Master Degree Thesis
Double bubble with small volume
in compact manifolds
Candidate: Advisor:
Gianmichele Di Matteo
Prof. Andrea Malchiodi
Academic Year 2016/2017
Abstract
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´e M., Ros A., Proof of the Double Bubble 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
