Deformations of multivalued harmonic functions

Donaldson, Simon

doi:10.48550/arxiv.1912.08274

Cited by 6 publications

(34 citation statements)

References 7 publications

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“…Part 1: This part of the proof explains why the infimums of E on T and T * are the same. To this end, fix once and for all a smooth, non-increasing function on R to be denoted by χ that equals 1 for t < 1 4 and equals 0 for t ≥ 3 4 . With χ in hand, fix a positive number to be called ρ with its upper bound being 1 1000 times the minimum of the distances between the points in Z.…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

“…This function is equal to 1 where the distance to p is less than 1 2 ρ and it is equal to 0 where the distance to p is greater than ρ. Note that its derivative has support only in the annulus where the distance to p is between 1 2 ρ and ρ; and that the norm of this derivative is bounded by a ρ-independent constant times 1 ρ . Next, given a positive number ǫ < ρ, define a second function using χ to be denoted by µ ǫ by the rule µ ǫ (•) ≡ χ 2 1 − dist(•,p) ǫ .…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

“…As explained by Takahashi (see [5] and [6]) and elaborated on by Donaldson [1], there is a well behaved moduli space of Z/2 harmonic 1-forms and spinors near any given (Z, I, v or s) in the case when Z is a C 1 embedded submanifold in X. But, it is not known a priori that this is always the case.…”

Section: Introductionmentioning

confidence: 98%

“…Set f to denote a section of I defined over S 2 − Z that obeys ∆ f = −E f and also the conditions in (1.3). Now let v denote the 1-form d(|x| α π * f) with α denoting 1 2…”

Section: Introductionmentioning

confidence: 99%

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Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

Taubes,

2020

Preprint

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We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a Z/2 harmonic 1-form or spinor in 3dimensions near a singular point in its zero loci. The local models are Z/2 harmonic 1-forms or spinors on R 3 that are homogeneous with respect to the rescaling of R 3 with their zero loci consisting of 4 or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example, and to those of a centered octahedron and a centered icosahedron in two others.

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Section: Proof Of Proposition 21mentioning

confidence: 99%

Section: Proof Of Proposition 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…Set f to denote a section of I defined over S 2 − Z that obeys ∆ f = −E f and also the conditions in (1.3). Now let v denote the 1-form d(|x| α π * f) with α denoting 1 2…”

Section: Introductionmentioning

confidence: 99%

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Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

Taubes,

2020

Preprint

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“…These examples are all Lagrangian surgeries obtained from Morse 1-forms on Q and our results have applications to Morse-Novikov theory. In comparison, Donaldson and He [16,29,30] have a new method of constructing branched special Lagrangians from Z 2 harmonic 1-forms.…”

Section: Introductionmentioning

confidence: 99%

Nearby Special Lagrangians

Abouzaid¹,

Imagi²

2021

Preprint

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Let X be a Calabi-Yau manifold and Q ⊂ X a closed connected embedded special Lagrangian. We use a combination of methods from Floer theory and from geometric analysis, to study those closed immersed special Lagrangians L which are C 0 -close to Q and unobstructed in the sense of Floer theory. Our results are easiest to state under the assumption that the fundamental group π1Q has no nonabelian free subgroup: we prove that every such nearby special Lagrangian L as above is unbranched (that is, the projection L → Q is a covering map); moreover, if π1Q is abelian then L is a union of smooth perturbations of Q ⊂ X. In comparison with Tsai-Wang's results, their Riemannian geometry assumptions on Q ⊂ X are replaced by the purely topological ones on π1Q, at the cost of introducing the Floer theory assumption on L.The π1Q conditions, the Floer theory condition and the special Lagrangian condition are all essential to the statement above, as we show by constructing counterexamples; they are branched C 0 -nearby Lagranigans of Maslov class zero, which fail to satisfy the three kinds of conditions respectively.Our results have also applications to Morse-Novikov theory.

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Bryant-Salamon $\mathrm{G}_2$ manifolds and coassociative fibrations

Karigiannis,

Lotay

2020

Preprint

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Bryant-Salamon constructed three 1-parameter families of complete manifolds with holonomy G2 which are asymptotically conical to a holonomy G2 cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on R 7 : the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of C 3 with a line, and the Harvey-Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere S 4 , and the cone on CP 3 , whose smooth fibres are T * S 2 , and whose singular fibres are R 4 /{±1}. We relate these fibrations to hypersymplectic geometry, Donaldson's work on Kovalev-Lefschetz fibrations, harmonic 1-forms and the Joyce-Karigiannis construction of holonomy G2 manifolds, and we construct vanishing cycles and associative "thimbles" for these fibrations. PreliminariesIn this section we review various preliminary results on G 2 manifolds, Riemannian conifolds, calibrated submanifolds, multimoment maps, coassociative fibrations, and hypersymplectic structures.

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Deformations of multivalued harmonic functions

Cited by 6 publications

References 7 publications

Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

Nearby Special Lagrangians

Bryant-Salamon $\mathrm{G}_2$ manifolds and coassociative fibrations

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