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William H. Meeks III

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

48 records · Page 3Linked to original sources
arXiv2010

Half-space theorems for minimal surfaces in Nil_3 and Sol_3

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil_3 and the Lie group Sol_3 endowed with their left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil_3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol_3 that lies on one side of a special plane, then S is another special plane.

math.DG↗
arXiv2010

Existence of regular neighborhoods for H-surfaces

In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in certain n-manifolds.

math.DG↗
arXiv2009

Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must be congruent to a homothetic scaling of one of the classical examples found by Riemann in 1860. These examples {\cal R}_s, 0 }, are singly-periodic and intersect each horizontal plane in R^3 in a circle or a line parallel to the x-axis. Earlier work by Collin, Lopez and Ros and Meeks and Rosenberg demonstrate that the plane, the catenoid and the helicoid are the only properly embedded minimal surfaces of genus zero with finite topology (equivalently, with a finite number of ends). Since the surfaces {\cal R}_s converge to a catenoid as s tends to 0 and to a helicoid as s tends to infinity, then the moduli space {\cal M} of all properly embedded, non-planar, minimal planar domains in R^3 is homeomorphic to the closed unit interval [0,1].

math.DG↗
arXiv2009

Calabi-Yau domains in three manifolds

We prove that given any compact Riemannian 3-manifold with boundary M, there exists a smooth properly embedded one-manifold G, included in M, each of whose components is a simple closed curve and such that the domain D=Int(M)-G does not admit any properly immersed open surfaces with at least one annular end, bounded mean curvature, compact boundary (possibly empty) and a complete induced Riemannian metric.

math.DG↗
arXiv2009

Existence of proper minimal surfaces of arbitrary topological type

Consider a domain D in R^3 which is convex (possibly all R^3) or which is smooth and bounded. Given any open surface M, we prove that there exists a complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and bounded, then we prove that the immersion f : M --> D can be chosen so that the limit sets of distinct ends of M are disjoint connected compact sets in the boundary of D.

math.DG↗
arXiv2008

The Dynamics Theorem for CMC surfaces in R^3

In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surfaces in minimal T-invariant sets of T(M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T-invariant set in T(M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one.

math.DG↗
arXiv2008

Limit leaves of a CMC lamination are stable

Suppose ${\cal L}$ is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature. We prove that every limit leaf of ${\cal L}$ is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of ${\cal L}$ has the structure of a lamination.

math.DG↗
arXiv2008

The rigidity of embedded constant mean curvature surfaces

We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.

math.DG↗
arXiv2005

Bending the Helicoid

We construct Colding-Minicozzi limit minimal laminations in open domains in $\rth$ with the singular set of $C^1$-convergence being any properly embedded $C^{1,1}$-curve. By Meeks' $C^{1,1}$-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination ${\cal L}$ is a locally finite collection $S({\cal L})$ of $C^{1,1}$-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding-Minicozzi limit minimal lamination. In the case the curve is the unit circle $\esf^1(1)$ in the $(x_1, x_2)$-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli $H_n$ of finite total curvature which contain the circle. The complete minimal surfaces $H_n$ contain embedded compact minimal annuli $\bar{H}_n$ in closed compact neighborhoods $N_n$ of the circle that converge as $n \to \infty$ to $\rth - x_3$-axis. In this case, we prove that the $\bar{H}_n$ converge on compact sets to the foliation of $\rth - x_3$-axis by vertical half planes with boundary the $x_3$-axis and with $\esf^1(1)$ as the singular set of $C^1$-convergence. The $\bar{H}_n$ have the appearance of highly spinning helicoids with the circle as their axis and are named {\em bent helicoids}.

math.DG↗
arXiv2005

Minimal surfaces with the area growth of two planes; the case of infinite symmetry

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface.

math.DG↗