Asymptotically cylindrical Calabi-Yau manifolds
Date
2015-01-01
Authors
Journal Title
Journal ISSN
Volume Title
Repository Usage Stats
views
downloads
Abstract
© 2015, International Press of Boston, Inc. All rights reserved. Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) 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¯ and a divisor D¯ ∈ |-KM¯| with torsion normal bundle such that M is biholomorphic to M¯ \ D¯, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M¯ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M¯, D¯) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M¯ \ D¯.
Type
Department
Description
Provenance
Subjects
Citation
Permalink
Collections
Scholars@Duke
Mark Haskins
My research concerns problems at the intersection between Differential Geometry and Partial Differential Equations, particularly special geometric structures that arise in the context of holonomy in Riemannian geometry.
In Fall 2024 I was the lead organizer of the program Special Geometric Structures and Analysis, at the Simons Laufer Mathematical Institute in Berkeley California. Many of the lectures given at the program are available to watch.
From 2016-2024 I was the Deputy Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics. My colleague here at Duke, Robert Bryant, was the Collaboration Director. During the course of the Collaboration we organized over 35 research meetings. Most of the lectures from these meetings are available to watch.
Currently, I am particularly interested in special types of 7-dimensional spaces called G2-holonomy manifolds, or G2-manifolds for short. These spaces also arise naturally in modern theoretical physics in the 11-dimensional theory known as M theory. To get from 11 dimensions down to 4 dimensions it is necessary to 'compactify' on a 7-dimensional space and to preserve the maximal degree of (super)symmetry this 7-dimensional space should have G2-holonomy. In fact, realistic 4-dimensional physics appears to demand singular G2-holonomy spaces and trying to construct compact singular G2-holonomy spaces is one of my current research projects.
Manifolds with special holonomy also come equipped with special submanifolds, called calibrated submanifolds, and special connections on auxiliary vector bundles, called generalised instantons. I am particuarly interested in associative and coassociative submanifolds in G2-holonomy spaces and special Lagrangian submanifolds in Calabi-Yau spaces. In the past I have also studied singular special Lagrangian n-folds.
Recently I have become interested in using geometric flow techniques to try to construct G2-holonomy manifolds. This has led me to study singularity formation in Laplacian flow and the structure of solitons in Laplacian flow. I have found new types of shrinking , steady and expanding solitons in Laplacian flow.
Unless otherwise indicated, scholarly articles published by Duke faculty members are made available here with a CC-BY-NC (Creative Commons Attribution Non-Commercial) license, as enabled by the Duke Open Access Policy. If you wish to use the materials in ways not already permitted under CC-BY-NC, please consult the copyright owner. Other materials are made available here through the author’s grant of a non-exclusive license to make their work openly accessible.