ETD

Archivio digitale delle tesi discusse presso l'Università di Pisa

Tesi etd-05242017-123512


Tipo di tesi
Tesi di laurea magistrale
Autore
DI MATTEO, GIANMICHELE
URN
etd-05242017-123512
Titolo
Double bubble with small volume in compact manifolds
Dipartimento
MATEMATICA
Corso di studi
MATEMATICA
Relatori
relatore Prof. Malchiodi, Andrea
Parole chiave
  • small volume
  • double bubble
Data inizio appello
09/06/2017
Consultabilità
Completa
Riassunto
In this work we would like to study the existence of small constant mean
curvature double bubbles in an ambient compact manifold. This work is mainly
based on the article [2], in which Pacard and Xu show the existence of small
constant mean curvature spheres, which are perturbations of small geodesic
spheres.
In the introduction we give our motivation to study this problem, with em-
phasis to the isoperimetric problem for small volume and to the double bubble
conjecture.
Afterwards, we recall shortly some basic facts about the elliptic regular-
ity and about the immersed riemannian geometry, and we study briefly some
geometric properties of the standard double bubble, following [1].
In the third chapter we present deeply the article of Pacard and Xu [2].
Then we present our perturbation argument, and we get an asymptotic ex-
pansion for the mean curvature of the perturbed bubble in function of the per-
turbation. Since one has to allow the presence of a tangential component in the
perturbation, we have to adapt all the expansion for the geometric quantities of
a perturbed sphere obtained in [2].
Subsequently, we provide a characterization of the kernel of the Jacobi oper-
ator associated to a standard double bubble, first in the two dimensional case,
then in the symmetric case in any dimension. This requires to deal with some
special functions, with the singular set of the standard double bubbles, and to
use strongly the geometric properties of them. It will turn out that this ker-
nel consists of normal perturbations generated by infinitesimal traslations and
rotations.
Finally, we provide proofs of some results used in the course of the work in
the Appendix.

References
[1] Hutchings M., Morgan F., Ritor ́ M., Ros A., Proof of the Double Bubble
e
Conjecture, Annals of Mathematics, Vol. 155 No. 2, 2002
[2] Pacard F., Xu X., Constant mean curvature spheres in Riemannian mani-
folds, manuscripta math. 128, 2009
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