On the existence of nontrivial extremal metrics on complete noncompact surfaces
The author generalizes the existence of extremal metrics to the case of complete noncompact smooth surfaces which is served as a generalization of the classical uniformization theorem on a closed smooth surface to a complete noncompact surface. More precisely, for a complete noncompact smooth surface \(\Sigma\) with a fixed conformal class \([g_0]\), choose an exhaustion \(\{\Omega_i\}\) of compact domains in \(\Sigma\) with \(\Omega_j\subset\Omega_{j+1}\), \(\cup\Omega_j=\Sigma\). Then the 2-dimensional local Calabi flow \[ \left\{ \begin{aligned} &\frac{\partial\lambda}{\partial t}= \chi_i^4\left(\bigtriangleup R+\frac{1}{2}R^2- \frac{1}{2}r_i\right),\\ &\lambda_0(p)=\lambda(p,0)\in C^\infty(\Omega_i,d\mu_0),\quad g=e^{2\lambda}g_0,\quad \int_{\Omega_i}e^{2\lambda_0}d\mu_0=\int_{\Omega_i}d\mu_0 \end{aligned} \right. \] on \((\Sigma,[g_0])\) is considered, where \(\lambda:\Sigma\times[0,\infty)\rightarrow\mathbb R\), \(R\) is the scalar curvature with respect to the metric \(g\), \(r_i= \int_{\Omega_i}\chi_i^4(2\Delta R+2R^2) d\mu/ \int_{\Omega_i}\chi_i^4 d\mu,\) \(d\mu\) is the volume element associated with \(g\), and \(\chi_i\) is a nonnegative smooth compactly supported function on \(\Sigma\) with \(\chi_i\in C_0^\infty(\Omega_i)\), \(0\leq\chi_i\leq 1\). The key Bondy-mass type estimate is derived from the Bochner formula and bound of energy. Based on the Harnack-type theorem, it is shown the long-time existence and asymptotic convergence of a subsequence of solutions of such a flow on \((\Sigma, g_0)\) with \(R_0 \leq 0\) and \(R_0\) bounded from above by a negative constant on a ball. For its applications, this leads to the existence of extremal metrics on a complete noncompact smooth surface of finite topological type. In particular, there exists an extremal metric of nonconstant Gaussian curvature on \(\Sigma=\mathbb H^2\) or \(\mathbb R^2\).
2-dimensional local Calabi flow on a complete noncompact surface
initial value problem of fourth order parabolic equation
scientific article; zbMATH DE number 1889142