The parabolic \(U(1)\)-Higgs equations and codimension-two mean curvature flows (Q6581839)

In this technical paper the authors carry out a detailed analysis of the weak coupling asymptotics of the gradient flow associated with the self-dual \(\mathrm{U}(1)\)-Higgs functional.\N\NLet \((M,g)\) be an \(n\)-dimensional closed Riemannian manifold, \(n\leq 3\), and \(L\) a complex Hermitian line bundle over it. Moreover take a smooth section \(u\) of \(L\) and a compatible connection \(\nabla\) on \(L\) with curvature \(F_\nabla\) and fix a real number \(\varepsilon >0\) (called the ``coupling constant''). Consider the following functional:\N\[\NE_\varepsilon (u,\nabla):=\int_M\left(\vert\nabla u\vert^2+\varepsilon^2\vert F_\nabla\vert^2+\frac{(1-\vert u\vert^2)^2}{4\varepsilon^2}\right) \mathrm{d}\:\mathrm{vol}_g=:\int_Me_\varepsilon (u,\nabla )\mathrm{d}\:\mathrm{vol}_g\:.\N\]\NThis so-called self-dual \(\mathrm{U}(1)\)-Higgs functional with its energy density \(e_\varepsilon (u,\nabla )\) can be regarded as the gauged and twisted (by a line bundle) version of the well-known Landau-Ginzburg functional from the theory of superconductivity. A \(1\)-parameter family of pairs \((u_t,\nabla_t)=(u_t,\nabla_0-i\alpha_t)\) with \(F_{\nabla_t}=-i\omega_t\) satisfying the parabolic system\N\[\N\left\{\begin{array}{cll} \partial_tu_t&=-\nabla_t^*\nabla_tu_t+\frac{1}{2\varepsilon^2}(1-\vert u_t\vert^2)u_t\\\N\partial_t\alpha_t&=-\mathrm{d}^*\omega_t+\frac{1}{\varepsilon^2}\langle iu_t\:,\:\nabla_tu_t\rangle \end{array}\right.\N\]\Nis formally the gradient flow of \(E_\varepsilon /2\) with respect to the natural \(L^2\)-product on the space of pairs \((u,\nabla )\). Note that the first equation of this system also can be viewed as the so-called Allen-Cahn regularization by the derivative of the quartic potential \(\frac{1}{4}(1-s^2)^2\) of the (twisted) mean curvature flow on manifolds. This remark is in order because the behaviour of mean curvature flows are in general very singular hence it demands appropriate regularization; the present article in part is devoted to the search of ``good'' regularizations (quite technical examples are the ``Brakke flow'' and the ``enhanced motion'', see below).\N\NThe authors' two main results are roughly as follows: \N\N(i) the energy density as a measure \(\mu_{t,\varepsilon}:=e_\varepsilon (u_t,\nabla_t)\mathrm{d}\:\mathrm{vol}_g\) converges for all \(t>0\) as a Radon measure to a measure \(\mu_t\) whenever \(\varepsilon\rightarrow 0\) such that these limit measures are so-called \textit{integer \((n-2)\)-rectifiable} and \(\{\mu_t\}_{t>0}\) defines a so-called Brakke flow (see Theorem 1.1 for a precise formulation; for the notion of Brakke flow see Definition 2.1 in the article); \N\N(ii) the limit measure \(\mu_0\) in the above family of measures can be characterized as the Poincaré-dual of \(c_1(L)\in H^2(M)\), and the family \(\{\mu_t\}_{t>0}\) induces a so-called enhanced motion (for the precise formulation of this technical statement see Theorem 1.4 and for the notion of an enhanced motion see Definition 2.2 in the article).