Abstract
We extend short-time existence and stability of the Dirichlet energy flow as proven in a previous article by the authors to a broader class of energy functionals. Furthermore, we derive some monotonely decreasing quantities for the Dirichlet energy flow and investigate an equation of soliton type. In particular, we show that nearly parallel G2-structures satisfy this soliton equation and study their infinitesimal soliton deformations.
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Alexandrov, B., Semmelmann, U.: Deformations of nearly parallel G2-structures. Preprint, arXiv: 1101.2143 (2011)
Besse A.: Einstein Manifolds. Springer, Berlin (1987)
Bourguignon, J.-P.: Déformations des métriques d’Einstein. Astérisque 80, 21–31, SMF (1980)
Bryant R.: Metrics with exceptional holonomy. Ann. Math. 126(3), 525–576 (1987)
Bryant, R.: Some remarks on G2-structures. In: Gökova Geometry/Topology Conference (GGT), Gökova, pp. 75–109 (2006)
Dai X., Wang X., Wei G.: On the stability of Riemannian manifolds with parallel spinors. Invent. Math. 161, 151–176 (2005)
DeTurck D., Kazdan J.: Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14(3), 249–260 (1981)
Ebin, D.: The moduli space of Riemannian metrics. In: Global Analysis, vol. 15 of Proceedings of Symposia in Pure Mathematics, pp. 11–40. AMS, Providence (1968)
Fernández M., Gray A.: Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 132, 19–45 (1982)
Friedrich T., Kath I., Moroianu A., Semmelmann U.: On nearly parallel G2-structures. J. Geom. Phys. 23, 259–286 (1997)
Gray A.: Weak holonomy groups. Math. Z. 123, 290–300 (1971)
Hitchin N.: Stable forms and special metrics. Contemp. Math. 288, 70–89 (2001)
Kazdan J.: Another proof of Bianchi’s identity in Riemannian geometry. Proc. AMS 81(2), 341–342 (1981)
Koiso N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983)
Tischler D.: On fibering certain foliated manifolds over S 1. Topology 9, 153–154 (1970)
Topping, P.: Lectures on the Ricci flow. LMS Lecture note series 325. Cambridge University Press, Cambridge (2006)
Weiss, H., Witt, F.: A heat flow for special metrics. Preprint, arXiv:0912.0421 (2010)
Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)
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Weiss, H., Witt, F. Energy functionals and soliton equations for G2-forms. Ann Glob Anal Geom 42, 585–610 (2012). https://doi.org/10.1007/s10455-012-9328-y
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DOI: https://doi.org/10.1007/s10455-012-9328-y