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William H. Meeks III

Publications and source records attributed to William H. Meeks III.

At least 19 recordsLinked to original sources
arXiv2023

Hierarchy structures in finite index CMC surfaces

Given $\varepsilon_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq \varepsilon_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M \looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ with index at most $I$. For such $M \looparrowright X$, we prove Structure Theorem 1.2 which describes how the interesting ambient geometry of the immersion is organized locally around at most $I$ points of $M$ where the norm of the second fundamental form takes on large local maximum values.

math.DG↗
arXiv2023

Geometry of CMC surfaces of finite index

Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M\looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ and with index at most $I$. We will obtain geometric estimates for such an $M\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [9]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of $M\looparrowright X$, especially results related to the area and diameter of $M$. By item E of Theorem 2.2, the area of such a non-compact $M\looparrowright X$ is infinite. We will improve this area result by proving the following when $M$ is connected; here $g(M)$ denotes the genus of the orientable cover of $M$: 1. There exists $C_1=C_1(I,r_0,K_0,H_0)>0$ such that Area$(M)\geq C_1(g(M)+1)$. 2. There exists $C>0,G(I)\in \mathbb{N}$ independent of $r_0,K_0,H_0$ and also $C$ independent of $I$ such that if $g(M)\geq G(I)$, then Area$(M)\geq \frac{C}{(\max\{1,\frac{1}{r_0},\sqrt{K_0}, H_0\})^2}(g(M)+1)$. 3. If the scalar curvature $ρ$ of $X$ satisfies $3H^2+\frac{1}{2}ρ\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\leq A$ and Diameter$(M)\leq D$. Hence, $M$ is compact and, by item 1, $g(M)\leq A/C -1$.

math.DG↗
arXiv2022

Geometry of branched minimal surfaces of finite index

Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of Fischer-Colbrie and Ros explain that such surfaces are precisely the complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an $M$ with estimates that are given in terms of $I$ and $B$. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for $m$-dimensional submanifolds $Σ$ of an $n$-dimensional Riemannian manifold $X$, where these area estimates depend on the geometry of $X$ and upper bounds on the lengths of the mean curvature vectors of $Σ$. We also describe a family of complete, finitely branched minimal surfaces in $\mathbb{R}^3$ that are stable and non-orientable; these examples generalize the classical Henneberg minimal surface.

math.DG↗
arXiv2020

Modifications Preserving Hyperbolicity of Link Complements

Given a link in a 3-manifold such that the complement is hyperbolic, we provide two modifications to the link, called the chain move and the switch move, that preserve hyperbolicity of the complement, with only a relatively small number of manifold-link pair exceptions, which are also classified. These modifications provide a substantial increase in the number of known hyperbolic links in the 3-sphere and other 3-manifolds.

math.GT↗
arXiv2020

Totally umbilic surfaces in hyperbolic 3-manifolds of finite volume

We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N(S,H)$ with mean curvature $H$. Conversely, we prove that a complete, totally umbilic surface with mean curvature $H \in [0,1)$ embedded in a hyperbolic 3-manifold of finite volume must be proper and have finite negative Euler characteristic.

math.DG↗
arXiv2019

Bounds on the topology and index of minimal surfaces

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.

math.DG↗
arXiv2018

The embedded Calabi-Yau conjecture for finite genus

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if $M$ has at least two simple limit ends, then $M$ has exactly two simple limit ends. Furthermore, we demonstrate that $M$ is properly embedded in $\mathbb{R}^3$ if and only if $M$ has at most two limit ends if and only if $M$ has a countable number of limit ends.

math.DG↗
arXiv2017

Properly immersed surfaces in hyperbolic 3-manifolds

We study complete finite topology immersed surfaces $Σ$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $Σ$ is bounded from above by $a$ and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $Σ$ must be proper in $N$ and its total curvature must be equal to $2πχ(Σ)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $Σ$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $Σ$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.

math.DG↗
arXiv2017

Chord arc properties for constant mean curvature disks

We prove a chord arc bound for disks embedded in $\mathbb{R}^3$ with constant mean curvature. This bound does not depend on the value of the mean curvature. It is inspired by and generalizes the work of Colding and Minicozzi in [2] for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in $\mathbb{R}^3$ with finite topology or with positive injectivity radius.

math.DG↗
arXiv2016

Structure theorems for singular minimal laminations

We apply the local removable singularity theorem for minimal laminations and the local picture theorem on the scale of topology to obtain two descriptive results for certain possibly singular minimal laminations of $\mathbb{R}^3$. These two global structure theorems will be applied in forthcoming papers to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in $\mathbb{R}^3$, and to prove that a complete, embedded minimal surface in $\mathbb{R}^3$ with finite genus and a countable number of ends is proper.

math.DG↗
arXiv2016

Triply periodic constant mean curvature surfaces

Given a closed flat 3-torus $N$, for each $H>0$ and each non-negative integer $g$, we obtain area estimates for closed surfaces with genus $g$ and constant mean curvature $H$ embedded in $N$. This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer $g$ with $g\neq 2$, connected closed embedded minimal surfaces of genus $g$ with arbitrarily large area.

math.DG↗
arXiv2016

The geometry of stable minimal surfaces in metric Lie groups

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem.

math.DG↗
arXiv2016

Finite topology minimal surfaces in homogeneous three-manifolds

We prove that any complete, embedded minimal surface $M$ with finite topology in a homogeneous three-manifold $N$ has positive injectivity radius. When one relaxes the condition that $N$ be homogeneous to that of being locally homogeneous, then we show that the closure of $M$ has the structure of a minimal lamination of $N$. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is $\mathbb{S}^3$ equipped with a homogeneous metric of nonnegative scalar curvature.

math.DG↗
arXiv2016

The local picture theorem on the scale of topology

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.

math.DG↗