2015
DOI: 10.1007/s00208-015-1315-8
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A spinorial energy functional: critical points and gradient flow

Abstract: Abstract. Let M be a compact spin manifold. On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M ≥ 3, are precisely the pairs (g, ϕ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor ϕ. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

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Cited by 24 publications
(57 citation statements)
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“…The Laplacian flow for co-closed G 2 -structures was introduced by Karigiannis-McKay-Tsui in [KMT12] and a modified co-flow was studied by Grigorian [Gri13]. An approach via gradient flow of energy-type functionals was introduced by Weiss-Witt [WW12] and Ammann-Weiss-Witt in [AWW16].…”
Section: Introductionmentioning
confidence: 99%
“…The Laplacian flow for co-closed G 2 -structures was introduced by Karigiannis-McKay-Tsui in [KMT12] and a modified co-flow was studied by Grigorian [Gri13]. An approach via gradient flow of energy-type functionals was introduced by Weiss-Witt [WW12] and Ammann-Weiss-Witt in [AWW16].…”
Section: Introductionmentioning
confidence: 99%
“…This energy functional has several important symmetries. We restrict here to the invariance under so-called spin diffeomorphisms and refer for the other symmetries to [1]. A spin diffeomorphism is a diffeomorphism f : M → M for which the induced map df : P → P lifts to the spin structureP .…”
Section: Introductionmentioning
confidence: 99%
“…Let M be a compact spin manifold and N the union of all pairs (g, ϕ) where g is a Riemannian metric on M and ϕ ∈ Γ(Σ(M, g)) is a spinor of the spin manifold (M, g) whose pointwise norm is constant and equal to 1. The spinorial energy functional E, introduced in [2], is defined by…”
Section: Introductionmentioning
confidence: 99%
“…Then, if the fibers are sufficiently short (i.e. ε is small enough), the spinor flow converges to a 2-dimensional sphere in infinite time.This theorem can be seen as a special case of the conjecture that S 1 -principal bundles with suitable Riemannian metrics and sufficiently short fibers together with S 1 -invariant spinors collapse to the base manifold under the spinor flow.In [2] it was observed that the volume-normalized standard metric on S 3 together with a Killing spinor is a critical point of the volume-normalized spinor flow. It is not clear whether this critical point is stable.…”
mentioning
confidence: 99%
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