On Calabi-Yau Manifolds with Maximal Volume Growth

We study complete noncompact Calabi-Yau manifolds with maximal volume growth. These manifolds arise as desingularizations of affine Calabi-Yau varieties, and as bubbles in the singularity formation of compact Kähler-Einstein manifolds. The rather explicit metric behavior of these manifolds enables us to understand the metric behavior of the seemingly impenetrable compact case. It is therefore of great importance to study the uniqueness/deformation problem of these manifolds and metrics. Our first result is that subquadratic harmonic functions on such manifolds must be pluriharmonic. This should be seen as the linear version of the uniqueness/deformation problem. Next, we show that if a ddbar-exact Calabi-Yau manifold with maximal volume growth has tangent cone at infinity which splits an Euclidean factor, then the metric is unique under subquadratic perturbation of the Kähler potential. Geometrically, this means that the Calabi-Yau metric is rigid under deformations that fix the tangent cone at infinity. It is therefore interesting to know if a quadratic perturbation of the Kähler potential, which should correspond to an automorphism of the tangent cone at infinity, really changes the metric. As a first step toward understanding this picture, we construct new Calabi-Yau metrics on C^3 with tangent cone at infinity given by C x A_2, and we show that these metrics are inequivalent in the sense that they are not related by isometries and scalings. This existence result provides the first example of families of Calabi-Yau metrics with the same tangent cone at infinity while the underlying complex structure is fixed.