ASYMPTOTICALLY CYLINDRICAL CALABI-YAU MANIFOLDS

作者：Haskins Mark^*; Hein Hans Joachim; Nordstroem Johannes

来源：Journal of Differential Geometry, 2015, 101(2): 213-265.

DOI：10.4310/jdg/1442364651

Let M be a complete Ricci-flat Kah ler manifold with one end and assume that this end converges at an exponential rate to [0, infinity) x X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M) = SU(n), where n is the complex dimension of M. If n > 2 we then show that there exists a projective orbifold (M) over bar and a divisor (D) over bar is an element of vertical bar-K-(M) over bar vertical bar with torsion normal bundle such that M is biholomorphic to (M) over bar\(D) over bar), thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where (M) over bar is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair ((M) over bar, (D) over bar) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on (M) over bar\(D) over bar.

• 出版日期2015-10