423 research outputs found
Proof of the Log-Convex Density Conjecture
We completely characterize isoperimetric regions in R^n with density e^h,
where h is convex, smooth, and radially symmetric. In particular, balls around
the origin constitute isoperimetric regions of any given volume, proving the
Log-Convex Density Conjecture due to Kenneth Brakke.Comment: 40 pages, 7 figure
Constructing monotone homotopies and sweepouts
This article investigates when homotopies can be converted to monotone
homotopies without increasing the lengths of curves. A monotone homotopy is one
which consists of curves which are simple or constant, and in which curves are
pairwise disjoint. We show that, if the boundary of a Riemannian disc can be
contracted through curves of length less than , then it can also be
contracted monotonously through curves of length less than . This proves a
conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian
-sphere through curves of length less than can be replaced with a
monotone sweepout through curves of length less than . Applications of these
results are also discussed.Comment: 16 pages, 6 figure
Optimal sweepouts of a Riemannian 2-sphere
Given a sweepout of a Riemannian 2-sphere which is composed of curves of
length less than L, we construct a second sweepout composed of curves of length
less than L which are either constant curves or simple curves.
This result, and the methods used to prove it, have several consequences; we
answer a question of M. Freedman concerning the existence of min-max embedded
geodesics, we partially answer a question due to N. Hingston and H.-B.
Rademacher, and we also extend the results of [CL] concerning converting
homotopies to isotopies in an effective way.Comment: 20 pages, 8 figures; Modified statements and proofs of theorems to
reflect changes in "Contracting loops on a Riemmanian 2-surface" by the first
author and R. Rotma
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