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On the Kähler manifolds with the largest infimum of spectrum of Laplace–Beltrami operators and sharp lower bound of Ricci or holomorphic bisectional curvatures

The paper studies the extremal or rigidity problem associated to the largest infimum of spectrum of Laplace-Beltrami operator Δg on Kähler manifolds (M, g) under the sharp lower bound assumption on either Ricci curvature or holomorphic bisectional curvature. The paper provides some conterexamples on those rigidity problems. In particular, we consider D(A) = {z ∈ C : |z|2 + Re ∑n j=1 Ajz 2 j < 1} a convex domain in C with n > 1 and Aj ∈ (−1, 1). Assuming g0 is the Kähler-Einstein metric on D(A), we prove that λ1(Δg0) = n 2 on (D(A), g0), but D(A) is not biholomorphic to the unit ball Bn when A = 0. Moreover, we prove that ρ(z) = −eu is strictly plurisubharmonic in D(A) where u is the potential function for Kähler-Einstein metric on D(A). We also construct a complete Kähler metric g1 on D(A) with holomorphic bisectional curvature Kg1 ≥ −1 and λ1(Δg1) = n, but D(A) is not biholomorphic to Bn when A = 0.