B. Daniel Titulo: On the area of minimal surfaces in a slab Abstract

B. Daniel
Titulo: On the area of minimal surfaces in a slab
Abstract: 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.
Martin Chuaqui Farr`
u
Titulo: Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts
Abstract: The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the Weierstrass-Enneper lift of a harmonic mapping of the disk to a minimal
surface, producing homeomorphic and quasiconformal extensions to space. The extension is defined through the family of best M¨obius approximations to the lift applied to
a bundle of Euclidean circles orthogonal to the disk. Extension of the planar harmonic
map is also obtained subject to additional assumptions on the dilatation. The hypotheses
involve bounds on a generalized Schwarzian derivative for harmonic mappings in terms
of the hyperbolic metric of the disk and the Gaussian curvature of the minimal surface.
Hyperbolic convexity plays a crucial role.
M. Del Pino
Titulo: Bubbling in the critical heat equation: the role of Green’s function
Abstract: We investigate the pointwise, infinite-time bubbling phenomenon for positive
solutions of the semilinear heat equation at the critical exponent in a bounded domain.
We build an invariant manifold for the flow which ends at k bubbling points of the domain
for any given k. The delicate role of dimension is described. This is joint work with C.
Cortzar and M. Musso.
J. M. Espinar
Titulo: Geometry and topology of f -extremal domains in Hadamard Manifold
Abstract: In this talk investigate the geometry and topology of f -extremal domains
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in a manifold with negative sectional curvature. A f -extremal domain is a domain that
supports a positive solution to the overdetermined elliptic problem

∆u + f (u) = 0



u>0
u=0



h∇u, ~v iM = α
in
Ω,
in
Ω,
on ∂Ω,
on ∂Ω,
(1)
where Ω is an open connected domain in a complete Hadamard n-manifold (M, g) with
boundary ∂Ω of class C 2 , f is a given Lipschitz function, h·, ·iM is the inner product on
M induced by the metric g, and α, ~v the unit outward normal vector of the boundary ∂Ω
and α a non-positive constant.
We will show narrow properties of such domains in a Hadamard manifolds and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of
its boundary at infinity. Later, we focus on f −extremal domains in the Hyperbolic Space
Hn . Symmetry and boundedness properties will be shown. Hence, we are able to prove
the Berestycki-Caffarelli-Nirenberg Conjecture in H2 . Specifically:
Theorem: Let Ω ⊂ H2 a domain with properly embbeded C 2 boundary such
that H2 \ Ω is connected. If there exists a (strictly) positive function u ∈ C 2 (Ω)
that solves the equation

in
Ω,

 ∆u + f (u) = 0

u>0
in
Ω,
u=0
on ∂Ω,



h∇u, ~v iH2 = α
on ∂Ω,
where f : (0, +∞) → R if Lipschitz, then Ω is either a geodesic ball or a
horoball.
Furthermore, if f : (0, +∞) → R satisfies f (t) ≥ λ t for some constant λ
2
, then Ω must be a geodesic ball.
satisfying λ > − (n−1)
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If time permits, we will generalize the above results to more general OEPs.
D. Henao
Titulo:Phase-field regularization of fracture surfaces in elasticity
Abstract: An overview will be given of the variational modelling of fracture in elasticity
theory, where determining where will a body break first and how will cracks propagate are
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cast as free-discontinuity problems in SBV. In order to compute the discontinuity surfaces
associated to fracture, the energy functional is regularized using Ambrosio and Tortorelli’s
approach for image segmentation in computer vision (in turn inspired in Modica-Mortola’s
approximation of the perimeter functional). Special emphasis will be given to the difficulties that appear when the phenomenon of microscopic void formation (known in the
literature as cavitation) is taken into account in the model. This is joint work with Carlos
Mora-Corral (Madrid - UAM) and X. Xu (Beijing - CAS).
A. Jimenez
Title: Isolated singularities of PDEs and applications.
Abstract: In this talk we will describe the asymptotic behavior of singular solutions of
the general Monge-Ampre equation det(D2 z + A(x, y, z, Dz)) = ϕ(x, y, z, Dz) > 0 under
certain hypothe sis. We also show some classification results that relate singular solutions
of the Monge-Ampre equation with regular, real analytic strictly convex Jordan curves
in R2 . Finally we show some geometric applications in the theory of surfaces of positive
extrinsic curvature. Joint work with Jos A. Glvez and Pablo Mira.
M. Kowalczyk
Title: Delaunay ends solutions to the Cahn-Hilliard equation
Abstract:In this talk I will describe a construction of new solutions to the Cahn-Hilliard
equation in the whole space. The zero level sets of these solutions are close to embedded,
non degenerate and non compact CMC surfaces, which are known to have Delaunay ends
at infinity.
J. Lira
Title: Applications of the maximum principle to minimal graphs
Abstract: We survey some recent results about uniqueness and non-existence of entire
minimal graphs in Riemannian manifolds endowed with a Killing vector field. These results
are based on gradient estimates and rely on a variant of the maximum principle obtained
in joint work with L. Alias and M. Rigoli.
Monica Musso
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Title: Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem
Abstract: We provide the first example of a sequence of nondegenerate, in the sense of
Duyckaerts-Kenig-Merle [?], nodal nonradial solutions to the critical Yamabe problem
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−∆Q = |Q| n−2 Q, Q ∈ D1,2 (Rn ).
J. Perez:
Title: Existence of CMC foliations in compact n-manifolds
Abstract: We will show that every closed, smooth n-manifold admits a Riemannian
metric together with a smooth, transversely oriented CMC foliation if and only if its
Euler characteristic is zero. This is in contrast with the behaviour of the similar problem
when we change ’CMC’ by ’minimal’: for instance, the 3-sphere does not admit minimal
foliations (for any metric), but it does admit CMC foliations. Joint work with Bill Meeks.
A. Malchiodi
Title: Embedded Willmore tori in three-manifolds with small area constraint
Abstract. While there are lots of contributions on Willmore surfaces in the threedimensional Euclidean space, the literature on curved manifolds is still relatively limited.
One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations. We construct embedded Willmore tori in manifolds with a small area
constraint by analysing how the Willmore energy under the action of the Mbius group is
affected by the curvature of the ambient manifold. The loss of compactness is then taken
care of using minimisation arguments or Morse theory.
L. Mazet
Title : Minimal hypersurfaces asymptotic to Simons cones.
Abstract : In this talk, I will study minimal hypersurfaces that are asymptotic to
Simons cone (i.e. minimal cone over product of spheres). I will explain that there are
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at most two such hypersurfaces up to similarity for any fixed Simons cone. This is a
generalization of a result of Simon and Solomon.
A. Quaas
Title : Liouville type theorems for nonlinear elliptic equations and systems involving
fractional Laplacian in the half-space
Abstract : We study the nonexistence of solutions for fractional elliptic problems via
a monotonicity result which obtained by the method of moving planes with an improved
Aleksandrov- Bakelman -Pucci type estimate for the fractional Laplacian in unbounded
domain.
R. Rodiac
Title : Harmonic maps with prescribed degrees on the boundary of an annulus and
bifurcation of catenoid.
Abstract: We will study critical points of the Dirichlet energy among maps from an
annulus to the disk with prescribed modulus |u| = 1 and with prescribed degrees on the
boundaries of the annulus. This is a problem with lack of compactness. We can prove
existence and non-existence results in some cases using a strong link with minimal surfaces
in R3 . In particular some solutions are obtained from bifurcation of a p covering of portion
of catenoid.
H. Rosenberg
Title : Minimal surfaces of least area and applications
M. Saez
Title : Mean curvature flow without singularities
Abstract: We study graphical mean curvature flow of complete solutions defined on
subsets of Euclidean space. We obtain smooth long time existence. The projections of the
evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly
flow through singularities by studying graphical mean curvature flow in one additional
dimension.
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T. Zolotareva
Title : Free Boundary Minimal Surfaces in the Unit 3-Ball
Abstract: In a recent paper A. Fraser and R. Schoen have proved the existence of free
boundary minimal surfaces Σn in B 3 which have genus 0 and n boundary components,
for all n ≥ 3. For large n, we give an independent construction of Σn and prove the
˜ n in B 3 which have genus 1 and n boundary
existence of free boundary minimal surfaces Σ
components. As n tends to infinity, the sequence Σn converges to a double copy of the
˜ n converges
unit horizontal (open) disk, uniformly on compacts of B 3 while the sequence Σ
to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of
B 3 − {0}.
D. Zhou
Title : Spectrum of the drifted Laplacian and applications
Abstract: In this talk, I will discuss some recent results on the spectrum of the Laplacian and drifted Laplacian on complete Riemannian manifolds.In particular,
I will present a generalization of Lichnerowicz-Obata theorem to to case when (M n , g, e−f dv)
´
is a complete smooth metric measure space with positive Bakry-Emery
Ricci curvature
tensor. We also generalize the Choi-Wang’s first eigenvalue estimate for minimal hypersurfaces to corresponding minimal hypersurface with respect to weighted measure. These
results can be naturally applied to study self-shrinkers for MCF and gradient shrinking
soliton for Ricci flow.
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