Grupo de investigación de la Junta de Andalucía (FQM 325).
Dpto de Geometría y Topología. Universidad de Granada
Research group of Junta de Andalucía (FQM 325).
Deparment of Geometry and Topology. Granada University
The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were $2$-dimensional surfaces in the $3$-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by Federer and Fleming in the sixties, following pioneering ideas of De Rham. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).
In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. In this talk I will try to give a feeling for the difficulties in the proof and how they can be overcome.
Teichmüller harmonic map flow is designed to evolve maps from a closed surface to a general target manifold towards (branched) minimal immersions. Defined as gradient flow of energy considered as a function of both a map and a metric on the domain, the flow enjoys the strong regularity properties known from harmonic map heat flow for as long as there is no degeneration in Teichmüller space but at the same time tries to make the map not only harmonic but also conformal and thus minimal. In this talk we will discuss the definition and properties of the flow and show in particular that global solutions, guaranteed to exist in certain settings, change (or decompose) arbitrary initial data into (a union of) branched minimal immersions, possibly parametrized over surfaces of lower genus. This is joint work with Peter Topping.
Almost flat manifolds are the solutions of bounded size perturbations of the equation $\mathrm{Sec} = 0$ ($\mathrm{Sec}$ is the sectional curvature). In a celebrated theorem, Gromov proved that the presence of an almost flat metric implies a precise topological description of the underlying manifold.
During this talk we will explain how, under lower sectional curvature bounds, to impose an $L_1$-pinching condition on the curvature is surprisingly rigid, leading indeed to the same conclusion as in Gromov's theorem under more relaxed curvature conditions (in particular, so weak that we are not allowed to use Ricci flow in the proof). We will describe which alternative techniques lead us to a successful proof, ans this will be sketched in detail. This is a joint work with B. Wilking.
We prove that, given $S$ an open oriented surface, then there exists a complete, proper, area minimizing embedding $f: S \to \mathbb{H}^3$. The main tool in the proof of the above result is a sort of bridge principle at infinity for properly embedded area minimizing surfaces in hyperbolic three space. This is a joint work with Brian White.
How the shape of the ends of a minimal surface determines the surface itself? We discuss the problem for minimal surfaces with two ends in $\mathbb{H}^2\times\mathbb{R}$. Namely, we prove a Schoen type theorem for such surfaces.
We study regular deformations of constant mean curvature (CMC) $1/ 2$ surfaces with vertical ends in $\mathbb{H}^2 \times \mathbb{R}$ using a suitable extension of the mean curvature operator at infinity. The two main results are the following :
We present a deformation of surfaces from a product space $M_1\times R$ into another product space $M_2\times \mathbb{R}$ such that the relation of the principal curvatures of the deformed surfaces can be controlled in terms of the curvatures of $M_1$ and $M_2$. Thus, starting from a known example, we obtain subsolutions for the existence or barriers for the non existence of surfaces with fixed mean curvature, extrinsic curvature or Gaussian curvature in $M\times \mathbb{R}$.
This is a joint work with Bruno Colbois. Consider a compact Riemannian manifold $M$ with an involutive isometry $\gamma$, and assume that the distance of any point to its image under $\gamma$ is bounded below by a positive constant $\beta$ (the smallest displacement). We observe that this simple geometric situation has a strong consequences on the spectrum of a large class of $\gamma$-invariant operators $D$ (including the Schrödinger operator acting on functions and the Hodge Laplacian acting on forms): roughly speaking, the gap $\lambda_2(D)-\lambda_1(D)$ between the first and the second eigenvalue of $D$ is uniformly bounded above by a constant depending only on the displacement $\beta$ (in particular, not depending on $D$).
The sequence of eigenvalues of the Dirichlet Laplacian on a
bounded Euclidean domain satisfies several restrictive conditions
such as : Faber-Krahn isoperimetric inequality, that is the
principal eigenvalue is bounded above in terms of the volume of
the domain, Payne-Pólya-Weinberger type universal inequalities,
that is the $k$-th eigenvalue is controlled in terms of the $k-1$
previous ones, etc.
The situation changes completely as soon as Euclidean domains are
replaced by compact manifolds. For example, according to results
by Colin de Verdière and Lohkamp, given any compact manifold $M$
of dimension $n\ge 3$, it is possible to prescribe arbitrarily and
simultaneously, through the choice of a suitable Riemannian metric
on $M$, a finite part of the spectrum of the Laplacian, the volume
and the integral of the scalar curvature. Hence, Faber-Krahn and
Payne-Pólya-Weinberger inequalities have no analogue in this
context.
In this talk, we will discuss the effect of the geometry on the
eigenvalues. We will try to understand what kind of geometric
situations lead to large eigenvalues for the Laplacian on
manifolds of fixed volume, and what does such a Riemannian
manifold look like once realized as a submanifold of a Euclidean
space.
On the other hand, we show that when the Laplacian is penalized by
the squared norm of the mean curvature, then we obtain a
Schrödinger type operator whose spectral behavior is similar to
that of the Dirichlet Laplacian on Euclidean domains.
To formulate Sobolev inequalities one needs to answer questions like: what is the role of dimension? What norms are appropriate to measure the integrability gains? Just to name a few… For example, in contrast to the Euclidean case, the integrability gains in Gaussian measure are logarithmic but dimension free (log Sobolev inequalities). So it is easy to understand the difficulties to derive a general theory. I will discuss some new methods to prove general Sobolev inequalities that unify the Euclidean and the Gaussian cases, as well as several important model manifolds.
Our aim is to give a simple characterization of the class of Lorentzian
manifolds which can be isometrically embedded in Lorentz-Minkowski $L^N$ for
some large $N$ (in the spirit of classical Nash theorem) and, then, to show
that this class includes the most relevant type of relativistic spacetimes,
i.e., the globally hyperbolic ones. This last result was claimed by CJS
Clarke (1970), but his proof was affected by the so-called folk
problems of smoothability in Lorentzian Geometry. These problems will be
specially discussed.
The talk is based in a joint work with O. Müller (arXiv:0812.4439v4)
We give an introduction to recent developments in the geometry of compact manifolds with exceptional holonomy, focusing on recent work with Corti, Nordstrom and Pacini; we prove the existence of many compact 7-manifolds with holonomy G2 that contain rigid associative submanifolds. The main ingredients in the proof are: an appropriate noncompact version of the Calabi conjecture, gluing methods and a certain class of complex projective 3-folds (weak Fano 3-folds).
A surface $S$ in a complete $3$-manifold $M$ is area-minimizing mod $2$ if it has least area among all surfaces, orientable or nonorientable in the same homology class. These surfaces present a rich and interesting geometry, even in flat or positively curved $3$-manifolds. For instance, if $M$ is flat, Fischer-Colbrie and Schoen, Do Carmo and Peng, and Pogorelov proved that complete two-sided stable minimal surfaces are flat, but Ross proved that some nonorientable quotient of the classic Schwarz P and D surfaces are estable, and we proved that an area minimizing surface in $\mathbb{R}^2\times S^1$ is either planar or a quotient of the Helicoid. We will review some results about this problem and we will prove that area minimizing surfaces in flat $3$-tori are planar.
On the one hand, given an open Riemann surface $\mathcal{N}$ and a real number $\theta \in ]0,\pi/4[$, we construct a conformal minimal immersion $X = (X_1, X_2, X_3): \mathcal{N} \to \mathbb{R}^3$ such that $X_3 + \tan \theta |X_1|: \mathcal{N} \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with arbitrarily prescribed flux map. This construction if related with a problem posed by Shoen and Yau. On the other hand, we produce properly immersed hyperbolic minimal surfaces with empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph. This construction can be linked to a conjecture by Meeks. The main tool used in the construction of the above examples is an approximation theorem by minimal surfaces. We will also remark some other applications of this result.
We prove the existence of constant curvature hypersurfaces in Hadamard manifolds subject satisfying topological/boundary conditions.
We will show that any (non-constant) harmonic map on an arbitrary open Riemannian $N$ surface can be realized as the third coordinate of a complete minimal immersion of $N$ in $\mathbb{R}^3$. As a consequence we will prove that any open Riemann surface admits a complete conformal minimal immersion whose Gauss map misses two points. This is a joint work with F.J. López and A. Alarcón.
In this talk I will discuss recent results on the geometry of constant mean curvature ($H\neq 0$) surfaces embedded in $\mathbb{R}^3$. Among other things I will prove a radius and curvature estimates for constant mean curvature disks embedded in $\mathbb{R}^3$. It follows from the radius estimate that the only complete constant mean curvature disk embedded in $\mathbb{R}^3$ is the round sphere. This is joint work with Bill Meeks.
The classical Faber-Krahn inequality compares the first Dirichlet eigenvalue of the Laplacian for a bounded domain to that of a ball.
I will discuss similar bounds for domains contained in wedges, both in Euclidean space and the sphere. If time allows, I will also discuss an application to Brownian pursuit. This is joint work with Andrejs Treibergs.
We first consider the problem of finding all the entire area-stationary graphs in the sub-Riemannian Heisenberg group $H_1$.This question has been well studied for $t$-graphs and for intrinsicgraphs. In particular, examples of unstable entire area-stationary graphs may be given in $H_1$.
After discussing some stability results for complete graphs we state a joint result with A. Hurtado and M. Ritoré, where we prove that a $C^2$ complete stable area-stationary surface in $H_1$ must be a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.
In this talk, we prove that, in a stable cmc surface in $\mathbb{R}^3$, the intrinsic distance from a point to the boundary is less than $\pi/(2H)$. This estimate is sharp and can be extended to $\mathbb{H}^3$ and $\mathbb{S}^3$.
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 $H>1/\sqrt{3}$, 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.
In this paper we deal with the uniqueness of the Helicoid and Enneper's surface as maximal immersions in the Lorentz-Minkowski space $\mathbb{R}^3_1.$ In both cases the surface contains a proper lightlike arc where the immersion folds back, and so the image via the immersion is a (double) surface without selfintersections with lightlike boundary of mirror symmetry.
El autor, en colaboración con Laurent Mazet, presenta la construcción de una superficie minimal embebida en el producto llano $\mathbb{R}^2\times \mathbb{S}^1$ que es casi-periódica pero no periódica.