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

Abstract

We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.

Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

“…Finally, we obtain the following self-improvement of Lemma 3.4. This is similar to [11, Lemma 2.5], although in [11] this improvement only applies to the θ-derivatives because the x α , y α -directions have length O(1), and here we already have a better estimate for the θ-derivatives. Lemma 3.6.…”

“…We expand u according to powers of x, which results in a formal power series solution differing from u by O(x ∞ ). This is similar to asymptotic expansion arguments as in [11,15,17,18,22], with our proof being closest in spirit to [11]. The coefficients of our expansion are actually constants, the first coefficient determines all the others, and the formal series converges to −(n + 1) log(1 + cx) for some c ∈ R. We think of this expression as the "tangent cone" of u.…”

“…However, during the proof of Theorem 3.1, it is helpful to work with a more specific O notation, which we will now define; cf. [11,Thm 1.1].…”

“…Thus, it suffices to choose µ ∈ (ε, 1). Sharp higher-order bounds for u are now almost immediate as in [11,Lemma 2.4]. Here we use the flatness of D although it seems likely that this can be avoided using the methods of [10].…”

“…Asymptotic expansion methods [11,15,17,18,22] stop at the statement that the difference between u and some formal (in this case, convergent) power series in x is O(x ∞ ). A new point here is that we go further and prove an explicit estimate, (1.1), for this difference, which is actually completely sharp as evidenced by the next generation of solutions to the homogeneous linearized PDE.…”