’ Theorem with density. Kodai Math. J. 29 (2006), 454-460.

Generalizations of theorems of Myers and others to Riemannian manifolds with density. Derdzinski (PAMS 134, 2006) has similar estimates on closed curves.

[M80] With César Rosales, Antonio Cañete, and Vincent Bayle, On the isoperimetric problem in Euclidean space with density. Calc. Var. PDE 31 (2008), 27-46; arXiv.org (2006).

In **R** with unimodal density we characterize isoperimetric regions. In **R*** ^{n}* with density we prove existence results and derive stability conditions, leading to the conjecture that for a radial, log-convex density, balls about the origin are isoperimetric. We prove this conjecture for the density exp(

*r*) by symmetrization.

^{2}[M81] With Manuel A. Fortes and M. Fatima Vaz, Pressures inside bubbles in planar clusters. Phil. Mag. Lett. 87 (2007), 561-565.

We provide theoretical estimates and Surface Evolver experiments on the pressures of bubbles in planar clusters.

[M82] In orbifolds, small isoperimetric regions are small balls. Proc. AMS 137 (2009), 1997–2004; arXiv.org (2006).

In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

[M83] The Levy-Gromov isoperimetric inequality in convex manifolds with boundary. J. Geom. Anal. 18 (2008), 1053-1057; arXiv.org (2007).

We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary and certain singularities. See vast generalization:

Emanuel Milman, Sharp isoperimetric inequalities and model spaces for curvature-dimension-diameter condition, http://arxiv.org/abs/1108.4609v3

[M84] Existence of least-perimeter partitions. Phil. Mag. Lett. 88 (Fortes mem. issue, Sept., 2008), 647-650; arXiv.org (2007).

We prove the existence of a perimeter-minimizing partition of **R*** ^{n}* into regions of unit volume.

[M85] with Quinn Maurmann, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE 36 (2009), 1-5.

We give several isoperimetric comparison theorems for manifolds with density, including a generalization of a comparison theorem from Bray and Morgan. We find for example that in the Euclidean plane with density exp(r^{α}) for α ≥ 2, discs about the origin minimize perimeter for given area, by comparison with Riemannian surfaces of revolution.

[M86] Isoperimetric balls in cones over tori. Ann. Global Anal. Geom. 35 (2009), 133-137.

In the cone over a cubic three-torus T^{3}, balls about the vertex are isoperimetric if the volume of T^{3} is less than π/16 times the volume of the unit three-sphere. The conjectured optimal constant is 1.

[M87] with Antonio Ros. Stable constant constant mean curvature hypersurfaces are area minimizing in small L^{1} neighborhoods, Interfaces Free Boundaries 12 (2010), 151–155; arXiv.org (2008).

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^{1} neighborhood. For a better proof without the dimension restriction see Theorem 1.1 of “Minimality via second variation for a nonlocal isoperimetric problem” by Acerbi, Fusco, and Morini (to appear).

[M88] with Steven J. Miller, Edward Newkirk, Lori Pedersen, and Deividas Seferis. Isoperimetric sequences. Math. Magazine 84 (Feb. 2011), 37-42.

We generalize the isoperimetric problem from geometry to numbers. For a correction of the graph of Figure 1, see the web post.

[M89] with Davide Carozza and Stewart Johnson, Baserunner’s optimal path, Math. Intelligencer 32 (2010), 10-15; online November, 2009 ; see Huffington Post blog. Featured in Live Science. See also “Computing Minimum Time Paths With Bounded Acceleration” by Johnson.

We compute the fastest path around the bases, assuming a bound on the magnitude of the acceleration.

[M90] with Sean Howe and Nate Harman, Steiner and Schwarz symmetrization in warped products and fiber bundles with density, Revista Mat. Iberoamericana 27 (2011), 909-918; arXiv.org (2009); see blog post.

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.

[M91] with Walter Filkins, Rebalance Every (15000/V)^{1/3} Years (July, 05 2010). Available at SSRN and Morgan blog.

[M92] The log-convex density conjecture. Christian Houdré, Michel Ledoux, Emanuel Milman, and Mario Milman, eds., Concentration, Functional Inequalities and Isoperimetry (Proc. intl. wkshp., Florida Atlantic Univ., Oct./Nov. 2009), Contemporary Mathematics 545, Amer. Math. Soc., 2011, 209-211.

We discuss the conjecture that in Euclidean space with radial log-convex density, balls about the origin are isoperimetric.

Note: The hypothesis in the fourth example should be *p* > 1 instead of *p* ≥ 1. Thanks to Ping Ngai Chung, Miguel Fernandez, Niralee Shah, and Luis Sordo for the correction.

[M93] with Ping Ngai Chung, Miguel A. Fernandez, Yifei Li, Michael Mara, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Isoperimetric pentagonal tilings, Notices Amer. Math. Soc. 59 (May, 2012), 632-640.

We identify least-perimeter unit-area tilings of the plane by convex pentagons, namely tilings by Cairo and Prismatic pentagons, find infinitely many such perimeter-minimizing Cairo-Prismatic tilings, and prove that they minimize perimeter among tilings by convex polygons with at most five sides.

Don Chakerian has identified the general dimensional case of Proposition 3.1 on isoperimetric polygons with prescribed angles as Lindelöf’s Theorem (Propriétés générales des polyèdres qui, sous une étendue superficielle donnée, renferment le plus grand volume, Math. Ann. 2 (1869), 150–159). See for example Theorem 34 (page 210) of A. Florian, Extremum problems for convex discs and polyhedra, P. M. Gruber and J. M. Wills, eds., Handbook of Convex Geometry, North-Holland, 1993, 177-221. Ivan Izmestiev has identified the 2D case as L’Huilier’s Theorem, cited in Lyusternik’s book Convex Figures and Polyhedra, §26, p. 118.

[M94] with Aldo Pratelli, Existence of isoperimetric regions in **R**^{n}** **with density, Ann. Global Anal. Geom. 43 (2013), 331–365; arXiv.org (2011).

We prove the existence of isoperimetric regions in **R**^{n}** **with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions. Theorem 5.10 and Corollary 5.11(3) need a positive lower bound on the density. See also Cinti and Pratelli. Main conjecture proved in 2014 by De Philippis, Franzina, and Pratelli.

[M95] with Isabel M. C. Salavessa, The isoperimetric problem in higher codimension, Manuscripta Mathematica 142 (2013), 369–382; arXiv.org (2012).

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.

The case of Conjecture 4.2, *m*=1, for fixed v(*S*), was proved by Giomi and Mahadevan, Minimal surfaces bounded by elastic lines, *Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.* 468 (2012), no. 2143, 1851–1864, between equations (3.5) and (3.6).

[M96] with S. J. Cox and F. Graner, Are large perimeter-minimizing two-dimensional clusters of equal-area bubbles hexagonal or circular? Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), 20120392, 10 pp.; arXiv.org (2012).

A computer study of clusters of 10000 equal-area bubbles shows for the first time that rounding conjectured optimal hexagon planar soap bubble clusters reduces perimeter.

[M97] with Wen-Tai Hsu and Thomas J. Holmes, Optimal city hierarchy: a dynamic programming approach to central place theory, preprint (2012), J. Econ. Theory 154 (2014), 245–273, and appendix.

We place cities of various sizes on the line to minimize set-up and transportation costs, and we provide a dynamic programming solution. We show that there must be one and only one immediate smaller city between two neighboring larger-sized cities. Often the smaller city takes a “central place” halfway between the next larger cities, but not always.

[M98] The space of planar soap bubble clusters, Imagine Math 6 – Mathematics and Culture, XXth Anniversary, Michele Emmer and Marco Abate, eds., Springer, 2018, arXiv.org (2016).

We provide some basic results on the space of planar clusters of *n* bubbles of fixed topology. We show for example that such a space of clusters with positive second variation is an *n*-dimensional manifold, although the larger space without the positive second variation assumption can have singularities.

[M99] with Wyatt Boyer, Bryan Brown, Alyssa Loving, and Sarah Tammen, Double bubbles in hyperbolic surfaces, Involve 11 (2018), 207-217.

We seek the least-perimeter way to enclose and separate two prescribed areas in certain hyperbolic surfaces.

[M100] Isoperimetric symmetry breaking: a counterexample to a generalized form of the log-convex density conjecture, Anal. Geom. Metr. Spaces 4 (2016), 314-316.

We give an example of a smooth surface of revolution for which all circles about the origin are strictly stable for fixed area but small isoperimetric regions are nearly round discs away from the origin.

## Popular or Expository Articles

[M101] *Can a wire bound infinitely many soap films?* The Link, MIT, September 1980.

An illustrated exposition is given of a curve in **R**^{3} bounding continua of minimal surfaces.

[M102] *Soap bubbles and soap films,*in Joseph Malkevitch and Donald McCarthy, ed., *Mathematical Vistas: New and Recent Publications in Mathematics from the New York Academy of Sciences, *Vol. 607, 1990, 98-106.

This talk (given at the New York Academy of Sciences) discusses soap bubbles and soap films: the structure of singularities and examples of nonuniqueness and nonfiniteness, with premier illustrations by James F. Bredt.

[M103] Review of *Hackers: Heroes of the Computer Revolution *Levy). Technology Review, May/June, l985.

[M104] *Soap films and problems without unique solutions.* Amer. Scientist, May, 1986, 232-236.

This article uses soap films and many illustrations to explain recent results on uniqueness or finiteness of the number of minimal surfaces with a given boundary in various dimensions.

[M105] Review of *Mathematical People *(Albers/Alexanderson). Technology Review, February/March, l986.

[M105A] With Dana Nance Mackenzie et al., Open problems list, Williamstown Calibrations Conference, August 15-17, 1988.

[M106] *Area-minimizing surfaces, faces of Grassmannians, and calibrations. *Amer. Math. Monthly 95 (1988), 813-822.

A survey of some current work in the theory of calibrations.

[M107] Review of *Mathematics and Optimal Form* (Hildebrandt/Tromba). Amer. Math. Monthly 95 (1988), 569-575.

[M108] *Calibrations and new singularities in area-minimizing surfaces: a survey,*in Henri Berestycki, Jean-Michel Coron, and Ivar Ekeland, ed., *Variational methods (Proc. Conf. Paris, June 1988).* Prog. Nonlinear Diff. Eqns. Applns. Vol. 4, Birkhäuser, Boston, 1990, 329-342.

This survey leads from major historical examples of calibrations to recent results. It discusses the proof by Lawlor and Nance of the Angle Criterion, Lawlor’s classification of area-minimizing cones over products of spheres, and Morgan’s example of a hypercone in **R**^{4} that is minimizing for certain smooth elliptic integrands.

[M109] *Soap films and mathematics, *R. E. Greene and S.-T. Yau, eds., Differential Geometry, Proc. Symp. Pure Math. 54 (1993), Part 1, 375-380.

This review of mathematics inspired by soap films and Plateau’s problem includes fundamental open questions and conjectures. Also see Ken Brakke’s applet of Adams’s deformation retract of a soap film onto its boundary.

[M110] with C. Adams and D. Bergstrand, *The Williams SMALL undergraduate research project. *UME Trends, January, 1991.

[M111] *Compound soap bubbles, shortest networks, and minimal surfaces, *write-up of invited AMS-MAA address, San Francisco (1991).

Open questions, new results, and undergraduate research.

[M112] *Compound soap bubbles, shortest networks, and minimal surfaces, *AMS video, May, 1992.

[M113] *Minimal surfaces, crystals, and norms on R^{n}*. Proc. 7th Annual Symposium on Computational Geometry (June, 1991).

Finding energy-minimizing surfaces hinges on open questions about the existence of many equidistant points in **R*** ^{n}* in the Euclidean and other norms.

[M114] *Minimal surfaces, crystals, shortest networks, and undergraduate research. *Math. Intelligencer 14 (1992), 37-44.

New results and methods on energy-minimizing surfaces and networks. Some important recent advances have been made by undergraduates.

[M115] *Mathematicians, including undergraduates, look at soap bubbles, *Amer. Math. Monthly 101 (1994), 343-351.

It is an open mathematical question whether the common double bubble succeeds in minimizing area or whether there is some as yet undiscovered configuration of less area enclosing and separating the same two volumes of air. The analogous planar problem recently has been solved by undergraduates.

[M116] *Calculus, planets, and general relativity*. SIAM Review 34 (June 1992), 295-299.

In explaining the motions of the planets, Newton invented the calculus, John Couch Adams predicted Neptune, and Einstein developed general relativity. The full story now includes a surprise appearance by Galileo. This article includes a very simplified explanation of general relativity and Mercury’s precession.

[M117] With Tom Garrity, *The Williams College SMALL Undergraduate Mathematics Research Project,* preprint.

The distinctive features of the project and an annotated bibliography of publications.

[M118] *Survey lectures on geometric measure theory. *Geometry and Global Analysis, report of the First MSJ International Research Institute, July 12-23, 1993, Tohoku University, Sendai, Japan, edited by Takeshi Kotake, Seiki Nishikawa, and Richard Schoen, Tohoku University Mathematics Institute, Sendai, Japan, 1993.

These survey lectures describe basic concepts and techniques of geometric measure theory, soap bubble clusters, and calibrations, including undergraduate research.

[M119] *Maxima Minima Problems* (video on YouTube), Views of Calculus, edited by J. Mazur, AK Peters, Wellesley, 1994.

A video lecture, including soap bubbles, now available on the web via my home page.

[M120] *What is a surface? *Amer. Math. Monthly 103 (May, 1996), 369-376.

A search for a good definition of surface leads to the rectifiable currents of geometric measure theory, with interesting advantages and disadvantages.

[M121] *The Williams College SMALL Undergraduate Research Project. *Geometric Optimization unit, Connected Geometry, Education Development Center, Newton, Massachusetts.

A brief report on the SMALL project in general and on the Geometry Group solution of the planar double bubble problem in particular, as a contribution to a high school curriculum development project.

[M122] *Calibrations and minimal surfaces. *Vorlesungsreihe, Analysis-Seminar 1994-1996, University of Bonn, May, 1997, 27-28.

A description of my talk in the Analysis seminar at the University of Bonn.

[M123] *New undergraduate research prize. *Notices AMS, January, 1995.

A news report on the new AMS-MAA-SIAM undergraduate research prize.

[M124] *100-year-old Kelvin Conjecture disproved by Weaire and Phelan.* Math. Horizons, September, 1999.

[M125] *Geometric measure theory, *Instructional Workshop on Analysis and Geometry, Tim Cranny and John Hutchinson, ed. Proc. Cent. Math. Appl., Australian Natl. Univ. 34 (1996), Part II, 51-66.

[M126] *The Double Bubble Conjecture.* FOCUS, Math. Assn. Amer., December, 1995.

A report on the recent computer proof by Hass and Schlafly of the Double Bubble Conjecture on the least-area way to enclose and separate two regions of equal volumes.

[M127] Edited with John Sullivan, *Open problems in soap bubble geometry. *International J. Math. 7 (1996), 833-842.

Open problems from the AMS special session on Soap Bubble Geometry organized by Morgan at the Burlington Mathfest in August, 1995.

[M128] *Dooppelseifenblasen und Studentenforschung,* Mittelungen DMV 1 (1997), 25-27.

A survey article on the double bubble and undergraduate research.

[M129] With Edward Burger, *Fermat’s Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools’ Day*. Amer. Math. Monthly 104 (March, 1997), 246-255.

Write-up of our April Fools’ Day celebration of famous wrong 19th Century proofs.

[M130] With Ted Melnick and Ramona Nicholson. *The soap bubble geometry contest.* The Mathematics Teacher 90 (December, 1997), 746-750.

Write-up of my famous contest, for use by high school teachers.

[M131] Review of *The Parsimonious Universe* (Hildebrandt/Tromba). The Amer. Math. Monthly 104 (April, 1997), 377-380.

[M132] With Hugh Howards and Michael Hutchings. *The isoperimetric problem on surfaces, *Amer. Math. Monthly 106 (1999), 430-439.

A survey of old and new results, including a proof that horizontal circles provide the least-perimeter way to enclose given area in a paraboloid of revolution.

[M133] *Coffee bubbles.* Why Is It? #149, Mutual Radio Network, February 4, 1997.

Bubbles in your coffee congregate around the edges to minimize surface energy. Radio program based on a telephone interview with me, produced by Justin Warner.

[M134] *Teaching mathematics at Williams.* Parents’ Newsletter, spring, 1997.

Williams students come not only with talent but also with a great capacity for growth, that soon makes them the mathematicians and teachers.

[M135] *On being a student of Almgren’s.* Exp. Math 6 (1997), 8-10.

Fred Almgren was my ideal of a thesis advisor.

[M136] *Recollections of Fred Almgren.*J. Geom. Anal. 8 (1998), 877-882.

Recollections of students and colleagues.

[M137] *Do Mathematicians Think Sideways?* Math Medley radio program with Dr. Pat Kenschaft, KFNX at 1100 AM (Phoenix) and WALE at 990 AM (Providence), Sept. 11, 1999.

[M138] *Does the millennium begin on Jan. 1, 2000? *Congressional Quarterly Researcher 9 (Oct. 15, 1999), 899.

[M139] *Math professor divulges truth about upcoming millennium.* The Williams Record, Williams College, Williamstown, MA, December 7, 1999.

[M140] *When and where does the new millennium begin?* Scientific American’s “Ask the Experts,” December 20, 1999.

[M141] *Is 2001 more worthy of celebrating than 2000? *The Daily Jeffersonian, Cambridge, Ohio, December 26, 1999.

This all began when an AP release quoted me as saying that, “The inexorable mathematical logic which cannot be refuted is that the year 2000 is the last year of this millennium and 2001 is the first of the next millennium.”

[M142] Guest on Ron Plock’s *Opinions* call-in radio show, Berkshire Broadcasting, FM radio WMNB100.1, AM radio WNAW 1230, North Adams, Massachusetts, 8:30-9 am, January 6, 2000.

[M143] *Hales’s proof of the hexagonal honeycomb conjecture and related recent results and open problems.* Pacelli Zitha, John Banhart, and Guy Verbist, ed., Proc. 3rd Euroconference on Foams, Emulsions and their Applications (Delft, The Netherlands, June 4-8, 1999), Metall Innovation Tech MIT, Bremen, Germany, 2000.

[M144] A mathematician at heaven’s gate. MathChat.org (archive), June 21, 2001.

A play which opens as an impeccable mathematician arrives at the Pearly Gates.

[M145] *Proof of the double bubble conjecture.* Amer. Math. Monthly, March, 2001, 193-205. Reprinted in Robert Hardt, ed., Six Themes on Variation, Amer. Math. Soc., 2004, 59-77.

[M146] *Double bubble no more trouble.* Math Horizons (November, 2000), 2, 30-31.

[M147] How it all fits. MathChat.org, April 5, 2002, October 4, 2001.

It’s a miracle the way the world fits together, lot against lot, road meeting road, one jagged property line meshing perfectly with the neighbor’s.

[M148] *The perfect shape for a rotating rigid body*. Mathematics Magazine 75 (February, 2002), 30-32.

The energy-minimizing shape for a rotating rigid body is not an oblate spheroid but a stationary ball with a small, distant planet.

[M149] Radio interview by Martha Foley, North Country Public Radio, Canton, NY, www.ncpr.org, January 29, 2001, on occasion of talk at SUNY Potsdam on “Soap Bubbles and the Universe.”

[M150] With Joseph Corneli, Paul Holt, Nicholas Leger, Eric Schoenfeld. *Mathematicians on Michael Feldman’s “Whad’Ya Know?”* FOCUS, Math. Assn. Amer., November, 2001.

An humorous account of the appearance of Morgan, his Geometry Group, and other mathematicians on the popular program on Public Radio International, in Madison during the MathFest, Saturday, August 4, 2001.

[M151] *Fractals and geometric measure theory: friends and foes.* Michel L. Lapidus and Michiel van Frankenhuijsen, ed., Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot [Jan., 2002], Proc. Symp. Pure Mathematics 72 (2004), 93-96.

Mandelbrot’s fractals, like good friends, inspire more general and realistic geometries. But later, like foes, they thwart efforts to prove that solutions to geometric problems are well behaved.

[M152] Geometric measure theory and the proof of the double bubble conjecture, lectures at Clay Math. Inst. 2001 Summer School, MSRI, available as streaming video at http://www.msri.org/publications/video/index02.html

[M153] With Manuel Ritoré. *Geometric measure theory and the proof of the double bubble conjecture,* Global Theory of Minimal Surfaces (Proc. Clay Research Institution 2001 Summer School, MSRI, David Hoffman, editor), Amer. Math. Soc., 2005. www.claymath.org/publications/Minimal_Surfaces

Notes by Ritoré based on Morgan’s course at MSRI.

[M154] *Soap on a hope.* The Last Word, New Scientist (Jan. 17-23, 2004) 57, www.newscientist.com

Illustrated response to a question about the existence of torus bubbles.

[M155] With Tom Garrity.* Teaching tips,* Amer. Math. Soc. (2005), www.ams.org/ams/ttips.pdf.

Easy ways to be a better teacher.

[M156] *“Kepler’s Conjecture” and Hales’s Proof.* Notices Amer. Math. Soc. 52 (2005), 44-47.

A review of G. Szpiro’s book on Kepler’s Conjecture and a discussion of Hales’s recent proof.

[M157] Compactness. Pro Math. 22 (2008), 121-131; Williams Math Blog (2009).

In my opinion, compactness is the most important concept in mathematics. This Williams College undergraduate colloquium talk tracks compactness from the one-dimensional real line in calculus to infinite dimensional spaces of functions and surfaces to see what it can do.

[M158] Review of Singular Sets of Minimizers for the Mumford-Shah Functional by Guy David, SIAM Review 48 (2006), 187-189.

[M159] Problem 852 on “Eigenvalues of a sum,” College Math. J. 38 (May, 2007), 227; solution Vol. 40 (January, 2009), 56-57.

This problem on the eigenvalues of a sum of two projection matrices is related to work by Victor Guillemin and Reyer Sjamaar on convexity theorems in symplectic geometry http://front.math.ucdavis.edu/math.SG/0504537.

[M160] Soap bubble clusters, Expeditions in Mathematics, Spectrum Series, Math. Assn. Amer., 2011, 165-173.

Planar soap bubble clusters continue to fascinate and perplex mathematicians. We report on some recent progress, including work by undergraduates. Based on a talk/contest for Bay Area Mathematics Adventures.

[M161] Geometric measure theory and soap bubbles, lecture at International Seminar on Applied Mathematics in Andalusia, September, 2006, posted at gigda.ugr.es/isaga06.

[M162] Soap bubble clusters. Rev. Mod. Phys. 79 (2007), 821-827.

Although soap bubble clusters and froths provide simple models of diverse physical phenomena, the underlying mathematics is deep and still not understood.

[M163] Geometry lessons, interview by Jeffrey Hildner, The Christian Sci. J. 124 (October 2006), 52-55; O Arauto Ciencia Crista 57 (Marco 2007), 19-23; El Heraldo Ciencia Christiana 57 (Mayo/Junio 2007), 8-11;

Mathematical principles governing the shape of soap bubbles provide an analogy for God as divine Principle governing the universe.

[M164] Review of Riemannian Geometry: A Modern Introduction by Isaac Chavel, SIAM Review 49 (2007), 536-537.

[M165] with Cesar Silva. The SMALL Program at Williams College. Proc. Conf. Promoting Und. Res. Math. (J. Gallian, ed.), Amer. Math. Soc., 2007.

[M166] Scientist in Heaven, Frank Morgan Blog, 7 October 2008.

A sketch of a movie idea about an excellent scientist and citizen, which I had had for some time, but which took further form during a cathedral mass in Granada in 1999.

[M167] Manifolds with density and Perelman’s proof of the Poincaré Conjecture. Amer. Math. Monthly 116 (Feb., 2009), 134-142.

Manifolds with density long have appeared in mathematics, with more recent attention to their differential geometry, including a generalization of Ricci curvature, which Perelman uses in exploring the Ricci flow. This note is based on Chapter 18 of the upcoming, fourth edition of my Geometric Measure Theory book.

[M168] Stochastic calculus and the Nobel Prize winning Black-Scholes equation, Math Horizons (Nov, 2009), 16-18; Williams Math Blog (2008).

The celebrated Black-Scholes partial differential equation for financial derivatives stands as a revolutionary application of stochastic or random calculus. Based on a short talk at a special “Stochastic Fantastic Day,” which my chair Tom Garrity organized to give his colleagues a chance to explore a compelling but unfamiliar topic and enjoy dinner at his home afterwards.

[M169] Fermat’s Last Theorem for Fractional and Irrational Exponents, College Math. J. 41 (2010), 182-185; see Morgan Blog (2008).

Fermat’s Last Theorem says that for positive integers *n* > 2, *x*, *y*,* z*, there are no solutions to x* ^{n}* + y

*= z*

^{n}*. What about rational exponents*

^{n}*n*? Irrational

*n*? Negative

*n*? See what an undergraduate senior seminar discovered.

[M170] with Ivan Corwin, The Gauss-Bonnet formula on surfaces with densities, Involve 4-2 (2011), 199-202. See blog post.

The celebrated Gauss-Bonnet formula has a nice generalization to surfaces with densities, in which both arclength and area are weighted by positive functions.

[M171] with Davide Carozza and Stewart Johnson, Baserunner’s optimal path, Collegiate Baseball Newspaper, 12 Feb. 2010.

An expository account of our paper [M89] about the fastest path around the bases, assuming a bound on the magnitude of the acceleration; see Morgan Blog.

[M172] Respecting our Mission, The Williams Record, May 5, 2010.

Our decisions process should reflect our mission as a liberal arts college.

[M173] Keep math inclusive, Notices Amer. Math. Soc. 58 (Sept. 2011), 1053-1054.

Arguments against the proposed AMS Fellows program.

[M174] Interview on Strongly Connected Components with Samuel Hansen, October 10, 2012.

[M175] With Paul Gallagher and Maggie Miller, Working up a Lather—Bubbles and Foam, Parts 3 and 4, Mathematical Moment interview by Michael Breen.

[M176] With Alissa S. Crans and Talithia Williams, Town Hall Meeting: Minority Participation in Math, MAA FOCUS (Dec. 2013/Jan. 2014).

[M177] Bubbles and tilings: art and mathematics, Proc. Bridges 2014.

The 2002 proof of the Double Bubble Conjecture on the ideal shape for a double soap bubble depended for its ideas and explanation on beautiful images of the multitudinous possibilities. Similarly recent results on ideal tilings depend on the artwork.

[M178] Academics Must Be Williams’s Top Priority, Williams Record, May 7, 2014.

[M179] Williams: Inclusive or Exclusive? WilliamsAlternative.com, May, 2014.

[M180] Interviewed on Rick Chrisman’s Berkshire Community College “1350 West Street” TV, Part 1 and Part 2, October 23, 2014.

[M181] Six milestones in geometry. Stephen F. Kennedy, ed., *A Century of Advancing Mathematics*, Math. Assn. Amer., 2015, 51-64.

My choices for the six biggest advances in geometry during the 100-year lifetime of the MAA.

[M182] Unsolved mathematical mysteries, Virginia Math. Teacher 42(1) (Fall, 2015), 19-20.

Write-up of talk on “Soap Bubbles and Mathematics” at the Connecting Mathematical Practices Conference at Radford University, May 8, 2015, including some open questions about soap bubble clusters.

[M183] Soap bubbles and mathematics. Eur. Math. Soc. Newsletter (Sept. 2015), 32-36.

Write-up of Abel Science Lecture, May 20, 2015, Oslo.

[M184] The isoperimetric problem with density, Math. Intelligencer 39 (2017) 2-8.

Enhanced write-up of talk at 2015 Lázló Fejes Tóth Centennial, including recent proofs of Log-Convex Density Theorem, the analog for perimeter and volume densities r* ^{k}* and r

*, and the isoperimetric solution for density r*

^{m}*.*

^{p}[M185] Isoperimetry with density.

Video of talk at CIRM Luminy on recent results and open questions on the isoperimetric problem in the presence of radial densities and metrics on **R*** ^{n}*.

[M186] To bring math to the world, start with mathematicians, Imagine Math 6 – Mathematics and Culture, XXth Anniversary, Michele Emmer and Marco Abate, eds., Springer, 2018.

Write-up of talk at Imagine Maths 6, Mathematics and Culture XX, Venice, March, 2017.

[M187] Bubbles inside bubbles, 2016 and 2017 SMALL undergraduate research Geometry Groups, Williams College Math Blog, June 25, 2018.

Report on discovery of a bubble inside a bubble as a minimizer under a density such as exp(r^{2}) by Morgan’s Geometry Groups: “Double bubbles on the line with log-convex density.”

[M188] edited with Pierre Pansu, “Open Problems” from the 2018 conference “Modern Trends in Differential Geometry,” University of São Paulo, special issue of the *São Paulo Journal of Mathematical Sciences*, to appear 2020.

## Books

[M190] *Geometric Measure Theory*: a *Beginner’s Guide.* Academic Press, 1988; Japanese translation, 1989; second edition, 1995; third edition, 2000; Russian edition, 2006; fourth edition, 2009; fifth edition, 2016.

An easy-going, illustrated introduction for the newcomer to this somewhat technical field. The fifth edition provides comprehensive updates and a new chapter on the Log Convex Density Theorem, a major new result in an area of mathematics—manifolds with density—that has exploded since its appearance in Perelman’s proof of the Poincaré conjecture.

[191] *Riemannian Geometry: a Beginner’s Guide.* A K Peters, Wellesley, 1993; second edition, 1998; revised printing 2001.

Starting with an extrinsic approach to curvature, this book provides a short, intuitive, direct introduction to Riemannian geometry, including topics from general relativity, global geometry, and current research on norms more general than area. The second edition includes many new problems and new sections on the isoperimetric problem and on double Wulff crystals.

[M192] *Calculus Lite.* A K Peters, Wellesley, 1995; second edition, 1997; third edition 2001; republished as *Calculus* (below), 2012.

This lean text covers single-variable calculus in under 300 pages by (1) getting right to the point, and stopping there, and (2) introducing some standard preliminary topics, such as trigonometry and limits, by using them in the calculus. The later editions include new exercises and a new section on multivariable calculus. Solutions available to instructors. Accompanying video on “Maxima Minima” on YouTube.

[M193] *The Math Chat Book. *Math. Assn. Amer., 2000.

A popular book based on the TV show and column (see below). Illustrated by James Bredt.

[M194] *Real Analysis. *Amer. Math. Soc., 2005. Author’s webpage.

Based on a one-semester core real analysis course at Williams.

[M195] *Real Analysis and Applications (including Fourier Series and the Calculus of Variations). *Amer. Math. Soc, 2005. Author’s webpage.

Streamlined, complete theory, plus applications in Fourier series and the calculus of variations, including physics (least action and Largrange’s equations), economics (optimal production and maximal utility), Riemannian geometry, and general relativity.

[M196] *Calculus*. CreateSpace.com, 2012.

A complete calculus text, including differentiation, integration, infinite series, and introductions to differential equations and multivariable calculus, published by the author at CreateSpace.com for $20. Solutions available to instructors. Accompanying video on “Maxima Minima” on YouTube.

## Math Chat TV & Column

[M197] Math Chat TV. Weekly, Williamstown (1996-97), Princeton (1997-98), Williamstown (January, 2000).

Weekly live call-in cable TV show with questions, answers, and prizes.

[M198] Math Chat. The Christian Science Monitor, biweekly, June 14, 1996-October 1, 1998; MAA web page 1998-2002.

Biweekly column with questions, answers, and prizes.

## Blogs

[M199] Columns posted on Math at Williams Blog.

[M200] Columns on Morgan’s own Blog.

[M201] Can math survive without the Bees? Huffington Post blog, 6 March 2012.

[M202] Alan Alda’s flame challenge and kids’ five most popular science questions, Huffington Post blog, 16 March 2012.

[M203] Soap bubbles in Scotland, Huffington Post blog, 23 March 2012.

[M204] Math Finds the Best Doughnut, Huffington Post blog, 2 April 2012.

[M205] Geometry Festival, Huffington Post blog, 30 April 2012.

[M206] Math Now—Commencement Can Wait, Huffington Post blog, 28 May 2012.

[M207] Why is Summer so Early (June 20)? Huffington Post blog, 17 June 2012.

[M208] I Win Soap-Bubble-Cluster Controversy, Huffington Post blog, 22 June 2012.

[M209] Spilled Orange Juice on My Way to a Math Conference in Spain, Huffington Post blog, 30 June 2012.

[M210] Why a Laptop is Not a Computer, Huffington Post blog, 24 July 2012.

[M211] U.S. Presidential Election Paradox, Huffington Post blog, 16 October 2012.

[M212] The Fastest Path Around the Bases, Huffington Post blog, 19 October 2012

[M213] Why I Don’t Like Energy-Efficient Light Bulbs, Huffington Post blog, 24 Nov 2012.

[M214] How Often Should I Rebalance My Investments? Huffington Post blog, 3 Dec 2012.

[M215] Dark Matter and Worst Packings, Huffington Post blog, 28 May 2013.

[M216] Are Smaller College Classes Really Better? Huffington Post blog, 26 August 2013.

[M217] Adding Fractions, Huffington Post blog, 14 March, 2014.

[M218] The Inferiorities of Mac Mail, Huffington Post blog, 15 September, 2015.

[M219] Sphere Packing in Dimension 8, Huffington Post blog, 21 March, 2016.

[M220] Math in Cuba, Huffington Post blog, 6 March, 2017.

**Newspaper, TV, Speaking, Service**

Morgan had a weekly live call-in Math Chat show on local cable TV and a biweekly Math Chat column in *The Christian Science Monitor* and at the MAA website. He gives over thirty talks a year, at venues ranging from research seminars to high schools.

Morgan has served on a number of visiting committees, including Bucknell, Colgate, Colby, Connecticut College, Dickinson, Hamilton, Harvey Mudd, Queens (CUNY), US Naval Academy, Vassar, Washington & Lee, and Wesleyan. He does his share of refereeing and reviewing, and has served on various National Science Foundation committees and panels.