Finite topology self-translating surfaces for the mean curvature flow in $\mathbb R^3$
Finite topology self translating surfaces to mean curvature flow of surfaces constitute a key element for the analysis of Type II singularities from a compact surface, since they arise in a limit after suitable blow-up scalings around the singularity. We find in a surface orientable, embedded and complete with finite topology (and large genus) with three ends asymptotically paraboloidal, such that the moving surface evolves by mean curvature flow. This amounts to the equation where denotes mean curvature, is a choice of unit normal to , and is a unit vector along the -axis. The surface is in correspondence with the classical 3-end Costa-Hoffmann-Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded, complete minimal surfaces with finite total curvature.
We say that a family of orientable, embedded hypersurfaces in evolves by mean curvature if each point of moves in the normal direction with a velocity proportional to its mean curvature at that point. More precisely, there is a smooth family of diffeomorphisms , , determined by the mean curvature flow equation
where designates the mean curvature of the surface at the point , , namely the trace of its second fundamental form, is a choice of unit normal vector.
The mean curvature flow is one of the most important examples of parabolic geometric evolution of manifolds. Relatively simple in form, it generates a wealth of interesting phenomena which is so far only partly understood. Extensive, deep studies on the properties of this equation have been performed in the last 25 years or so. We refer for instance the reader to the surveys  and .
A classical, global-in-time definition of a weak solution to mean curvature flow is due to Brakke. These solutions typically develop finite time singularities. When they arise, the evolving manifold loses smoothness, and a change of topology of the surface may occur as the singular time is crossed.
The basic issue of the theory for the mean curvature flow, is to understand the way singularities appear and to achieve an accurate description of the topology of the surface obtained after blowing-up the manifold around the singularity.
Singuarities are usually classified as types I and II. If is a time when a singularity appears, type I roughly means that the curvatures grow no faster than . In such a case, a blowing-up procedure, involving a time dependent scaling and translation leads in the limit to a “self shrinking” ancient solution, as established by Huisken in . The appearance of these singularities turns out to be generic under suitable assumptions, see Colding and Minicozzi .
Instead, if the singularity is not of type I, it is called type II. In that case, a suitable normalization leads in the limit to an eternal solution to MC flow. See Colding and Mincozzi , Huisken and Sinestrari [16, 17]. An eternal solution to (1.1) is one that is defined at all times .
The simplest type of eternal solutions are the self-translating solutions, surfaces that solve (1.1), do not change shape and travel at constant speed in some specific direction. A self-translating solution of the mean curvature flow (1.1), with speed and direction is a hypersurface of the form
that satisfies (1.1). Equivalently, such that
This problem is nothing but the minimal surface equation in case . A result by Hamilton  states that in the case of a compact convex surface, the limiting scaled singularity does indeed take place in the form of a self-translating solution. This fact makes apparent the importance of eternal self-translating solutions in the understanding of singularity formation. On the other hand, the result in  is not known without some convexity assumptions. An open, challenging issue is to understand whether or not a given “self-translator” (convex or non-convex) can arise as a limit of a type II singularity for (1.1).
A situation in which strong insight has been obtained is the mean convex scenario (namely, surfaces with non-negative mean curvature, a property that is preserved under the flow). In fact under quite general assumptions, mean convexity in the singular limit becomes full convexity for the blown-up surface, as it has been established by B. White [34, 35], and by Huisken and Sinestrari [16, 17].
In spite of their importance in the theory for the mean curvature flow, relatively few examples of self-translating solutions are known, and a theory for their understanding, even in special classes is still far from achieved. Since for , equation (1.3) reduces to the minimal surface equation, it is natural to look for analogies with minimal surface theory in order to obtain new nontrivial examples. On the other hand, it has been proven by Ilmanen [21, 22], that the genus of a surface is nonincreasing along the mean curvature flow. Therefore, self-translators originated from a singularity in the flow of a compact surface must have finite genus, or finite topology.
The purpose of this paper is to construct new examples of self-translating surfaces to the mean curvature flow with the finite topology in . More precisely, we are interested in tracing a parallel between the theory of embedded, complete minimal surfaces in with finite total Gauss curvature (which are precisely those with finite topology) and self-translators with positive speed. Before stating our main result, we recall next some classical examples of self-translators.
If is a travelling graph , namely
then equation (1.3) reduces to the elliptic PDE for ,
For instance for and , Grayson  found the explicit solution, so-called the grim reaper curve , given by the graph
In other words, solves (1.1).
For dimensions , there exist entire convex solutions to Equation (1.4). Altschuler and Wu  found a radially symmetric convex solution to (1.4) by blowing-up a type II singularity of mean curvature flow. This solution can be explicitly found by solving the radial PDE (1.4) which becomes simply
We shall denote by the graph of this entire graphical self-translator (which is unique up to an additive constant) which we shall refer to as the travelling paraboloid. Of course this means that solves (1.1).
Xu-Jia Wang  proved that for , solutions of (1.4) are necessarily radially symmetric about some point, thus in particular they are convex. Suprprisingly, for dimensions , Wang was able to construct nonradial convex solutions of (1.4).
In dimension , , Angenent and Velázquez  constructed an axially symmetric solution to (1.1) that develops a type II singularity with a tip that blows-up precisely into the paraboloid . On the other hand, B. White proved that the convex surface in given by the where is the grim reaper curve (1.5), cannot arise as a blow-up of a type II singularity for (1.1).
A non-graphical, two-end axially symmetric self translating solutions of (1.1) for has been found by direct integration of the radial PDE (1.4) by Clutterbuck, Schnurer and Schulze . It can be described as as follows:
Given any number , there is a self-translating solution of (1.1)
where is a two-end smooth surface of revolution of the form
where the functions solve (1.6) for and , with and . It is shown in  that the functions have the asymptotic behavior (1.7) of up to an additive constant. See Figure 2. We call the two-end translating surface the travelling catenoid. The reason is natural: when equation (1.6) is nothing but the minimal surface equation for an axially symmetric minimal surface around the -axis. When the equation leads (up to translations) to the plane , or the standard catenoid . The catenoid is exactly the parallel to . The plane is actually in correspondence with the paraboloid .
These simple, however important examples, are the only ones available with finite topology. On the other hand, the third author has constructed self translating surfaces with infinite topology, periodic in one direction in [25, 26, 27].
Embedded minimal surfaces of finite total curvature in .
The theory of embedded, minimal surfaces of finite total curvature in has reached a spectacular development in the last 30 years or so. For about two centuries, only two examples of such surfaces were known: the plane and the catenoid. The first nontrivial example was found in 1981 by C. Costa [6, 7]. The Costa surface is a genus one minimal surface, complete and properly embedded, which outside a large ball has exactly three components (its ends), two of which are asymptotically catenoids with the same axis and opposite directions, the third one asymptotic to a plane perpendicular to that axis. Hoffman and Meeks [12, 13, 14] built a class of three-end, embedded minimal surfaces, with the same look as Costa’ÃÂs far away, but with an array of tunnels that provides arbitrary genus . These are known as the Costa-Hoffman-Meeks surfaces, see Figure 1. Many other examples of multiple-end embedded minimal surfaces have been found since.
All surfaces of this kind are constituted, away from a compact region, by the disjoint union of ends ordered along one coordinate axis, which are asymptotic to planes or to catenoids with parallel symmetry axes, as established by Osserman , Schoen  and Jorge and Meeks . The topology of such a surface is thus characterized by the genus of a compact region and the number of ends, having therefore finite topology.
Main result: the travelling CHM surface of large genus
In what follows we restrict ourselves to the the case .
Our purpose is to construct new complete and embedded surfaces in which are self translating under mean curvature flow. After a rotation and dilation we can assume that and that the travelling direction is the that of the positive -axis. Thus we look for orientable, embedded complete surfaces in satisfying the equation
where . In other words, the moving surface satisfies equation (1.1). A major difficulty to extend the theory of finite total curvature minimal surfaces in Euclidean 3d space to equation (1.8) is that much of the theory developed relies in the powerful tool given by the Weierstrass representation formula, which is not available in our setting. Unlike the static case, the travelling catenoid for instance is not asymptotically flat and does not have total finite total Gauss curvature.
What we establish in our main result is the existence of a 3-end surface that solves (1.8), homeomorphic to a Costa-Hoffmann-Meeks surface with large genus, whose ends behave like those of a travelling catenoid and a paraboloid.
More precisely, let us consider the union of a travelling paraboloid and and a travelling catenoid , which intersect transversally on a circle for some . See Figure 2.
Let and be respectively a travelling paraboloid and catenoid, which intersect transversally. Then for all small, there is a complete embedded 3-end surface satisfying equation (1.8), which lies within an -neighborhood of . Besides we have that
The construction provides much finer properties of the surface . Let us point out that the CHM with large genus was found in  to approach in the multiple-tunnel zone, a Scherk singly periodic minimal surface. See Figure 1.
Kapouleas [18, 19, 20], Traizet [31, 32] and Hauswirth and Pacard  established a method for the reverse operation, namely, starting with a union of intersecting catenoids and planes, they desingularize them using Scherk surfaces to produce smooth minimal surfaces (complete and embedded). A key element in those constructions is a fine knowledge of the Jacobi operator of the Scherk surface and along the asymptotically flat ends. This approach was used by the third author to construct translating surfaces in built from a 2d picture of intersecting parallel grim reapers and vertical lines, trivially extended in an additional direction, desingularized in that direction by infinite Scherk surfaces, see [25, 26, 27]. We shall use a similar scheme in our construction. The context here is considerably more delicate, since no periodicity is involved (the ultimate reason why the topology resulting is finite), and the fine interplay between the slowly vanishing curvatures and the Jacobi operators of the different pieces requires new ideas. Our method extends to the construction of more general surfaces built upon desingularization of intersection of multiple travelling catenoids and travelling paraboloids, but for simplicity in the exposition we shall restrict ourselves to the basic context of Theorem 1. Before proceeding into the detailed proof, we sketch below the core ingredients of it.
1.1. Sketch of the proof of Theorem 1
After a change of scale of , the problem is equivalent to finding a complete embedded surface that satisfies
The first step is to construct a surface that is close to being a solution to this equation. This is accomplished by desingularizing the union of and using singly periodic Scherk surfaces. At a large distance from , the approximation is and in some neighborhood of , it is a slightly bent singly periodic Scherk surface. We call the core of the region where the desingularization is made. The actual approximation will depend on four real parameters: , which are going to be small, of order .
Let denote a choice of unit normal of . We search for a solution of (1.9) in the form of the normal graph over of a function , that is, of the form
Let and denote the mean curvature and normal vector of , respectively, while and denote those of . Then
where is the Laplace-Beltrami operator on , is the tangential component of the gradient, and , are quadratic functions in . This allows us to write equation (1.9) as
To solve (1.11), we linearize around , and the following linear operator becomes relevant:
We work with the following norms for functions defined on , where , are fixed:
Here is a small fixed parameter. The function measures geodesic distance to the core of and will be defined precisely later on, and is the geodesic ball centered at with radius 1.
The term in (1.11) that does not depend on is
We have the following approximation for it.
The following claim illustrates de invertibility of the linear operator , although it will not be used directly. Let us fix by fixing the parameters sufficiently small and consider the problem
Then, for small, there is a linear operator that produces for a solution of (1.14) with
where is independent of .
Assume ( and . Then, for small,
with independent of and .
These results can be used to prove Theorem 1 by the contraction mapping principle, which is done in Section 6. The preparatory steps are the construction of an initial approximate solution in Section 2 and some geometric computations in Section 3, which lead to the estimate of in Proposition 2 and the estimate of in Proposition 3. In Section 4, we analyze the Jacobi equation for the Scherk surface and in Section 5.1, we study the Jacobi operator on the ends, which are the regions far from the desingularization.
2. Construction of an initial approximation
The purpose of this section is to construct a surface that will serve as an initial approximation to (1.8).
Let the unique radially symmetric solution of
and let be the corresponding surface . Let be a catenoidal self-translating solution of MCF, which can be written as where is given by and satisfies (2.1) for , with , , .
We assume that and intersect transversally at a unique circle of radius . To quantify the transversality, we fix a small constant so that all the intersection angles are greater than . In this section, we are going to replace in a neighborhood of with an appropriately bent Scherk surface. The number of periods used, and thus the number of handles, is of order . Two of the three ends of the resulting approximate solution will differ slightly from the original ends.
2.1. Self-translating rotationally symmetric surfaces
We briefly recall some properties of self-translating rotationally symmetric surfaces. Let be a small constant, let , be a smooth planar curve parametrized by arc length and let and be the surfaces of revolution parametrized by
where , and , . (The reason for introducing in (2.2) is to make the parametrization conformal at .)
The surface ( respectively) is a self-translating surface under mean curvature flow with velocity ( respectively) if and only if , parametrized by arc length, satisfies the differential equation
Another way to represent an axially symmetric self-translating solution is through the graph of a radial function, , where satisfies (2.1) on some interval . Then satisfies
Using , , with and the asymptotic behavior (2.5), we can deduce the following estimates.
For a smooth planar curve , parametrized by arc length with and satisfying (2.3), we have
as tends to infinity.
2.2. The Scherk surfaces
Let be Euclidean coordinates in and consider the one parameter family of minimal surfaces given by the equation
Outside of a large cylinder around the -axis, has four connected components. We call these components the wings of and number them according to the quadrant where they lie. Each wing of is asymptotic to a half-plane forming an angle with the -plane (note that the asymptotic half-planes do not contain the -axis unless ). Here, we will restrict the parameter to so that the geometry on all the ’s can be uniformly bounded as stated in the following lemma.
Let be the half-plane . Note that the parameter here is on a different scale than the one used in the previous section. We construct approximate solutions satisfying (1.9) here, while the rotationally symmetric surfaces in Section 2.1 satisfy (1.8).
is a singly periodic embedded complete minimal surface which depends smoothly on . There is a constant and smooth functions so that the wings of can be expressed as the graph of over half-planes. More precisely, the half-plane asymptotic to the first wing can be parametrized by , with
where and . The wing itself is parametrized by , which is defined by
The functions and depend smoothly on . Moreover, we have
for any .
The function satisfies the minimal surface equation
for , .
Let us denote by the reflection across the -plane and the reflection across the -plane. The parametrizations of the second, third, and fourth wings are given by
The th wing of is given by and is denoted by . The parametrizations of the corresponding asymptotic half-planes are obtained by replacing by in the above formulas. We use to denote the parametrization of the th asymptotic half-plane as well as its image, . The inner core of is the surface without its four wings.
2.3. Dislocation of the Scherk surfaces
We now perform dislocations on the first and fourth wings of . These perturbations will help us to deal with the kernel of the linear operator associated to normal perturbations of solutions to (1.9). Indeed, because translated solutions of (1.9) remain solutions, the functions , , and are in the kernel of . Here we have taken the normal component of the translations because we are considering normal perturbations. The last function, does not satisfy our imposed symmetries. Therefore, we can discard it from the kernel. The other two remain. In Section 4, we will show that the Dirichlet problem for the linear operator can be solved on a truncated piece of , up to constants at the boundary. By adding a linear combination of the functions in the kernel, we can obtain a solution that vanishes on the boundary of two adjacent wings, say the second and third wings. To obtain a solution that vanishes on all the connected pieces of the boundary, we will artificially translate the first and fourth wing by constants and .
The linear operator is close to linear operator associated to the equation , so we have small eigenvalues due to changes of Scherk angle and rotation. Indeed, because there is a one parameter family of Scherk surfaces, we expect a function in the kernel of the Jacobi operator , namely, the normal component of the motion associated to changing the angle . One more dimension is generated by rotation of the Scherk surfaces around the -axis. To summarize, besides the translations, we have two more dimensions in the kernel of generated by linear functions along the wings. This is reasonable since the is close to the Laplace operator along the wings. By adding a linear combination of these two linear eigenfunctions, we can force exponential decay along the second and third wings again. As before, we will generate linear functions on the first and fourth wings through rotations by angles and respectively.
For , we define the map to be the rotation of angle (counterclockwise in the -plane) around the -axis:
In what follows, we will confine to , where is a small fixed number.
We consider two constants , , and a family of smooth transition functions such that , for , and for . The numbers , will be fixed later to be large.
Given , , we modify the first and fourth wings in the following way: the th wing is shifted by at around , then it is rotated by an angle at distance . The parametrization of the new th wing, for , is given by , where
and is the reflection across the -plane. Note that the th wing is moved away from the -axis for positive constants and . We denote the new wings by , (see Figure 4).
The wings have natural coordinates given by the parametrizations and . The surface (or for short) is defined to be the union of the inner core of with the four wings , , , and . We will call the region of for which the outer core.
The maps and , can be used to pullback tensors defined on to and vice versa: in the case of a function defined on , the composition is the corresponding pullback function on . Taking each wing at a time, these maps transport functions and tensors between and . This is very useful as it lets us work on a fixed surface, usually . We will use the same notation for functions and tensors on or their pullback to . For example, could denote the mean curvature of as a function on or its pullback to . The same notation convention applies to the unit normal vector , the metric , and the second fundamental form .
Let us define the following functions on , which capture the contribution of the dislocations to the mean curvature:
These functions are compactly supported because rotations and translations do not change the mean curvature. They will later help us solve the Dirichlet problem associated to the Jacobi operator on the Scherk surfaces in Section 4.
Because the parameters are associated to rotations, the functions and can be written explicitly as the Jacobi operator on the normal component of rotation at a point :
where is defined as on wing 1 and zero elsewhere and similarly for . We also have,
on , where on wing 1 and zero elsewhere, and similarly for .
2.4. Wrapping the dislocated Scherk surfaces around a circle
We first rotate our new surface so that its second and third wings match the directions of two chosen pieces of catenoid or paraboloid coming out of the intersection circle. The wrapping is performed by simply using a smooth map from a tubular neighborhood of the -axis to a neighborhood of a large circle. The scaling factor is so our target circle will have a radius of order .
For and , we define
This map takes a segment of length on the -axis to the circle of radius .
We can not wrap the whole surface , so we cut its four wings at and denote the new surface by , with a “bar” on top to indicate that it has a boundary. Our desingularizing surface is a dislocated rotated wrapped Scherk surface
where the angle has yet to be fixed and is the closest number in to (the radius associated to the original intersection )
We wish to prolong the wings of the desingularizing surface with pieces of self-translating catenoids or paraboloids. At this point, it will be useful to record the boundary, not of the surface itself, but of the asymptotic plane underneath at . We will extend the asymptotic pieces first, then construct the approximate surface by adding the graph of .
2.5. Fitting the Scherk surface
It is now time to examine the initial configuration in detail. We will work with cross-sections in the -plane. Let denote the intersection of the par