On the Kahler 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 Delta(g) on Kahler manifolds (M-n, 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 is an element of C-n : |z|(2) + Re Sigma(n)(j=1) Lambda(j)z(j)(2) < 1} a convex domain in C-n with n > 1 and Lambda(j) is an element of (-1, 1). Assuming g0 is the Kahler-Einstein metric on D(A), we prove that lambda(1)(Delta(g0)) = n(2) on (D(A), g0), but D(A) is not biholomorphic to the unit ball B-n when A not equal 0. Moreover, we prove that rho(z) = -e(u) is strictly plurisubharmonic in D(A) where u is the potential function for Kahler-Einstein metric on D(A). We also construct a complete Kahler metric g1 on D(A) with holomorphic bisectional curvature kappa(g1) >= -1 and lambda(1)(Delta(g1)) = n(2), but D(A) is not biholomorphic to B-n when A not equal 0.