Given x = (x1, x2, x3) ∈ Γt ⊂ R3, we consider the entropy-type functional with respect to
For the general and related arguments about the entropy-type functional, refer to [4, 7, 8].
Lemma 4. If a multiphase surface Γ is a regular self-expanding solution to the mean cur-vature flow, then it is a critical point of the entropy-type functional (6.1).
Proof. Consider a surface α ⊂ Γ that is smoothly embedded in R3. We define a smooth family of diffeomorphisms {Φs}s∈[0,1] on an open neighborhood U ⊂ α.
Fix s ∈ [0, 1].
The area functional is
is the Jacobian of the metric g, J Φs is the Jacobian of the Euclidean metric, and H2 is the 2-dimensional Hausdorff measure. where ~T is the unit tangent vector at point x.
If a variational vector ~X has a compact support on each surface, then the first integral in (6.2) obviously equals to zero and the second integral vanishes since α is a self-expanding solutions to the mean curvature flow i.e., ~H = 2λ(t)Cx⊥ for every point x ∈ α. If ~X is compactly supported on a neighborhood of each triple junction (αi(0, t))3i=1, then the second integral vanishes for the same reason mentioned above and the first one equals to zero which is guaranteed by the balancing condition i.e.,
3
X
i=1
νi = 0 where νi is an unit normal vector to the surface αi.
If ~X is compactly supported around a regular quadruple point (αi(0, t))6i=1 = O, then the skewness in Definition 6 or the balancing condition in [13] i.e.,
νij + νjk + νkl+ νli = 0 1 ≤ i 6= j 6= k 6= l ≤ 4
is enough to make the first integral zero and the second one vanishes for the self-similarity of each surface around the point.
Remark 11. In one dimensional case, the balancing condition at the triple point provides a sufficient relation on any two curves clustering at the triple point. In two dimensional case, we take Figure 6.1 for example. When we consider the surface G, the balancing condition on the triple junction curves enclosing the surface G offers the direct information of the sur-faces R, B, Y , and C. After we finish the following argument, the skewness property at the quadruple point provides a direct connection between the surfaces G and P and a interaction between triple junctions and subregions around a quadruple point.
Consider a region near the quadruple point O, there exist three half-planes and triple junctions enclosing this region. See Figure 6.3 for example, we let u, v, and w be the outward-pointing unit tangent vectors along the triple-junctions and denote the angles between any two of them by
θ1 = ∠(u, v) θ2 = ∠(v, w) θ3 = ∠(w, u).
With the same argument in the proof of Lemma 3, we impose the balancing condition on the triple junction which is the intersection of half-planes determined by the unit normal vectors ˜νuv, ˜νvw, and ˜νwu. The balancing condition shows that
˜
νuv+ ˜νvw+ ˜νwu= 0.
Applying the triple product expansion of cross product to the above equation, we derive the following equalities
cos2θ1 = cos2θ2 = cos2θ3.
Using the balancing condition again, it forces that all angles are all equal and they belong to one of the intervals (0,π2) or (π2, π). Because the same argument is valid for the other regions, we can conclude that the angles between any two unit tangent vectors are all the same i.e.,
∠(u, v) = ∠(v, w) = ∠(w, u) = ∠(z, u) = ∠(z, v) = ∠(z, w) := θ. (6.3)
Since the region around O is separated into four equal subregions, the angle θ in (6.3) must belong to an interval (π2, π). In addition, each subregion has the regular spherical triangle as its boundary on S2. These results implies that the quadruple point O clustered by the triple junctions determined by u, v, w, and z is a regular quadruple point.
Figure 6.1: Six half-planes
(i) Smooth (ii) Brakke spoon (iii) Island
(iv) Eyeglasses
(v) Lens (vi) Theta (vii) Tree
Table 6.1: Classification.
Figure 6.2: The stereographic compactification
(i) Pseudosphere at the origin.
(ii) The inscribed tangent cone at some point above the origin.
Table 6.2: Schematical pictures
Figure 6.3:
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