Structural and optical properties of MOCVD AllnN epilayers

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Another remark on a result of Ding-Jost-Li-Wang

Let $(M,g)$ be a compact Riemann surface, $h$ be a positive smooth function on $M$. It is well known the functional $$J(u)=\frac{1}{2}\int_M|\nabla u|^2dv_g+8\pi\int_M udv_g-8\pi\log\int_Mhe^{u}dv_g$$ achieves its minimum under Ding-Jost-Li-Wang condition. This result was generalized to nonnegative $h$ by Yang and the author. Later, Sun and Zhu (arXiv:2012.12840) showed Ding-Jost-Li-Wang condition is also sufficient for $J$ achieves its minimum when $h$ changes sign, which was reproved later by Wang and Yang (J. Funct. Anal. 282: Paper No. 109449, 2022) and Li and Xu (Calc. Var. 61: Paper No. 143, 2022) respectively using flow approach. The aim of this note is to give a new proof of Sun and Zhu's result. Our proof is based on the variational method and the maximum principle.Comment: 13 pages. To appear on Proc. AM

Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay

We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang