# «Conjecture (Convex Curve Conjecture, Meeks) Two convex Jordan curves in parallel planes cannot bound a compact minimal surface of positive genus. ...»

Conjecture (Convex Curve Conjecture, Meeks)

Two convex Jordan curves in parallel planes cannot bound a

compact minimal surface of positive genus.

Results of Meeks and White indicate that the Convex

Curve Conjecture holds in the case where the two convex

planar curves lie on the boundary of their convex hull; in this

case, the planar Jordan curves are called extremal.

Results of Ekholm, White and Wienholtz imply every

compact minimal surface that arises as a counterexample to the Convex Curve Conjecture is embedded, and that for a ﬁxed pair of extremal, convex planar curves, there is a bound on the genus of such a minimal surface.

Meeks has conjectured that if Γ = {α, β1, β2,.., βn } ⊂ R3 is a ﬁnite collection of planar, convex, simple closed curves with α in one plane and with {β1, β2,.., βn } bounding a pairwise disjoint collection of disks in a parallel plane, then any compact minimal surface with boundary Γ must have genus 0.

Conjecture (4π-Conjecture, Meeks, Yau, Nitsche) If Γ is a simple closed curve in R3 with total curvature at most 4π, then Γ bounds a unique compact, orientable, branched minimal surface and this unique minimal surface is an embedded disk.

Nitsche proved that a regular analytic Jordan curve in R3 whose total curvature is at most 4π bounds a unique minimal disk.

Meeks and Yau demonstrated the conjecture if Γ is a C 2 -extremal curve (they even allowed the minimal surface spanned by Γ to be non-orientable).

**Ekholm, White and Wienholtz conjecture:**

Besides the unique minimal disk given by Nitsche’s Theorem, only one or two M¨bius strips can occur;

o and if the total curvature of Γ is at most 3π, then there are no such M¨bius strip examples.

o Conjecture (Isolated Singularities Conjecture, Gulliver, Lawson) If M ⊂ B − {(0, 0, 0)} is a smooth properly embedded minimal surface with ∂M ⊂ ∂B and M = M ∪ {(0, 0, 0)}, then M is a smooth compact minimal surface.

Conjecture (Fundamental Singularity Conjecture, Meeks, P´rez, Ros) e If A ⊂ R3 is a closed set with zero 1-dimensional Hausdorﬀ measure and L is a minimal lamination of R3 − A, then L extends to a minimal lamination of R3.

The related Local Removable Singularity Theorem for H-laminations by Meeks, P´rez and Ros is a cornerstone e

**for the proofs of:**

the Quadratic Curvature Decay Theorem the Dynamics Theorem the Finite Topology Closure Theorem, which illustrate the usefulness of removable singularities results.

Conjecture (Connected Graph Conjecture, Meeks) A minimal graph in R3 with zero boundary values over a proper, possibly disconnected domain in R2 can have at most two non-planar components. If the graph also has sublinear growth, then such a graph with no planar components is connected.

Consider a proper, possibly disconnected domain D in R2 and a solution u : D → R of the minimal surface equation with zero boundary values, such that u is non-zero on each component of D.

In 1981 Mikljukov proved that if each component of D is simply-connected with a ﬁnite number of boundary components, then D has at most three components.

Current tools show that his method applies to the case that D has ﬁnitely generated ﬁrst homology group.

Li and Wang proved that the number of disjointly supported minimal graphs with zero boundary values over an open subset of R2 is at most 24.

Next Tkachev proved the number of disjointly supported minimal graphs is at most three.

In the discussion of the conjectures that follow, it is helpful to ﬁx some notation for certain classes of complete embedded minimal surfaces in R3.

Notation C = the space of connected, complete, embedded minimal surfaces.

P ⊂ C = the subspace of properly embedded surfaces.

M ⊂ P = the subspace of surfaces with more than one end.

Conjecture (Finite Topology Conjecture I, Hoﬀman, Meeks) An orientable surface M of ﬁnite topology with genus g and k ends, k = 0, 2, occurs as a topological type of a surface in C if and only if k ≤ g + 2.

The method of Weber and Wolf indicates that the existence implication in the Finite Topology Conjecture holds when k 2.

Meeks, P´rez and Ros proved that for each positive e genus g, there exists an upper bound e(g) on the number of ends of an M ∈ M with ﬁnite topology and genus g.

Hence, the non-existence implication follows if one can show that e(g) can be taken as g + 2.

Concerning the case k = 2, the only examples in M with ﬁnite topology and two ends are catenoids (Collin, Schoen, Colding-Minicozzi).

If M has ﬁnite topology, genus zero and at least two ends, then M is a catenoid (Lopez, Ros).

Conjecture (Finite Topology Conjecture II, Meeks, Rosenberg) For every non-negative integer g, there exists a unique non-planar M ∈ C with genus g and one end.

Hoﬀman, Weber and Wolf and Hoﬀman and White proved existence of a genus one helicoid.

This existence proof is based on the earlier computational construction by Hoﬀman, Karcher and Wei.

For genera g = 2, 3, 4, 5, 6, there are computational existence results.

Conjecture (Inﬁnite Topology Conjecture, Meeks) A non-compact, orientable surface of inﬁnite topology occurs as a topological type of a surface in P if and only if it has at most one or two limit ends, and when it has exactly one limit end, then its limit end with inﬁnite genus.

In the inﬁnite topology case, either M has inﬁnite genus or M has an inﬁnite number of ends.

Such an M must have at most two limit ends (Collin, Kusner, Meeks and Rosenberg).

Such an M cannot have one limit end and ﬁnite genus (Meeks, P´rez and Ros).

e Conjecture (Liouville Conjecture, Meeks) If M ∈ P and h : M → R is a positive harmonic function, then h is constant.

If M ∈ P has ﬁnite genus, a limit end of genus 0 (Meeks, Perez, Ros) or two limit ends (Collin, Kusner, Meeks, Rosenberg), then M is recurrent for Brownian motion, which implies M satisﬁes the Liouville Conjecture.

By work of Meeks, P´rez and Ros, the above conjecture e holds for all of classical examples.

Below is a related conjecture.

Conjecture (Multiple-End Recurrency Conjecture, Meeks) If M ∈ M, then M is recurrent for Brownian motion.

Conjecture (Isometry Conjecture, Choi, Meeks, White) If M ∈ C, then every intrinsic isometry of M extends to an ambient isometry of R3. Furthermore, if M is not a helicoid, then it is minimally rigid, in the sense that any isometric minimal immersion of M into R3 is congruent to M.

**The Isometry Conjecture is known to hold if:**

M ∈ M (Choi, Meeks and White), M is doubly-periodic (Meeks and Rosenberg), M is periodic with ﬁnite topology quotient (Meeks and P´rez) e M has ﬁnite genus (Meeks and Tinaglia).

One can reduce the Isometry Conjecture to checking that whenever M ∈ P has one end and inﬁnite genus, then there exists a plane in R3 that intersects M in a set that contains a simple closed curve. The reason for this reduction is that the ﬂux of M along such a simple closed curve is not zero, and hence, none of the associate surfaces to M are well-deﬁned; but Calabi proved that the associate surfaces are the only isometric minimal immersions from M into R3, up to congruence.

Conjecture (Scherk Uniqueness Conjecture, Meeks, Wolf) If M is a connected, properly immersed minimal surface in R3 and Area(M ∩ B(R)) ≤ 2πR2 holds in balls B(R) of radius R, then M is a plane, a catenoid or one of the singly-periodic Scherk minimal surfaces.

By the Monotonicity Formula, any connected non-ﬂat, properly immersed minimal surface in R3 with

is embedded.

Meeks and Wolf proved the Scherk Uniqueness Conjecture holds under the assumption that the surface is periodic.

Conjecture (Unique Limit Tangent Cone Conjecture, Meeks) If M ∈ P is not a plane and has quadratic area growth, then limt→∞ 1 M exists and is a minimal, possibly non-smooth cone t over a ﬁnite balanced conﬁguration of geodesic arcs in the unit sphere, with common ends points and integer multiplicities.

Meeks and Wolf’s proof of the Scherk Uniqueness Conjecture in the periodic case uses that the Unique Limit Tangent Cone Conjecture above holds in the periodic setting; this approach

**suggests to solve the Scherk Uniqueness Conjecture by:**

First to prove the uniqueness of the limit tangent cone of M, from which it should follow that M has two Alexandrov-type planes of symmetry.

Next use these planes of symmetry to describe the Weierstrass representation of M. Meeks and Wolf claim this would be suﬃcient to complete the proof of the conjecture.

Conjecture (Injectivity Radius Growth Conjecture, Meeks, P´rez, Ros) e An M ∈ C has ﬁnite topology if and only if its injectivity radius function grows at least linearly with respect to the extrinsic distance from the origin.

If M ∈ C has ﬁnite topology, then M has ﬁnite total curvature or is asymptotic to a helicoid. So there exists a constant CM 0 such that the injectivity radius function IM : M → (0, ∞] satisﬁes

Work Meeks, P´rez and Ros indicates that this linear e growth property of the injectivity radius function characterizes the ﬁnite topology examples in C.

Conjecture (Negative Curvature Conjecture, Meeks, P´rez, Ros) e If M ∈ C has negative curvature, then M is a catenoid, a helicoid or one of the singly or doubly-periodic Scherk minimal surfaces.

Suppose M ∈ C has ﬁnite topology. M either has ﬁnite total curvature or is a helicoid with handles. Such a surface has negative curvature if and only if it is a catenoid or a helicoid.

Suppose M ∈ C is invariant under a proper discontinuous group G of isometries of R3 and M/G has ﬁnite topology.

Then M/G is properly embedded in R3 /G (Meeks, P´rez, e Ros) and M/G has ﬁnite total curvature (Meeks, Rosenberg). If M/G has negative curvature and the ends of M/G are helicoidal or planar, then M is easily proven to have genus zero, and so, it is a helicoid. If M/G is doubly-periodic, then M is a Scherk minimal surface. In the case M/G is singly-periodic, then M must have Scherk-type ends but we do not know if the surface must be a Scherk singly-periodic minimal surface.

Conjecture (Four Point Conjecture, Meeks, P´rez, Ros) e

**Suppose M ∈ C. Then:**

If the Gauss map of M omits 4 points on S (1), then M is a singly or doubly-periodic Scherk minimal surface.

If the Gauss map of M omits exactly 3 points on S (1), then M is a singly-periodic Karcher saddle tower whose ﬂux polygon is a convex unitary hexagon. (note that any three points in a great circle are omitted by one of these examples).

If the Gauss map of M omits exactly 2 points, then M is a catenoid,

or doubly-periodic Scherk minimal surface.

If the Gauss map of M omits exactly 3 points on S (1), then M is a singly-periodic Karcher saddle tower whose ﬂux polygon is a convex unitary hexagon. (note that any three points in a great circle are omitted by one of these examples).

If the Gauss map of M omits exactly 2 points, then M is a catenoid, a helicoid, one of the Riemann minimal examples or one of the KMR doubly-periodic minimal tori. In particular, the pair of points missed by the Gauss map of M must be antipodal.

A classical result of Fujimoto is that the Gauss map of any orientable, complete, non-ﬂat, minimally immersed surface in R3 cannot exclude more than 4 points.

If one assumes that a surface M ∈ C is periodic with ﬁnite topology quotient, then Meeks, P´rez and Ros have solved the e ﬁrst item in the above conjecture.

Conjecture (Finite Genus Properness Conjecture, Meeks, P´rez, Ros) e If M ∈ C and M has ﬁnite genus, then M ∈ P.

Colding and Minicozzi proved the conjecture for surfaces of ﬁnite topology.

Meeks, P´rez and Ros proved the Finite Genus e Properness Conjecture under the additional hypothesis that M has a countable number of ends or even a countable number of limit ends.

Meeks, P´rez and Ros had conjectured that if M ∈ C e has ﬁnite genus, then M has bounded Gaussian curvature, which they proved is equivalent to the above conjecture.

Conjecture (Embedded Calabi-Yau Conjecture Martin, Meeks, Nadirashvili; Meeks, Perez, Ros) Let M be open surface.

There exists a complete proper minimal embedding of M in every smooth bounded domain D ⊂ R3 iﬀ M is orientable and every end has inﬁnite genus.

There exists a complete proper minimal embedding of M in some smooth bounded domain D ⊂ R3 iﬀ every end of M has inﬁnite genus and M has a ﬁnite number of nonorientable ends.

There exists a complete proper minimal embedding of M in some particular non-smooth bounded domain D ⊂ R3 iﬀ every end of M has inﬁnite genus.

Conjecture (Embedded Calabi-Yau Conjectures) There exists an M ∈ C whose closure M has the structure of a minimal lamination of a slab, with M as a leaf and with two planes as limit leaves.

In particular, P = C (Meeks).

A connected, complete, embedded surface of non-zero constant mean curvature in R3 with ﬁnite genus is properly embedded (Meeks, Tinaglia).

Conjecture (Stable Minimal Surface Conjectures) A complete, non-orientable, stable minimal surface in R3 with compact boundary has ﬁnite total curvature (Ros).

If A ⊂ R3 is a closed set with zero 1-dimensional Hausdorﬀ measure and M ⊂ R3 − A is a connected, stable, minimally immersed surface which is complete outside of A, then the closure of M is a plane (Meeks).

If M ⊂ R3 is a minimal graph over a proper domain in R2 with

then M is δ-parabolic (Meeks, Rosenberg).

A complete, embedded, stable minimal surface in R3 with boundary a straight line is a half-plane, a half of the Enneper minimal surface or a half of the helicoida (P´rez, Ros, White).

e a P´rez has proved this conjecture under the additional assumption that the e surface is proper and has quadratic area growth.

Since if some ﬂux vector of a minimally immersed M in R3 is non-zero, then the inclusion map is the unique isometric minimal immersion of M into R3 up to congruence, the One-Flux Conjecture below implies the Isometry Conjecture.