Boundary expansions for minimal graphs in the hyperbolic space

Optimal regularity of minimal graphs in the hyperbolic space

“…Lemma 2.2 follows from Theorem 1.1 in [4] by taking ℓ = k = n + 1. In fact, c 2 , • • • , c n and c n+1,1 are coefficients for local terms and have explicit expressions in terms of ϕ.…”

“…We now discuss briefly the global regularity of solutions of (4.1) if the boundary ∂Ω is smooth. First, we recall a result established in [4].…”

“…We also note that n + 1 is the power of the first global term in the expansions of minimal graphs in the hyperbolic space. See [4]. However, if certain combinations of principal curvatures of the boundary do not vanish, then we can improve the global Hölder regularity.…”

“…The proof of Theorem 1.1 is based on the maximum principle and the recent work of Han and Jiang [4] on the boundary expansions for minimal graphs in the hyperbolic space.…”

Asymptotic expansions of complete Kähler-Einstein metrics with finite volume on quasi-projective manifolds

“…In this paper, we will give another proof of Rochon-Zhang's theorem by elementary tools, namely, besides rescaled interior Schauder estimates, the key tools are spectral decomposition for Laplacian operators on closed manifolds and the elementary theory of second order ordinary differential equations with constant coefficients. See also L. Andersson, P. Chruściel and H. Friedrich [1], H. Jian and X. Wang [11], and Han-Jiang [8] [9] for the ODE iteration method. Even though our result is not new, this elementary approach is interesting in itself, and the authors expect it to be useful in other geometric problems.…”