Journal of Differential Geometry

Mean curvature flows in manifolds of special holonomy

Chung-Jun Tsai and Mu-Tao Wang

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Abstract

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant–Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

Note

Supported in part by Taiwan MOST grants 102-2115-M-002-014-MY2 and 104-2115-M-002-007 (C.-J. Tsai). This material is based upon work supported by the National Science Foundation under Grant Numbers DMS 1405152 (Mu-Tao Wang). The second-named author is supported in part by a grant from the Simons Foundation (#305519 to Mu-Tao Wang). Part of this work was carried out when Mu-Tao Wang was visiting the National Center of Theoretical Sciences at National Taiwan University in Taipei, Taiwan.

Article information

Source
J. Differential Geom., Volume 108, Number 3 (2018), 531-569.

Dates
Received: 2 January 2016
First available in Project Euclid: 2 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1519959625

Digital Object Identifier
doi:10.4310/jdg/1519959625

Mathematical Reviews number (MathSciNet)
MR3770850

Zentralblatt MATH identifier
1385.53061

Citation

Tsai, Chung-Jun; Wang, Mu-Tao. Mean curvature flows in manifolds of special holonomy. J. Differential Geom. 108 (2018), no. 3, 531--569. doi:10.4310/jdg/1519959625. https://projecteuclid.org/euclid.jdg/1519959625


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