arXiv subjects

Frank Morgan

Publications and source records attributed to Frank Morgan.

16 recordsLinked to original sources

arXiv2016

The Space of Planar Soap Bubble Clusters

Soap bubbles and foams have been extensively studied by scientists, engineers, and mathematicians as models for organisms and materials, with applications ranging from extinguishing fires to mining to baking bread. Here 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. Earlier work of Moukarzel showed how to realize a cluster as a generalized Voronoi partition, though not canonically.

math.MG↗

arXiv2012

The Isoperimetric Problem in Higher Codimension

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

math.DG↗

arXiv2011

Isoperimetric Pentagonal Tilings

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

math.MG↗

arXiv2011

Existence of isoperimetric regions in $\R^n$ with density

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.

math.FA↗

arXiv2009

Steiner and Schwarz symmetrization in warped products and fiber bundles with density

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.

math.DG↗

arXiv2008

Stable constant mean curvature hypersurfaces are area minimizing in small L^1 neighborhoods

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.

math.DG↗

arXiv2008

In orbifolds, small isoperimetric regions are small balls

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.

math.MG↗

arXiv2008

Existence of Least-perimeter Partitions

We prove the existence of a perimeter-minimizing partition of R^n into regions of unit volume. We conclude with a short tribute to the late Manuel A. Fortes.

math.MG↗

arXiv2007

The Levy-Gromov Isoperimetric Inequality in Convex Manifolds with Boundary

We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.

math.DG↗

arXiv2006

In polytopes, small balls about some vertex minimize perimeter

In (the surface of) a convex polytope P^n in R^n+1, for small prescribed volume, geodesic balls about some vertex minimize perimeter. This revision corrects a mistake in the mass bound argument in the proof of Theorem 3.8.

math.MG↗

arXiv2006

When Soap Bubbles Collide

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.

math.DG↗

arXiv2006

On the isoperimetric problem in Euclidean space with density

We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.

math.DG↗

arXiv2004

Planar Clusters

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 .

math.MG↗

arXiv2004

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 \Bbb R^3.

math.DG↗

arXiv2004

Hexagonal surfaces of Kapouleas

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.

math.MG↗

arXiv2004

Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces in R^n

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 for 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 treat other categories of surfaces such as rectifiable currents modulo nu and soap films.

math.MG↗