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# Unicidad del cilindro grim reaper en $R^n$

## Eddy-Gledson Souza Gama Universidade Federal do Ceará

Obtenemos un teorema de caracterización del cilindro construido sobre una curva grim reaper como el único solitón de traslación del flujo por la curvatura media en el espacio euclídeo $R^n$ que es asintótico a dos hiperplanos fuera de un cilindro. Esto generaliza a dimensión arbitraria un resultado previo de Martin-Perez-Savas-Smoczyk para solitones de $R^3$.

Seminario 2ª Planta, IEMATH

# Asymptotic Dirichlet problems for the mean curvature operator

## Esko Heinonem Universidad de Helsinki

In $R^n$ ($n$ at most 7) the famous Bernstein's theorem states that every entire solution to the minimal graph equation must be affine. Moreover, entire positive solutions in $R^n$ are constant in every dimension by a result due to Bombieri, De Giorgi and Miranda. If the underlying space is changed from $R^n$ to a negatively curved Riemannian manifold, the situation is completely different. Namely, if the sectional curvature of $M$ satisfies suitable bounds, then $M$ possess a wealth of solutions.
One way to study the existence of entire, continuous, bounded and non-constant solutions, is to solve the asymptotic Dirichlet problem on Cartan-Hadamard manifolds. In this talk I will discuss about recent existence results for minimal graphs and f-minimal graphs. The talk is based on joint works with Jean-Baptiste Casteras and Ilkka Holopainen.

Seminario 1ª Planta, IEMATH

# On the topology of surfaces with the simple lift property

## Francesca Tripaldi

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface $\Sigma\subset\Omega$ with the simple lift property must have genus zero, if $\Omega$ is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that $\Omega$ cannot contain closed minimal surfaces. In this work, I generalize this result by taking an arbitrary orientable three-manifold $\Omega$ and proving that one is able to restrict the topology of an arbitrary surface $\Sigma\subset\Omega$ with the simple lift property. Among other things, I prove that the only possible compact surfaces with the simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where $\Sigma\subset\Omega$ is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.

Seminario 1ª planta, IEmath

# Eddy-Gledson Souza Gama

Despacho: IEMATH, B-8C

# Gabriel Ruiz-Hernández

## Instituto de Matemáticas, Universidad Nacional Autónoma de México

Despacho: D4, IEMath

# Marcos Paulo Tassi

## Universidade Federal de São Carlos

Despacho: IEMath B8-B

# Álvaro Pámpano

## UPV/EHU

Despacho: 5, segunda planta

# Rukmini Dey

## International Centre for Theoretical Sciences

Despacho: 5, segunda planta

# Jimmy Lamboley

## Université Paris 6

Despacho: IEMath, D11

# Franc Forstneric

## Univerza v Ljubljani

Despacho: D2, IEMath

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