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# On the area of minimal surfaces in a slab

## Benoît Daniel Université Henri Poincaré - Nancy 1

Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S) >= Area(W), provided the intersections of S with horizontal planes have the same orientation.This is joint work with Jaigyoung Choe.

Seminario, 1ª Planta, IEMath

# Existence and uniqueness of constant mean curvature spheres in $Sol(3)$

## Benoît Daniel Université Henri Poincaré - Nancy 1

We study the classification of immersed constant mean curvature spheres in the homogeneous 3-manifold $Sol(3)$, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every $H>\sqrt{1/3}$, there exists a unique (up to translations) immersed CMC $H$ sphere $S_H$ in $Sol(3)$. Moreover, this sphere $S_H$ is embedded, and is therefore the unique compact embedded CMC $H$ surface in $Sol(3)$.
The same results are obtained for all real numbers $H$ such that there exists a solution to the isoperimetric problem with mean curvature $H$.
Joint work with Pablo Mira.

# Benoît Daniel

## Université Henri Poincaré - Nancy 1

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