Regularity theory for -dimensional almost minimal currents II: branched center manifold
We construct a branched center manifold in a neighborhood of a singular point of a -dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of -dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of -dimensional area minimizing cones.
This paper is the third in a series of works aimed at establishing an optimal regularity theory for -dimensional integral currents which are almost minimizing in a suitable sense. Building upon the monumental work of Almgren , Chang in  established that -dimensional area minimizing currents in Riemannian manifolds are classical minimal surfaces, namely they are regular (in the interior) except for a discrete set of branching singularities. The argument of Chang is however not entirely complete since a key starting point of his analysis, the existence of the so-called “branched center manifold”, is only sketched in the appendix of  and requires the understanding (and a suitable modification) of the most involved portion of the monograph .
An alternative proof of Chang’s theorem has been found by Rivière and Tian in  for the special case of -holomorphic curves. Later on the approach of Rivière and Tian has been generalized by Bellettini and Rivière in  to handle a case which is not covered by , namely that of special Legendrian cycles in (see also  for a further generalization).
Meanwhile the first and second author revisited Almgren’s theory giving a much shorter version of his program for proving that area minimizing currents are regular up to a set of Hausdorff codimension , cf. [6, 8, 7, 9, 10]. In this note and its companion papers [11, 12] we build upon the latter works in order to give a complete regularity theory which includes both the theorems of Chang and Bellettini-Rivière as special cases. In order to be more precise, we introduce the following terminology (cf. [13, Definition 0.3]).
Let be a submanifold and an open set.
An -dimensional integral current with finite mass and is area minimizing in if for any -dimensional integral current with .
A semicalibration (in ) is a -form on such that at every , where denotes the comass norm on . An -dimensional integral current with is semicalibrated by if for -a.e. .
An -dimensional integral current supported in is a spherical cross-section of an area minimizing cone if is area minimizing.
In what follows, given an integer rectifiable current , we denote by the subset of consisting of those points for which there is a neighborhood such that is a (constant multiple of) a submanifold. Correspondingly, is the set . Observe that is relatively open in and thus is relatively closed. The main result of this and the works [11, 12] is then the following
Let and be as in (a), (b) or (c) of Definition 0.1. Assume in addition that is of class (in case (a) and (b)) and of class (in case (b)) for some positive . Then is discrete.
Clearly Chang’s result is covered by case (a). As for the case of special Lagrangian cycles considered by Bellettini and Rivière in  observe that they form a special subclass of both (b) and (c). Indeed these cycles arise as spherical cross-sections of -dimensional special L.agrangian cones: as such they are then spherical cross sections of area minimizing cones but they are also semicalibrated by a specific smooth form on .
Following the Almgren-Chang program, Theorem 0.2 will be established through a suitable “blow-up argument” which requires several tools. The first important tool is the theory of multiple valued functions, for which we will use the results and terminology of the papers [6, 8]. The second tool is a suitable approximation result for area minimizing currents with graphs of multiple valued functions, which for the case at hand has been established in the preceding note . The last tool is the so-called “center manifold”: this will be constructed in the present paper, whereas the final argument for Theorem 0.2 will then be given in . We note in passing that all our arguments use heavily the uniqueness of tangent cones for . This result is a, by now classical, theorem of White for area minimizing -dimensional currents in the euclidean space, cf. . Chang extended it to case (a) in the appendix of , whereas Pumberger and Rivière covered case (b) in . A general derivation of these results for a wide class of almost minimizers has been given in : the theorems in there cover, in particular, all the cases of Definition 0.1.
The proof of Theorem 0.2 is based, as in , on an induction statement, cf. Theorem 1.8 below. This and the next paper  can be thought as the two main steps in its proof. For this reason, before detailing the construction of the branched center manifold, which is the main object of this note, we will state Theorem 1.8, show how Theorem 0.2 follows from it and give a rough outline of the contributions of this and the next note .
The research of Camillo De Lellis and Luca Spolaor has been supported by the ERC grant RAM (Regularity for Area Minimizing currents), ERC 306247.
1. Preliminaries and the main induction statement
1.1. Basic notation and first main assumptions
For the notation concerning submanifolds we refer to [7, Section 1]. With and we denote, respectively, the open ball with radius and center in and the open ball with radius and center in . and will always denote the cylinder , where . We will often need to consider cylinders whose bases are parallel to other -dimensional planes, as well as balls in -dimensional affine planes. We then introduce the notation for and for . will denote the unit vectors in the standard basis, the (oriented) plane and the -vector orienting it. Given an -dimensional plane , we denote by and the orthogonal projections onto, respectively, and its orthogonal complement . For what concerns integral currents we use the definitions and the notation of . Since is used recurrently for -dimensional planes, the -dimensional area of the unit circle in will be denoted by .
By [11, Lemma 1.1] in case (b) we can assume, without loss of generality, that the ambient manifold coincides with the euclidean space . In the rest of the paper we will therefore always make the following
is an integral current of dimension with bounded support and it satisfies one of the three conditions (a), (b) or (c) in Definition 0.1. Moreover
In case (a), is a submanifold of dimension , which is the graph of an entire function and satisfies the bounds
where is a positive (small) dimensional constant and .
In case (b) we assume that and that the semicalibrating form is .
In case (c) we assume that is supported in for some with , so that . We assume also that is (namely and we let be a smooth extension to the whole space of the function which describes in . We assume then that (1.1) holds, which is equivalent to the requirement that be sufficiently small.
In addition to Assumption 1.1 we assume the following:
and the tangent cone at is given by where ;
is irreducible in any neighborhood of in the following sense: it is not possible to find , non-zero integer rectifiable currents in with (in ), and .
In order to justify point (iii), observe that we can argue as in the proof of [13, Theorem 3.1]: assuming that in a certain neighborhood there is a decomposition as above, it follows from [13, Proposition 2.2] that both and fall in one of the classes of Definition 0.1. In turn this implies that and thus . We can then replace with either or . Let and argue similarly if it is not irreducible: obviously we can apply the argument above one more time and find a which satisfies all the requirements and has . This process must stop after at most steps: the final current is then necessarily irreducible.
1.2. Branching model
We next introduce an object which will play a key role in the rest of our work, because it is the basic local model of the singular behavior of a -dimensional area minimizing current: for each positive natural number we will denote by the flat Riemann surface which is a disk with a conical singularity, in the origin, of angle and radius . More precisely we have
is topologically an open -dimensional disk, which we identify with the topological space . For each in we consider the connected component of which contains . We then consider the smooth manifold given by the atlas
where is the function which gives the real and imaginary part of the first complex coordinate of a generic point of . On such smooth manifold we consider the following flat Riemannian metric: on each with the chart the metric tensor is the usual euclidean one . Such metric will be called the canonical flat metric and denoted by .
When we can extend smoothly the metric tensor to the origin and we obtain the usual euclidean -dimensional disk. For the metric tensor does not extend smoothly to , but we can nonetheless complete the induced geodesic distance on in a neighborhood of : for the distance to the origin will then correspond to . The resulting metric space is a well-known object in the literature, namely a flat Riemann surface with an isolated conical singularity at the origin (see for instance ). Note that for each and the set consists then of nonintersecting -dimensional disks, each of which is a geodesic ball of with radius and center for some with . We then denote each of them by and treat it as a standard disk in the euclidean -dimensional plane (which is correct from the metric point of view). We use however the same notation for the distance disk , namely for the set , although the latter is not isometric to the standard euclidean disk. Since this might be create some ambiguity, we will use the specification (or ) when referring to the standard disk in .
1.3. Admissible -branchings
When one of (or both) the parameters and are clear from the context, the corresponding subscript (or both) will be omitted. We will always treat each point of as an element of , mostly using and for the horizontal and vertical complex coordinates. Often will be identified with and thus the coordinate will be treated as a two-dimensional real vector, avoiding the more cumbersome notation .
Definition 1.4 (-branchings).
Let , , and . An admissible -smooth and -separated -branching in (shortly a -branching) is the graph
of a map satisfying the following assumptions. For some constant we have
is continuous, on and ;
for every with ;
If , then there is a positive constant such that
The map will be called the graphical parametrization of the -branching.
Any -branching as in the Definition above is an immersed disk in and can be given a natural structure as integer rectifiable current, which will be denoted by . For a map as in Definition 1.4 is a (single valued) map . Although the term branching is not appropriate in this case, the advantage of our setup is that will not be a special case in the induction statement of Theorem 1.8 below. Observe that for the map can be thought as a -valued map , setting for and . The notation and is then coherent with the corresponding objects defined in  for general -valued maps.
1.4. The inductive statement
Before coming to the key inductive statement, we need to introduce some more terminology.
Definition 1.5 (Horned Neighborhood).
Let be a -separated -branching. For every we define the horned neighborhood of to be
where is the constant in (1.3).
Definition 1.6 (Excess).
Given an -dimensional current in with finite mass, its excess in the ball and in the cylinder with respect to the -plane are
For cylinders we omit the third entry when , i.e. . In order to define the spherical excess we consider as in Assumption 1.1 and we say that optimizes the excess of in a ball if
In case (b)
In case (a) and (c) and
Note in particular that, in case (a) and (c), differs from the quantity defined in [10, Definition 1.1], where, although does not coincide with the ambient euclidean space, is allowed to vary among all planes, as in case (b). Thus a notation more consistent with that of  would be, in case (a) and (c), . However, the difference is a minor one and we prefer to keep our notation simpler.
Our main induction assumption is then the following
Assumption 1.7 (Inductive Assumption).
If , is -separated for some ; a choice of some is fixed also in the case , although in this case the separation condition is empty.
for some ;
There exist and a with the following property. Let and , let be the connected component of containing and let be the plane tangent to at the only point of the form which is contained in . Then
The main inductive step is then the following theorem, where we denote by the rescaled current , through the map .
Theorem 1.8 (Inductive statement).
1.5. Proof of Theorem 0.2
As already mentioned, without loss of generality we can assume that Assumption 1.1 holds, cf. [13, Lemma 1.1] (the bounds on and can be achieved by a simple scaling argument). Fix now a point in . Our aim is to show that is regular in a punctured neighborhood of . Without loss of generality we can assume that is the origin. Upon suitably decomposing in some neighborhood of we can easily assume that (Sep) in Assumption 1.7 holds, cf. the argument of Step 4 in the proof of [13, Theorem 3.1]. Thus, upon suitably rescaling and rotating we can assume that is the unique tangent cone to at , cf. [13, Theorem 3.1]. In fact, by [13, Theorem 3.1] satisfies Assumption 1.7 with : it suffices to chose as admissible smooth branching. If were not regular in any punctured neighborhood of , we could then apply Theorem 1.8 inductively to find a sequence of rescalings with which satisfy Assumption 1.7 with for some strictly increasing sequence of integers. It is however elementary that the density bounds from above (see for instance the argument of the next section leading to Lemma 2.1), which is a contradiction.
2. The branched center manifold
2.1. The overall approach to Theorem 1.8
From now on we fix satisfying Assumption 1.7. Observe that, without loss of generality, we are always free to replace by with sufficiently small (and ignore whatever portion falls outside ). Indeed we will do this several times. Hence, if we can prove that something holds in a sufficiently small neighborhood of , then we can assume, without loss of generality, that it holds on . For this reason we can assume that the constant in Definition 1.4 and Assumption 1.7 is as small as we want. In turns this implies that there is a well-defined orthogonal projection , which is a map.
By the constancy theorem, coincides with the current (again, we are assuming in Definition 1.4 sufficiently small), where . If were , condition (Dec) in Assumption 1.7 and a simple covering argument would imply that , where is a constant depending on and . In particular, when is sufficiently small, this would violate, by the monotonicity formula, the assumption . Thus . On the other hand condition (Dec) in Assumption 1.7 implies also that must be positive (again, provided is smaller than a geometric constant).
Now, recall from [13, Theorem 3.1] that the density is a positive integer at any . Moreover, the rescaled currents converge to . It is easy to see that the rescaled currents converge to and that converges to . We then conclude that .
We summarize these conclusions in the following lemma, where we also claim an additional important bound on the density of outside , which will be proved later.
Let and be as in Assumption 1.7 for some and sufficiently small . Then the nearest point projection is a well-defined map. In addition there is such that and the unique tangent cone to at is . Finally, after possibly rescaling , for every and, for every , each connected component of contains at least one point of .
Since we will assume during the rest of the paper that the above discussion applies, we summarize the relevant conclusions in the following
satisfies Assumption 1.7 for some and with sufficiently small. is an integer, and for all .
The overall plan to prove Theorem 1.8 is then the following:
We construct first a branched center manifold, i.e. a second admissible smooth branching on , and a corresponding -valued map defined on the normal bundle of , which approximates with a very high degree of accuracy (in particular more accurately than ) and whose average is very small;
Assuming that alternative (a) in Theorem 1.8 does not hold, we study the asymptotic behavior of around and use it to build a new admissible smooth branching on some where is a factor of : this map will then be the one sought in alternative (b) of Theorem 1.8 and a suitable rescaling of will lie in a horned neighborhood of its graph.
The first part of the program is the one achieved in this paper, whereas the second part will be completed in : in the latter paper we then give the proof of Theorem 1.8. Note that, when , from (BU) we will conclude that alternative (a) necessarily holds: this will be a simple corollary of the general case, but we observe that it could also be proved resorting to the classical Allard’s regularity theorem.
2.2. Smallness condition
In several occasions we will need that the ambient manifold is suitably flat and that the excess of the current is suitably small. This can, however, be easily achieved after scaling. More precisely we have the following
is a complete submanifold of ;
and, , is the graph of a map .
Under these assumptions, we denote by and the following quantities
We postpone the proof of this (simple) technical lemma to a later section.
2.3. Conformal parametrization
In order to carry on the plan outlined in the previous subsection, it is convenient to use parametrizations of -branchings which are not graphical but instead satisfy a suitable conformality property. To simplify our notation, the map will be simply denoted by .
If we remove the origin, any admissible -branching is a Riemannian submanifold of : this gives a Riemannian tensor (where denotes the euclidean metric on ) on the punctured disk . Note that in the difference between the metric tensor and the canonical flat metric can be estimated by (a constant times) : thus, as it happens for the canonical flat metric , when it is not possible to extend the metric to the origin. However, using well-known arguments in differential geometry, we can find a conformal map from onto a neighborhood of which maps the conical singularity of in the conical singularity of the -branching. In fact, we need the following accurate estimates for such a map.
Proposition 2.4 (Conformal parametrization).
Given an admissible -separated -smooth -branching with there exist a constant , a radius and functions and such that
is a homeomorphism of with a neighborhood of in ;
, with the estimates
is a conformal map with conformal factor , namely, if we denote by the ambient euclidean metric in and by the canonical euclidean metric of ,
The conformal factor satisfies
A map as in Proposition 2.4 will be called a conformal parametrization of an admissible -branching.
2.4. The center manifold and the approximation
We are finally ready to state the main theorem of this note.
Theorem 2.6 (Center Manifold Approximation).
Let be as in Assumptions 1.7 and 2.2 and assume in addition that the conclusions of Lemma 2.3 apply (in particular we might need to replace by for sufficiently small). Then there exist , , an admissible -separated -smooth -branching , a corresponding conformal parametrization and a -valued map with the following properties:
in particular, if we denote by the second fundamental form of ,
is orthogonal to the tangent plane, at , to .
If we define , then is contained in a suitable horned neighborhood of the -branching , where the orthogonal projection onto it is well-defined. Moreover, for every we have
If we define
then the following estimates hold for every :
Finally, if we set
The rest of this note is dedicated to prove the above theorem. We first outline how the center manifold is constructed. We then construct an approximating map taking values on its normal bundle. Finally we change coordinates using a conformal parametrization and prove the above theorem for the map .
3. Center manifold: the construction algorithm
3.1. Choice of some parameters and smallness of some other constants
As in  the construction of the center manifold involves several parameters. We start by choosing three of them which will appear as exponents of (two) lenghtscales in several estimates.
Let be as in Assumptions 1.7 and 2.2, assume in addition that the conclusions of Lemma 2.3 apply (we might therefore need to replace with for a sufficiently small ) and in particular recall the exponents and defined therein. We choose the positive exponents , and (in the given order) so that
(where is the constant of [11, Theorem 5.2] and in this paper we assume it is smaller than )
Having fixed , and we introduce five further parameters: and . We will impose several inequalities upon them, but following a very precise hierarchy, which ensures that all the conditions required in the remaining statements can be met. We will use the term “geometric” when such conditions depend only upon and , whereas we keep track of their dependence on and using the notation and so on. is always the last parameter to be chosen: it will be small depending upon all the other constants, but constants will never depend upon it.
Assumption 3.2 (Hierarchy of the parameters).
In all the statements of the paper
is larger than a geometric constant and is a natural number larger than ; one such condition is recurrent and we state it here:
is larger than ;
is larger than ;
is smaller than .
3.2. Whitney decomposition of
From now on we will use for , since the positive natural number is fixed for the rest of the paper. In this section we decompose in a suitable way. More precisely, a closed subset of will be called a dyadic square if it is a connected component of for some euclidean dyadic square with
, , and ;
Observe that is truly a square, both from the topological and the metric point of view. is the sidelength of both and . Note that consists then of distinct squares . is the center of the square . Each lying over will then contain a point , which is the center of . Depending upon the context we will then use rather than for the first (complex) component of the center of .
The family of all dyadic squares of defined above will be denoted by . We next consider, for , the dyadic closed annuli
Each dyadic square of is then contained in exactly one annulus and we define . Moreover for some . We then denote by the family of those dyadic squares such that and . Observe that, for each , is a covering of and that two elements of can only intersect at their boundaries. Moreover, any element of can intersect at most 8 other elements of . Finally, we set . Observe now that covers a punctured neighborhood of and that if , then
intersects at most other elements ;
If , then and is either a vertex or a side of the smallest among the two.
More in general if the intersection of two distinct elements and in has nonempty interior, then one is contained in the other: if we then say that is a descendant of and an ancestor of . If in addition , then we say that is a son of and is the father of . When and intersect only at their boundaries, we then say that and are adjacent.
Next, for each dyadic square we set . Note that, by our choice of , we have that:
|if and , then .||(3.5)|
In particular consists of connected components and we can select the one containing , which we will denote by . We will then denote by the current . According to Lemma 2.1, contains at least one point of : we select any such point and denote it by . Correspondingly we will denote by the ball .
The height of a current in a set with respect to a plane is given by
If we will then set . If , is as in Assumption 1.1 and (in the cases (a) and (c) of Definition 0.1), then where gives the minimal height among all for which (and such that in case (a) and (c) of Definition 0.1). Moreover, for such we say that it optimizes the excess and the height in .
We are now ready to define the dyadic decomposition of .
Definition 3.4 (Refining procedure).
We build inductively the families of squares and their subfamilies , and so on. First of all, we set for . For we use a double induction. Having defined for all and