Existence

Let C₀ be a finite union of m surfaces meeting at the origin and separating R³ into k regions i.e., C₀ =Sm

i=1σ_i and Tm

i=1σ_i = {0}. We impose the following conditions on C₀:

(A₁) Each surface in C₀ is simply-connected and embedded in R³.

(A₂) Each region induced by C₀ is enclosed by at least two surfaces σ_i, 2 ≤ i ≤ m, which scales down homothetically to a limit curve Σ_i of finite length on S² i.e., lim_r→∞ ¹_r(σ_i∩ S^r) = Σ_i < ∞, 2 ≤ i ≤ m.

(A₃) The regular triple point is the one and only type of point junction where the curves in (ii) intersect on S². In other words, the permitted one dimensional junctions of surfaces are the regular triple junctions. The regular triple point is the intersection of three curves meeting at angle 120^◦ and the regular triple junction is the same aspect for surfaces.

Remark 3. Almgren [1] introduced (M, , δ) minimal set to model soap films, soap bubble clusters, and combination bubble-films which are the problems of partitioning space into regions of prescribed volumes in such a way as to minimize total interface area. Taylor [23] showed that (M, , δ) minimal surface in R³ have precisely the singularities observed in Plateau’s problem [20]. In accordance with this minimality around the singularities, we consider no point junctions except the regular triple point on S². Furthermore, the embeddedness of surface is for reasons of no interfaces created in the same region.

Remark 4. We give two conventions about the interface problem of immiscible fluids. Based on the topological classification of a network with two triple junctions in [14], we list locally the possible curves with no more than two triple points on S² in Table 6.1.

ˆ If a triple point occures, there exist three immiscible fluids concurring at a point. On the other hand, a interface cannot be created in the interior of a fluid.

ˆ When a curve touches the boundary, it means that this curve connects another triple point.

We exclude the cases (i) and (ii) because the former has no singular structures and the other violates the first convention. More specifically, the outside region in the case (ii) has an interface in its interior. Since the cases (iii) and (iv) have the structures like the Brakke spoon, we exclude these cases with the same reason. As the second convention is concerned,

the case (vii) has no regions enclosed by two surfaces. Hence, the lens-shaped and theta-shaped curves are the possible curves induced by a region endowed with two interfaces.

Define the metric

g(x) = exp^x21+x

22+x2 3

4 (dx²₁+ dx²₂+ dx²₃), (2.1) which is complete and negatively curved. In the following argument, we let Σ₁, · · · , Σ_m be the prescribed boundary curves of C₀∩ S².

Theorem 1 (Main Theorem). Let C₀ be a finite union of m surfaces meeting at the origin and separating R³ into k regions. Suppose C₀ satisfy the condition (A₁), (A₂), and (A₃) with its boundary curves Σ₁, · · · , Σ_m on S², then there exists a regular multiphase surface which is the connected self-expanding solution to mean curvature flow with Σ₁, · · · , Σ_m as its boundary and each surface of the solution is a minimal surface for the metric g.

Remark 5. Because we use the relation in the next chapter to prove Theorem 1, the regular connected self-expanders to mean curvature flow in R³ have a one-to-one correspondence with the possibly disconnected regular multiphase surfaces in B₁³(0). The “one-to-one cor-respondence” in Theorem 1 depends on the following idea: given a deformation from metric g to the standard hyperbolic metric in the class of complete negatively curved metrics, the regular multiphase surface found with respect to g should be continuously deformable to the regular multiphase surface with respect to the standard hyperbolic metric. Similarly, the regular multiphase surface in H³ should continuously produce a similar structure for g.

Because the problem regarding the deformation of the metric is difficult, we do not focus on this here, but simply use the correspondence among similar structures in these metrics.

Chapter 3

Relation between Poincar´ e ball model and Euclidean space

Definition 3. The Poincar´e ball model of hyperbolic space is the open submanifold B₁³(0) := {x = (x1, x2, x3) ∈ R³ : |x| < 1}

with the Riemannian metric

g_B = 4dx · dx (1 − |x|²)². Besides, we introduce a hyperbolic space

H³ := {x = (x₁, x₂, x₃, x₄) ∈ R^3,1 : hx, xi = x²₁ + x²₂+ x²₃− x²₄ = −1, x₄ ≥ 1}

to define a map between B₁³(0) and H³.

Definition 4. Hyperbolic stereographic projection is the map S : H³ → B³₁(0), S((x₁, x₂, x₃, x₄)) = 1

1 + x₄(x₁, x₂, x₃) := y, Remark 6. We differentiate S and then obtain

dS_x= dx⁰ 1 + x4

− x⁰

(1 + x4)²dx₄. We then obtain

S^∗g_B = 4dy · dy

(1 − |y|²)² = hdx, dxi Therefore, S is isometric onto (B₁³(0), g_B) and conformal to R³.

In the following arguments, we will identify each unit tangent sphere of B₁³(0) with

∂B³₁(0), and then define a conformal diffeomorphism between S²(∞) and ∂B₁³(0) i.e., ∂B₁³(0) is the sphere at infinity of the Poincar´e ball model.

If x ∈ B₁³(0) and S²(x) ⊂ R³ is the unit sphere in the tangent space at x, we define a boundary point from an interior point connected by the geodesic in B₁³(0) as below

B : S²(x) → ∂B₁³(0) B(u) = lim

t→∞γ_u(t)

where γ_u is the geodesic in B₁³(0) starting at x in the direction u.

Proposition 1. Consider a point n = n⁰ + n⁴e₄ in a subset {x ∈ R^3,1 : hx, xi > 0}. Define the map

S∞: S²(∞) → ∂B₁³(0), v = S∞[n] = 1 n₄n⁰. It is a conformal diffeomorphism with its inverse map S_∞⁻¹(v) = [v + e₄]

Remark 7. These two maps S and S∞ define a bijective map from H³∪ S²(∞) to a closed unit ball in R³. More discussions about Minkowski space and the proof of conformal diffeo-morphism between S²(∞) and ∂B³ can refer to the chapter 6 in [10] but we here just use these facts. Furthermore, we can use the Figure 6.2 to know how R³ is stereographically compactified onto B₁³(0). Because these maps are all conformal, the planes passing through the origin are invariant and the angles between two clustering surfaces are fixed under the stereographic compactification. More specifically, given a multiphase surface in R³, we can find a multiphase surface in Poincar´e ball B₁³(0) whose singular structures are one-to-one correspondence and induce the same (ideal) boundary curves.

Chapter 4

Self-expanding Solutions to the Multiphase Mean Curvature Flow

4.1 Smooth Case

Definition 5. (Mean Curvature Flow)

A family of smoothly embedded hypersurfaces (α_t)_t∈I in Rⁿ⁺¹ moves according to the mean curvature if

∂x

∂t = ~H(x) (4.1)

for x ∈ αt and t ∈ I, where I ⊂ R² is an open interval. Here, ~H(x) is the mean curvature vector at x ∈ α_t.

Theorem 2. Let (α_t)_t∈I be a family of smoothly embedded hypersurfaces in Rⁿ⁺¹. If (α_t)_t∈I is a self-similar solution to (4.1), then

H(x) =~ Cx^⊥

2λ²(t), (4.2)

where λ(t) = p1 + C(t − t0) for x ∈ α_t as long as 1 + C(t − t₀) > 0. This describes expanding self-similar solutions about 0 for C > 0 and contracting self-similar solutions about 0 for C < 0.