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Preprint
Harmonic maps from post-critically finite fractals to the circle
arXiv.org
23 Jul 2024
Abstract
Every continuous map between two compact Riemannian manifolds is homotopic to
a harmonic map (HM). We show that a similar situation holds for continuous maps
between a post-critically finite (p.c.f.) fractal and a circle. Specifically,
we provide a geometric proof of Strichartz's theorem stating that for a given
degree and appropriate boundary conditions there is a unique HM from the
Sierpinski Gasket (SG) to a circle. Furthermore, we extend this result to HMs
on p.c.f. fractals.
Our method uses covering spaces for the SG, which are constructed separately
for HMs of a given degree, thus capturing the topology intrinsic to each
homotopy class. After lifting continuous functions on the SG with values in the
unit circle to continuous real-valued functions on the covering space, we use
the harmonic extension algorithm to obtain a harmonic function on the covering
space. The desired HM is obtained by restricting the domain of the resultant
harmonic function to the fundamental domain and projecting the range to the
circle.
We show that with suitable modifications the method applies to p.c.f.
fractals, a large class of self-similar domains. We illustrate our method of
constructing the HMs using numerical examples of HMs from the SG to the circle
and discuss the construction of the covering spaces for several representative
p.c.f. fractals, including the 3-level SG, the hexagasket, and the pentagasket.
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Details
- Title
- Harmonic maps from post-critically finite fractals to the circle
- Creators
- Georgi S MedvedevMatthew S Mizuhara
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021894520104721