notation

notes/paper-notes/q4-Orientiation-Equivalence.md

@@ -75,7 +75,7 @@ minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by
 Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
 ```
 
-### Notation
+## Notation
 - $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
 - $`𝒬 := \{Q = s(n ⊗ n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
 - $`𝒬_2 := \{Q = s((n_1, n_2, 0) ⊗ (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
@@ -94,6 +94,7 @@ Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê
   $`⇒ 𝒬`$ is Riemannian manifold (p. 12 (=504) section "Orientability Issues"
 
   With this identification, $`P`$ is a covering map (details p. 12)
+- Jordan curve: continous, injective (non-self-intersecting) loop in $`ℝ^2`$ (Jordan curve theorem: it has an outside and an inside)
 - in chapter 3.1 firstly for continous $`Q`$ since that's standard from topology
   - *Theorem 1*: $`𝒬_2`$ (one-dim), orientable on boundary of holes, $`Q`$ continous $`⇒`$ orientable (long proof with a lot of fiddling!)
 - chapter 3.2 Theorem 2: $`Ω`$ simply connected, $`Q ∈ W^{1, p}`$, $`p \geq 2`$ (!) $`⇒`$ orientable with Sobolev-Seminorm estimate. Counterexample for $`p < 2`$