Persistence of superconductivity in thin shells beyond \(H_{c_1}\) (Q2809259)

The authors consider the Ginzburg-Landau functional \(G_{M,k} : H^1(M;\mathbb{C}) \to \mathbb{R}^+\), \(G_{M,k}(\psi)=\int_{M}^{}[|(\nabla_M-ihA)\psi|^2+\frac{k^2}{2}(|\psi|^2-1)^2]dH^2_M(x)\), where \(M\) is a compact surface homeomorphic to \(\mathbb{S}^2\), embedded in \(\mathbb{R}^3\), \(k,h > 0\), and \(A\) is a vector field on \(M\). More specifically, they consider the regime \(k\to +\infty\) when the function \(H\) defined via the relation \(H dH^2_M = dA\) (viewing \(A\) as a 1-form) changes sign (in a nondegenerate sense). It is known that there exists a first critical number \(H_c\sim C \ln k\) as \(k\to \infty\) for some \(C>0\) such that vortices appear in the minimizers when \(h\geq H_c\). Assuming that there exists \(b > 0\) such that \(\lim_{k\to \infty} \frac{\ln k}{h} = b\), they show that the region of persistence of superconductivity in the \(b \to 0\) limit is \(O(b^{1/3})\) close to the zero set of \(H\).