arXiv subjects

Mijia Lai

Publications and source records attributed to Mijia Lai.

23 records · Page 2Linked to original sources

arXiv2015

On convergence to a football

We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov. We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in $S^{3}$. Analytically, regarding the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.

math.DG↗

arXiv2013

The J-flow on Kahler surfaces: a boundary case

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of negative self-intersection on the surface. We discuss an application to the Mabuchi energy functional on Kahler surfaces with ample canonical bundle.

math.DG↗

arXiv2012

Convergence of general inverse $σ_k$-flow on Kähler manifolds with Calabi Ansatz

We study the convergence behavior of the general inverse $σ_k$-flow on Kähler manifolds with initial metrics satisfying the Calabi Ansatz. The limiting metrics can be either smooth or singular. In the latter case, interesting conic singularities along negatively self-intersected sub-varieties are formed as a result of partial blow-up.

math.DG↗

arXiv2012

On the geometric flows solving Kählerian inverse $σ_k$ equations

In this note, we extend our previous work on the inverse $σ_k$ problem. Inverse $σ_{k}$ problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.

math.DG↗

arXiv2010

On a class of fully nonlinear flow in Kähler geometry

In this paper, we study a class of fully nonlinear metric flow on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the corresponding Euler equation, which is fully nonlinear of Monge-Ampère type. As an application, we also discuss a complex Monge-Ampère type equation including terms of mixed degrees, which was first posed by Chen.

math.DG↗