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Conferencias impartidas por William H. Meeks III

CMC foliations of closed manifolds

University of Massachusetts, Amherst

We prove that a neccesary and sufficient condition for a closed, smooth $n$-manifold $X$ to admit a Riemannian metric together with a codimension-one, smooth, transversely oriented foliation with leaves of constant mean curvature, is that the Euler characteristic of $X$ is zero, Furthermore, this CMC foliation of $X$ can be chosen so that the constant values of the mean curvatures of its leaves change sign. Joint work with Joaquin Perez.

Seminario 1ª Planta, IEMath-Gr

The geometry of isoperimetric domains of large volume in simply connected homogeneous 3-manifolds

University of Massachusetts, Amherst

Consider a simply connected non-compact homogeneous 3-manifold $X$. For each $V\gt 0$, there exists a smooth compact domain $D(V)$ with volume $V$ whose boundary surface area is minimizing and we call such a boundary area minimizing domain an isoperimetric domain of $X$. One would like to classify all such isoperimetric domains in $X$ for a fixed $V$ up to ambient isometry and to understand the properties of such domains, such as the slope of the "isoperimitric profile of $X$" as volume $V$ tends to infinity. In a recent paper with Mira, Pérez and Ros, the speaker has obtained an number of such geometric results on these questions. We have been able to show that the Cheeger constant $\mathrm{Ch}(X)$ of $X$ is equal to twice the critical mean curvature $H(X)$ of the $X$. As a consequence of our proof of this relation, for any sequence of isopermetric domains in $X$ with volumes tending to infinity, the constant mean curvatures of their boundary surfaces tend to $H(X)$ and their radii tend to infinity. The difficult case in this study is when $X$ is isometric to universal cover of the Lie group $\mathrm{Sl}(2,\mathbb{R})$ equipped with a left invariant metric with a 3-dimensional isometry group. These results together with our recent classification of constant mean curvature spheres in $X$ leads to an isoperimetric inequality for smooth compact immersed surfaces in $X$ with 1 boundary component and with absolute mean curvature function less than or equal to $\mathrm{Ch}(X)/2$.

Open problems in the theory of constant mean curvature surfaces in 3-dimensional homogeneous manifolds (II)

University of Massachusetts, Amherst

Mini curso dentro del doctorado en Matemáticas impartido por el profesor William H. Meeks III de la universidad de Amherst

Open problems in the theory of constant mean curvature surfaces in 3-dimensional homogeneous manifolds (I)

University of Massachusetts, Amherst

Mini curso dentro del doctorado en Matemáticas impartido por el profesor William H. Meeks III de la universidad de Amherst

A survey of results in the theory of constant mean curvature surfaces in 3-dimensional homogeneous manifolds

University of Massachusetts, Amherst

Plateau problem in three-dimensional metric Lie groups

University of Massachusetts, Amherst

Seminario de Matemáticas. 1ª Planta, sección de matemáticas

Constant mean curvature surfaces in homogeneous 3-manifold III

University of Massachusetts, Amherst

I will present some of the basic results on the geometry of constant mean curvature H≥0 surfaces M in a complete homogeneous 3-manifold X; such an M will be called an H-surface. We will call a foliation F of X a CMC foliation if all of the leaves of F are H-surfaces with H possibly varying. The key results in the mini-course will include:

  • General theory of H-surfaces M in Riemannian 3-manifolds with an emphasis on the case M is complete and embedded, including work described in papers of Colding-Minicozzi and Meeks-Perez-Ros and Meeks-Tinaglia.
  • Curvature estimates for CMC foliations of X. Based on joint work with Perez and Ros.
  • Curvature estimates for H-disks with H>0 in X. Based on joint work with Tinaglia.
  • Uniqueness results for H-spheres in X, which generalize the classical result of Hopf that for each H>0, there is a unique H-sphere in R3. Based on joint work with Mira, Perez and Ros.

Constant mean curvature surfaces in homogeneous 3-manifold II

University of Massachusetts, Amherst

I will present some of the basic results on the geometry of constant mean curvature H≥0 surfaces M in a complete homogeneous 3-manifold X; such an M will be called an H-surface. We will call a foliation F of X a CMC foliation if all of the leaves of F are H-surfaces with H possibly varying. The key results in the mini-course will include:

  • General theory of H-surfaces M in Riemannian 3-manifolds with an emphasis on the case M is complete and embedded, including work described in papers of Colding-Minicozzi and Meeks-Perez-Ros and Meeks-Tinaglia.
  • Curvature estimates for CMC foliations of X. Based on joint work with Perez and Ros.
  • Curvature estimates for H-disks with H>0 in X. Based on joint work with Tinaglia.
  • Uniqueness results for H-spheres in X, which generalize the classical result of Hopf that for each H>0, there is a unique H-sphere in R3. Based on joint work with Mira, Perez and Ros.

Constant mean curvature surfaces in homogeneous 3-manifold Is

University of Massachusetts, Amherst

I will present some of the basic results on the geometry of constant mean curvature H≥0 surfaces M in a complete homogeneous 3-manifold X; such an M will be called an H-surface. We will call a foliation F of X a CMC foliation if all of the leaves of F are H-surfaces with H possibly varying. The key results in the mini-course will include:

  • General theory of H-surfaces M in Riemannian 3-manifolds with an emphasis on the case M is complete and embedded, including work described in papers of Colding-Minicozzi and Meeks-Perez-Ros and Meeks-Tinaglia.
  • Curvature estimates for CMC foliations of X. Based on joint work with Perez and Ros.
  • Curvature estimates for H-disks with H>0 in X. Based on joint work with Tinaglia.
  • Uniqueness results for H-spheres in X, which generalize the classical result of Hopf that for each H>0, there is a unique H-sphere in R3. Based on joint work with Mira, Perez and Ros.

H-surfaces in homogeneous 3-manifolds and H-spheres in Sol3.

University of Massachusetts, Amherst

I will go over recent work with Tinaglia on curvature estimates for H-surfaces (constant mean curvature surfaces) in homogeneous 3-manifolds and the classification of CMC-foliations of $R^3$. Next I will go over my recent proof that for each $H>0$, there exists a unique sphere $S_H$ in Sol3 with constant mean curvature H, which is based on previous work of Daniel and Mira.

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

The local and global geometry of embedded minimal and CMC surfaces in 3-manifolds.

University of Massachusetts, Amherst

I will give a survey of some of the exciting progress in the classical theory of surfaces M in 3-manifolds with constant mean curvature H greater than or equal to zero; we call such a surface an H-surface. The talk will cover the following topics: The classification of properly embedded genus 0 minimal surfaces in $\mathbb{R}^3$. (joint with Perez and Ros) The theorem that for any $c>0$, there exists a constant $K=K(c)$ such that for $H>c$, and any compact embedded H-disk D in $\mathbb{R}^3$ (joint with Tinaglia): the radius of D is less than K. the norm of the second fundamental form of D is less than K for any points of D of intrinsic distance at least c from the boundary of D is less than K. item 2.2 works for any compact embedded H-disk ($H>c$) in any complete homogeneous 3-manifold with absolute sectional curvature less than 1 for the same K. For $c>0$, there exists a constant K such that for any complete embedded H-surface M with injectivity radius greater than $c>0$ in a Riemannian 3-manifold with absolute sectional curvature <1 has the norm of its second fundamental form less than K. (joint with Tinaglia) Complete embedded finite topology H-surfaces in $\mathbb{R}^3$ have positive injectivity radius and are properly embedded with bounded curvature. Complete embedded simply connected H-surfaces in $\mathbb{R}^3$ are spheres, planes and helicoids; complete embedded H-annuli are catenoids and Delaunay surfaces. Complete embedded simply-connected and annular H-surfaces in $H^3$ with H less than or equal to 1 are spheres and horospheres, catenoids and Hsiang surfaces of revolution; the key fact here is that complete + connected implies proper. Classification of the conformal structure and asymptotic behavior of complete injective H-annuli $f: S^1 \times [0,1)\rightarrow \mathbb{R}^3$; there is a 2-parameter family of different structures for $H=0$. (joint with Perez when $H=0$) Solution of the classical proper Calabi-Yau problem for arbitrary topology (even with disjoint limit sets for distinct ends!!). (joint with Ferrer and Martin)

Q32. 3ª Planta, sección de Química.

Curvature estimates for CMC foliations

University of Massachusetts, Amherst

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

CMC dynamics theorem in $\mathbb{R}^3$

University of Massachusetts, Amherst

Motivated by the dynamics theorem for minimal surfaces by Meeks, Perez and Ros, then Meeks and Tinaglia proved a related result for CMC surfaces in $\mathbb{R}^3$. This theorem has important applications to the study of complete embedded CMC surfaces of finite topology in complete locally homogeneous three-manifolds. I will only be discussing the statements and proofs of the CMC dynamics theorem in $\mathbb{R}^3$, rather than applications.

The geometry of a CMC or minimal disk in a neightborhook of a point of almost maximal curvature and applications

University of Massachusetts, Amherst

Removable singularity results for minimal laminations and applications to the classical theory of minimal surfaces

University of Massachusetts, Amherst

Applications of minimal surfaces to the topology of three manifolds

University of Massachusetts, Amherst

Periodic minimal surfaces

University of Massachusetts, Amherst

The Geometry of Minimal Surfaces of Infinite Topology

University of Massachusetts, Amherst

Applications of Random Walks on Minimal Surfaces

University of Massachusetts, Amherst

William H. Meeks III

University of Massachusetts, Amherst

Conferencias impartidas
19
Visitas al departamento
29
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