Are circles isoperimetric in the plane with density er?
RHIT Undergrad. Math. J.
Rose-Hulman Undergraduate Mathematics Journal
Miguel A. Fernandez Truman State University 0 1 2
Niralle Shah 0 1 2
0 Ping Ngai Chung University of Cambridge
1 Luis Sordo Vieira University of Kentucky , USA
2 Willams College
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RoseHulman
Undergraduate
Mathematics
Journal
Are circles isoperimetric in the
plane with density er?
Ping Ngai Chung a
Niralee Shah c
Miguel A. Fernandez b Luis Sordo Vieira d
Sponsored by
Rose-Hulman Institute of Technology
Department of Mathematics
Terre Haute, IN 47803
Email:
http://www.rose-hulman.edu/mathjournal
aUniversity of Cambridge
bTruman State University
cWilliams College
dUniversity of Kentucky
Are circles isoperimetric in the plane with
density er?
Ping Ngai Chung
Miguel A. Fernandez
Luis Sordo Vieira
Niralee Shah
Acknowledgements: We thank our advisor Frank Morgan for his patience and
invaluable input. We also thank Diana Davis, Sean Howe and Michelle Lee for their help. For his
helpful comments, we would like to thank our anonymous referee. For funding, we thank
the National Science Foundation for grants to Prof. Morgan and to the Williams
College \SMALL" Research Experience for Undergraduates, Williams College for additional
funding, and the Mathematical Association of America for supporting our trip to speak at
MathFest 2011.
Introduction
There has been a recent surge of interest in Riemannian manifolds with a positive \density"
function that weights volume and area (see recent works by Morgan [M1-4]). In fact,
Perelman [P, Section 1.1] used the concept of manifolds with density in his beautiful proof of the
Poincare conjecture (see [M4] for an intuitive explanation). Although one might choose to
weight area and perimeter di erently, the case where both the area and the perimeter are
weighted by the same function is of most interest. A fundamental problem in manifolds with
density is the isoperimetric problem, which seeks to enclose given area or volume with the
least amount of perimeter. For example, it is well known that in the Euclidean plane, a circle
encloses given area with least perimeter (for a historical treatment, see [B]). Due to its many
applications in probability theory and in Perelman's proof of the Poincare conjecture, one of
the most important examples is Rn with Gaussian density e r2: In 1975, C. Borell [B1] and
V.N. Sudakov and B.S. Tsirel'son [ST] proved that half-spaces are perimeter minimizing for
the Gaussian density. On the other hand, in Rn with density e+r2, C. Borell ([B2], see [Ro,
Introduction and Theorem 5.2]) proved that balls about the origin are perimeter minimizing.
Rosales et al. gave a conjecture on when balls about the origin are isoperimetric:
Log-Convex Density Conjecture 1.1. [Ro, Conjecture 3.12] Consider Rn with a smooth
radial density. If the log of the density is convex, then balls about the origin provide
isoperimetric regions of any given volume.
The borderline case of density er, whose log is linear and hence just barely convex, is a
pivotal case and the subject of the following study. Although we are not able to prove that
the isoperimetric solution is a circle about the origin, we obtain the following partial result:
Theorem 3.19. In the plane with density er, an isoperimetric region is convex and contains
the origin in its interior.
The proof uses symmetrization, the Four-Vertex Theorem, and the equation for geodesic
curvature.
We conclude Section 3 with some numerical analysis. Relative to a given density e
there is a generalization of curvature . Just as classically an isoperimetric curve (circle)
has constant curvature, in the presence of a density an isoperimetric curve has constant
generalized curvature. Corollary 3.23 provides a simple di erential equation for curves r( )
of constant generalized curvature :
dr
d
= r
s
r
1
(r
e rC)2
1
for some constant C (
1
)e1=(
1
)= . Given , C, and initial position r(0), there are
two solutions to the di erential equation: a nondegenerate solution that looks like one of the
curves in Figures 1 and 2, and a degenerate solution that corresponds to a circle about the
= 1:2,
origin that does not have constant generalized curvature (Figure 3). When C reaches its
minimum, the nondegenerate solution coincides with the degenerate solution and gives the
unique constant generalized curvature curve, which is precisely a circle about the origin. To
show that discs about the origin are isoperimetric, it su ces to show that a nondegenerate
solution is not symmetric with respect to the line through the origin and a maximum point
of r when C is strictly greater than its minimum, and therefore cannot be isoperimetric
(Proposition 2.5). Thus the only remaining candidate for an isoperimetric curve is a circle
about the origin, as desired.
The structure of the paper is as follows:
Section 2 provides basic de nitions and known results. Section 3 begins (...truncated)