| Résume | In this talk, we will explore several results regarding constructions of minimal
and CMC annuli in the unit ball of R3.
First, we will present a countable number of real analytic, 1-parameter families
of embedded free boundary CMC annuli in the unit ball that are not rotational.
These examples provide the first non-rotational annular solutions to the
partitioning problem in the ball, answering negatively a question posed by
Wente in 1995.
Next, we will prove a conjecture by Fernández, Hauswirth, and Mira [Arch.
Rat. Mech. Anal. 247 (2023)] concerning the existence of non-rotational
embedded capillary minimal annuli in the unit ball. While these annuli were
previously constructed as local bifurcations of catenoids, we explore the global
behavior of this family. Specifically, we show that for every n ≥ 2, there exists
a 1-parameter family of embedded, non-rotational capillary minimal annuli,
which can be viewed as bifurcations either from catenoids or from a
configuration of n vertical disks.
As a corollary, we deduce the existence of non-rotational embedded capillary
minimal annuli that intersect the unit ball with angles arbitrarily close to, but not
equal to, either π/2 or 0.
This is a joint work with I. Fernández and P. Mira.
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