Lemma 1. If Γ is the regular multiphase surface in R3, then its (ideal) boundary is the regular triple points connected by curves. More specifically, only the surfaces and the triple junctions can connect to infinity.
Proof. As Γ is the regular multiphase surface, it contains the regular triple junctions and the regular quadruple points. Away from these singular structures, Γ is a finite union of disjoint surfaces that only induce curves on S2.
Around the singularities, we first consider the regular triple junction and then the regular quadruple point. The possible structures connecting to infinity are the end of regular triple junction and the 1-dimensional subset of regular triple junction. The end of regular triple junction induces a triple point on S2, but the 1-dimensional subset of regular triple junction cannot attach to S2. If this 1-dimensional subset wholly connects to infinity then we get a bi-junction on S2; in other words, two surfaces intersect along a curve with angle 120◦. Using the relation between the Poincar´e ball model and Euclidean space, the angle between two surfaces with the same bounday on S2 is 0◦ in the Poincar´e ball model, so the degeneracy of regular triple junction into bi-junction cannot happen.
We next consider the regular quadruple point. Because it is the intersection of four regular triple junctions, the possible structures connecting to infinity are almost the same in the triple junction case, but with one more structure. This one can be imagined by wholly degenerating one of the regular triple junctions into a point on S2 until the quadruple point coincides with the triple point on S2. We introduce the following definition to prove the nonexistence of this case.
Definition 7. A steradian is defined in R3 as the solid angle subtended at the center of a unit sphere by a unit area on its surface. For a sphere of radius r, any portion of its surface with area r2 subtends one steradian.
At the regular quadruple point x, we use the point x accompanied with four unit vectors τ1(x), τ2(x), τ3(x), and τ4(x) to represent the quadruple point with tetrahedral structure i.e., (x, τ1(x), τ2(x), τ3(x), τ4(x)) where τi are the unit tangent vector fields along their regular
triple junctions. Without loss of generality, we argue the degeneracy in τ1 direction. Sup-pose (x, τ1(x), τ2(x), τ3(x), τ4(x)) degenerates into (y, 0, τ2(y), τ3(y), τ4(y)). As the tetrahedral structure is connected, we can find three points on S2 in the directions of (τ2(y) + τ3(y)), (τ2(y) + τ4(y)), and (τ3(y) + τ4(y)). We connect these three points to y with Poincar´e arcs and denote the radii of the Poincar´e arcs by r1, r2, and r3. Now, we use r = min{r1, r2, r3} to construct a pseudosphere at y. This pseudosphere prevents a regular quadruple point x from degenerating into y on S2 because the steradian of this pseudosphere at y is zero.
Hence, we conclude that only the end of the triple junction and the surfaces can induce the boundaries on S2.
Remark 8. The steradian at the quadruple point with a tetrahedral structure has a lower bound of cos−1(√1
3). Because the tetrahedron has an inscribed spherical cone with angle cos−1(√1
3), this inscribed spherical cone serves as a barrier against degeneracy.
Lemma 2. Suppose Γ is a area-minimizing multiphase surface in R3 and all triple junctions in Γ connect to each other by surfaces. Let τ : U ⊂ R2 → R3 be a regular triple junction in Γ connecting to infinity. If τ locally induces its (ideal) boundary, a triple point with three curves, on S2, then τ is at infinity asymptotically closed to a half-line l passing through the origin i.e.,
dH(τ (U ) ∩ (R3\Br(0)), l(x) ∩ (R3\Br(0)) → 0 for r → ∞.
In other words, a triple point induced by τ is in fact a regular triple point on S2.
Proof. Let α ⊂ Γ be a surface connecting two regular triple junctions which connect to infinity, it induces a boundary curve Σ on S2 with triple points as its endpoints. We choose
M1 → S2
as a smooth immersion of simple closed oriented curve containing Σ. Before continuing the arguments, we need to check Σ is a simple curve so that the choice of M1 is feasible: Suppose Σ is not a simple curve on S2, we choose an open neighborhood U in a Poincar´e ball such that U containing a self-intersection or corner of Σ. Besides, we can find in U a four-junction or bi-junction generating Σ on S2. After taking the inverse hyperbolic stereographic projection on U , we obtain a four-junction or bi-junction locally existing in α ⊂ R3. However, the
results in [23, 17] show that the 1-dimensional area-minimizing singular structure in R3 is the regular triple junction. If the surface α ⊂ Γ induces a non-simple curve Σ on S2, then Γ cannot be the area-minimizer in R3.
By applying theorem 3 in [2] to this simple closed oriented curve M1, we obtain a com-plete area-minimizing locally integral 2-current σM1 in B3 with asymptotic boundary M1. Then using the remark of theorem 3 in [2], we can determine that σM1 is a smooth and properly embedded complete hypersurface in B3. Because σM1 is a complete surface of finite topological type and with well-defined limiting normal planes on its ends, the inverse image σ of the projection on σM1 is also a complete surface of finite topological type and with well-defined normal planes on each end. Applying theorem 3 in [11] to the inverse image σ, it looks from infinity like a plane passing through the origin. Because Σ and σ induce the same boundary α ∩ M1 on S2, Σ is asymptotic to a plane passing through the origin in R3. Hence, the asymptote of the regular triple junction is a line emanating from the origin, i.e., the regular triple junction intersects S2 orthogonally.
Remark 9. In Lemma 2, we impose the balancing condition on the triple junction τ :=T3 i=1σi
without further assumptions on the surfaces σi away from the singularity τ , so the regular triple junction τ may not have the well-defined limiting normal planes on its whole ends.
Nevertheless, the balancing condition on τ locally guarantees the normal plane of each surface σi near τ is well-defined; namely, the (ideal) boundary of σi behaves well near a triple point induced by τ on S2. Therefore, we may not have a simple closed oriented curve containing the whole (ideal) boundary of σi but a circle on S2 containing a triple point and a portion of the (ideal) boundary of σi. Following the arguments in the proof of Lemma 2, we know that a portion of the surface σi near τ approaches to a flat plane through the origin. Hence, the triple junction τ is the intersection of three flat planes through the origin and then we finish the proof. If each surface σi behaves well at infinity, for instance,“regular at infinity” which is given as a definition in [21], we can choose a great circle on S2 as simple closed oriented curve containing the (ideal) boundary of σi
Chapter 5
Proof of Main Theorem
5.1 Flat 2-dimensional Chains F
2(R
3, Z
k+1)
5.1.1 Introduction
Let C0 be a finite union of m surfaces meeting at the origin and separating R3 into k regions.
Each surface or region induced by C0 satisfies the conditions (A1), (A2), and (A3) in the section 2.3. The main theorem proves the existence of a regular multiphase surface Γ in R3 where each surface α is a minimal surface for the metric g with C0 as an initial condition.
Specifically, Γ spans tha same boundary as C0 on S2. When m equals to two or three, the existence and uniqueness are clear. However, the existence and uniqueness become more complicated as m is greater than three. For instance, given the four boundary curves Σ1, Σ2, Σ3, and Σ4, there exist two surfaces αi connecting Σi and Σi+2, i = 1, 2, but there exists no surface σi with the triple junction or quadruple point. Hence, the main argument here is to instead prove the existence of at least one connected regular multiphase surface that induces the specified boundary curves.
We consider the minimizing problem in the class F2(R3, Zk+1) of flat 2-dimensional chains in R3 with coefficients in Zk+1 and the norm on each nonzero element in Zk+1 equals to one.
For a complete discussion of flat chain and multiplicity, refer to [24, 18]. In the following, we briefly describe how the flat chains in F2(R3, Zk+1) representing the multiphase surface problem in R3.
The space of flat 2-dimensional chains is the completion of the space of polygons with respect to the flat norm.
The interior of each region in R3 is assigned to a nonzero element in Zk+1 and the zero element in Zk+1 represents the points not belonging to the interior of any regions in R3.
The norm equals to 1 on each nonzero coefficient aα ∈ Zk+1, which is part of the definition of the size of a flat chain, i.e.,
S(Σ) =
where M(α) is the mass with respect to g. The mass is equal to the area when α is the surface of class C1.
5.1.2 Example
Let Σ be a multiphase surface mentioned in Example 2. Using the flat 2-dimensional chains in F2(R3, Z5), we give a representation of Σ as below. Let R, G, B, Y, C, and P be the surfaces defined in Example 2, we call (·, ·, ·) the region enclosed by three surfaces and define a norm on Z5 by
The identities eij represent the surfaces separating the regions i and j where eij equals to eji.
We assign an element [j − i] ∈ Z5 to the identities eij and eji.
According to the above arguments, the correspondences between the surfaces and elements in Z5 are given by
R = [4] − [1] = [3] G = [4] − [3] = [1] B = [3] − [1] = [2]
Y = [4] − [2] = [2] C = [3] − [2] = [1] P = [2] − [1] = [1].
Althrough there may be some surfaces assigned literally to the same element in Z5, it rep-resents the different things. Nevertheless, it is independent of the assignments that the size of Σ is always defined by
S(Σ) = M(R) + M(G) + M(B) + M(Y ) + M(C) + M(P ).