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# CMC foliations of closed manifolds

## William H. Meeks III 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

## William H. Meeks III 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)

## William H. Meeks III 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)

## William H. Meeks III University of Massachusetts, Amherst

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

# Plateau problem in three-dimensional metric Lie groups

## William H. Meeks III University of Massachusetts, Amherst

Seminario de Matemáticas. 1ª Planta, sección de matemáticas

# Constant mean curvature surfaces in homogeneous 3-manifold III

## William H. Meeks 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

## William H. Meeks 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 Is

## William H. Meeks 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.

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

## William H. Meeks III 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.

## William H. Meeks III 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

## William H. Meeks III University of Massachusetts, Amherst

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

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

## William H. Meeks III 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.

# William H. Meeks III

## University of Massachusetts, Amherst

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