Publications
24 results found
Foscolo L, Haskins M, 2017, New G(2)-holonomy cones and exotic nearly Kahler structures on S-6 and S-3 x S-3, Annals of Mathematics, Vol: 185, Pages: 59-130, ISSN: 0003-486X
There is a rich theory of so-called (strict) nearly K¨ahler manifolds,almost-Hermitian manifolds generalising the famous almost complex structureon the 6-sphere induced by octonionic multiplication. Nearly K¨ahler6-manifolds play a distinguished role both in the general structure theoryand also because of their connection with singular spaces with holonomygroup the compact exceptional Lie group G2: the metric cone over a Riemannian6-manifold M has holonomy contained in G2 if and only if M isa nearly K¨ahler 6-manifold.A central problem in the field has been the absence of any completeinhomogeneous examples. We prove the existence of the first completeinhomogeneous nearly K¨ahler 6-manifolds by proving the existence of atleast one cohomogeneity one nearly K¨ahler structure on the 6-sphere andon the product of a pair of 3-spheres. We conjecture that these are theonly simply connected (inhomogeneous) cohomogeneity one nearly K¨ahlerstructures in six dimensions.
Haskins M, Hein H-J, Nordstroem J, 2015, ASYMPTOTICALLY CYLINDRICAL CALABI-YAU MANIFOLDS, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 101, Pages: 213-265, ISSN: 0022-040X
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- Citations: 25
Corti A, Haskins M, Nordstroem J, et al., 2015, G(2)-manifold and associative submanifolds via semi-fano 3-folds, Duke Mathematical Journal, Vol: 164, Pages: 1971-2092, ISSN: 0012-7094
We construct many new topological types of compact G2G2-manifolds, that is, Riemannian 77-manifolds with holonomy group G2G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 33-folds built from semi-Fano 33-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 77-manifolds completely; we find that many 22-connected 77-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G2G2-metrics. Many of the G2G2-manifolds we construct contain compact rigid associative 33-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G2G2-metrics. By varying the semi-Fanos used to build different G2G2-metrics on the same 77-manifold we can change the number of rigid associative 33-folds we produce.
Degeratu A, Haskins M, Weiß H, 2015, Mini-Workshop: Singularities in $\mathrm G_2$-geometry, Oberwolfach Reports, Vol: 12, Pages: 449-488, ISSN: 1660-8933
Corti A, Haskins M, Nordström J, et al., 2013, Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds, Geometry & Topology, Vol: 17, Pages: 1955-2059, ISSN: 1465-3060
We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–foldsstarting with (almost) any deformation family of smooth weak Fano 3–folds. Thisallow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previouslyonly a few hundred ACyl Calabi–Yau 3–folds were known. We pay particularattention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano3–folds satisfy stronger cohomology vanishing theorems and enjoy certaintopological properties not satisfied by general weak Fano 3–folds, but are far morenumerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P1s withnormal bundle O.1/˚ O.1/,giving rise to compact rigid holomorphic curves inthe associated ACyl Calabi–Yau 3–folds.We introduce some general methods to compute the basic topological invariants ofACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a smallnumber of representative examples in detail. Similar methods allow the computationof the topology in many other examples.All the features of the ACyl Calabi–Yau 3–folds studied here find application in [17]where we construct many new compact G2 –manifolds using Kovalev’s twistedconnected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano3–folds are particularly well-adapted for this purpose.
Haskins M, Kapouleas N, 2012, Closed twisted products and SO(p) X SO(q)-invariant special Lagrangian cones, COMMUNICATIONS IN ANALYSIS AND GEOMETRY, Vol: 20, Pages: 95-162, ISSN: 1019-8385
Haskins M, Hein H-J, Nordström J, 2012, Asymptotically cylindrical Calabi-Yau manifolds
Haskins M, Kapouleas N, 2008, Gluing Constructions of Special Lagrangian Cones, Handbook of geometric analysis, Editors: Ji, Publisher: International Press, Pages: 77-145, ISBN: 978-1-57146-130-8
We survey our recent work constructing new special Lagrangian cones in complex n-space for all n greater than 3 by gluing methods.
Haskins M, Kapouleas N, 2007, Special Lagrangian cones with higher genus links, INVENTIONES MATHEMATICAE, Vol: 167, Pages: 223-294, ISSN: 0020-9910
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- Citations: 20
Haskins M, Pacini T, 2006, Obstructions to special Lagrangian desingularizations, and the Lagrangian prescribed boundary problem, Geometry and Topology
Haskins M, Pacini T, 2006, Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem, GEOMETRY & TOPOLOGY, Vol: 10, Pages: 1453-1521, ISSN: 1465-3060
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- Citations: 5
Haskins M, 2005, Special Lagrangian T^2-cones via spectral curves and spectral geometry, Fukuoka, Japan, Integrable Systems, Geometry and Visualization, Publisher: Kyushu University, Pages: 13-28
Haskins M, 2004, Special Lagrangian cones with higher genus links, AM J MATH, Vol: 126, Pages: 845-871, ISSN: 0002-9327
We study special Lagrangian cones in C-n with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in C-3 each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in C-3 with a spherical link-any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone.
Haskins M, 2004, The geometric complexity of special Lagrangian T-2-cones, INVENTIONES MATHEMATICAE, Vol: 157, Pages: 11-70, ISSN: 0020-9910
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- Citations: 31
Haskins M, Speight JM, 2003, The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps, JOURNAL OF MATHEMATICAL PHYSICS, Vol: 44, Pages: 3470-3494, ISSN: 0022-2488
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- Citations: 7
Haskins M, Speight JM, 2002, Breather initial profiles in chains of weakly coupled anharmonic oscillators, PHYSICS LETTERS A, Vol: 299, Pages: 549-557, ISSN: 0375-9601
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- Citations: 7
Haskins M, 1998, Breathers in the weakly coupled topological discrete sine-Gordon system, Nonlinearity, Vol: 11, Pages: 1651-1671
Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered in studies of discrete breathers, the TDSG system does not decouple into independent oscillator units in the weak coupling limit. The results of a systematic numerical study of these breathers are presented, including breather initial profiles and a portrait of their domain of existence in the frequency-coupling parameter space. It is found that the breathers are uniformly qualitatively different from those found in conventional spatially discrete systems.
Haskins M, Nordström J, Cohomogeneity-one solitons in Laplacian flow: local, smoothly-closing and steady solitons
We initiate a systematic study of cohomogeneity-one solitons in Bryant'sLaplacian flow of closed G_2-structures on a 7-manifold, motivated by theproblem of understanding finite-time singularities of that flow. Here we focuson solitons with symmetry groups Sp(2) and SU(3); in both cases we prove theexistence of continuous families of local cohomogeneity-one gradient Laplaciansolitons and characterise which of these local solutions extend smoothly overtheir unique singular orbits. The main questions are then to determine which ofthese smoothly-closing solutions extend to complete solitons and furthermore tounderstand the asymptotic geometry of these complete solitons. We provide complete answers to both questions in the case of steady solitons.Up to the actions of scaling and discrete symmetries, we show that the set ofall smoothly-closing SU(3)-invariant steady Laplacian solitons defined on aneighbourhood of the zero-section of the anti-self-dual bundle of CP^2 isparametrised by the set of nonnegative reals. An open interval I=(0,c)corresponds to complete nontrivial gradient solitons that are asymptotic to theunique SU(3)-invariant torsion-free G_2 cone. The boundary point 0 of Icorresponds to the well-known Bryant--Salamon asymptotically conicalG_2-manifold, while the other boundary point c corresponds to an explicitcomplete gradient steady soliton with exponential volume growth and novelasymptotic geometry. The open interval (c, oo) consists entirely of incompletesolutions. In addition, we find an explicit complete gradient shrinking soliton on theanti-self-dual bundle of S^4 and CP^2. Both these shrinkers are asymptotic toclosed but non-torsion-free G_2 cones. Like the nontrivial AC gradient steadysolitons on the anti-self-dual bundle of CP^2, these shrinkers appear to bepotential singularity models for finite-time singularities of Laplacian flow.
Foscolo L, Haskins M, Nordström J, Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds
We develop a powerful new analytic method to construct complete non-compactG2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is thecompact exceptional Lie group G2. Our construction starts with a completenon-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle Mover B satisfying a necessary topological condition. Our method then produces a1-parameter family of circle-invariant complete G2-metrics on M that collapsesto the original Calabi-Yau metric on the base B as the parameter converges to0. The G2-metrics we construct have controlled asymptotic geometry at infinity,so-called asymptotically locally conical (ALC) metrics, and are the naturalhigher-dimensional analogues of the ALF metrics that are well known in4-dimensional hyperk\"ahler geometry. We give two illustrations of the strengthof our method. Firstly we use it to construct infinitely many diffeomorphismtypes of complete non-compact simply connected G2-manifolds; previously only ahandful of such diffeomorphism types was known. Secondly we use it to prove theexistence of continuous families of complete non-compact G2-metrics ofarbitrarily high dimension; previously only rigid or 1-parameter families ofcomplete non-compact G2-metrics were known.
Foscolo L, Haskins M, Nordström J, Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces
We construct infinitely many new 1-parameter families of simply connectedcomplete noncompact G_2-manifolds with controlled geometry at infinity. Thegeneric member of each family has so-called asymptotically locally conical(ALC) geometry. However, the nature of the asymptotic geometry changes at twospecial parameter values: at one special value we obtain a unique member ofeach family with asymptotically conical (AC) geometry; on approach to the otherspecial parameter value the family of metrics collapses to an AC Calabi-Yau3-fold. Our infinitely many new diffeomorphism types of AC G_2-manifolds areparticularly noteworthy: previously the three examples constructed by Bryantand Salamon in 1989 furnished the only known simply connected AC G_2-manifolds. We also construct a closely related conically singular G_2 holonomy space:away from a single isolated conical singularity, where the geometry becomesasymptotic to the G_2-cone over the standard nearly K\"ahler structure on theproduct of a pair of 3-spheres, the metric is smooth and it has ALC geometry atinfinity. We argue that this conically singular ALC G_2-space is the naturalG_2 analogue of the Taub-NUT metric in 4-dimensional hyperKaehler geometry andthat our new AC G_2-metrics are all analogues of the Eguchi-Hanson metric, thesimplest ALE hyperK\"ahler manifold. Like the Taub-NUT and Eguchi-Hansonmetrics, all our examples are cohomogeneity one, i.e. they admit an isometricLie group action whose generic orbit has codimension one.
Foscolo L, Haskins M, New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres
There is a rich theory of so-called (strict) nearly Kaehler manifolds,almost-Hermitian manifolds generalising the famous almost complex structure onthe 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifoldsplay a distinguished role both in the general structure theory and also becauseof their connection with singular spaces with holonomy group the compactexceptional Lie group G2: the metric cone over a Riemannian 6-manifold M hasholonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any completeinhomogeneous examples. We prove the existence of the first completeinhomogeneous nearly Kaehler 6-manifolds by proving the existence of at leastone cohomogeneity one nearly Kaehler structure on the 6-sphere and on theproduct of a pair of 3-spheres. We conjecture that these are the only simplyconnected (inhomogeneous) cohomogeneity one nearly Kaehler structures in sixdimensions.
Haskins M, Speight JM, Breather initial profiles in chains of weakly coupled anharmonic oscillators, Phys. Lett. A., Vol: 299, Pages: 549-557
A systematic correlation between the initial profile of discrete breathersand their frequency is described. The context is that of a very weaklyharmonically coupled chain of softly anharmonic oscillators. The results arestructurally stable, that is, robust under changes of the on-site potential andare illustrated numerically for several standard choices. A precise genericitytheorem for the results is proved.
Haskins M, Khan I, Payne A, Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow
We prove a uniqueness result for asymptotically conical (AC) gradientshrinking solitons for the Laplacian flow of closed G_2-structures: If twogradient shrinking solitons to Laplacian flow are asymptotic to the same closedG_2-cone, then their G_2-structures are equivalent, and in particular, the twosolitons are isometric. The proof extends Kotschwar and Wang's argument foruniqueness of AC gradient shrinking Ricci solitons. We additionally show thatthe symmetries of the G_2-structure of an AC shrinker end are inherited fromits asymptotic cone; under a mild assumption on the fundamental group, thesymmetries of the asymptotic cone extend to global symmetries.
Haskins M, Kapouleas N, Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones
We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL)cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues ofthe $\sorth{2}$-invariant SL cones constructed previously by MH and used in ourgluing constructions of higher genus SL cones in $\C^{3}$. We study in detailthe geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, inpreparation for their application to our higher dimensional special Legendriangluing constructions. In particular the symmetries of these cones and theirasymptotics near the spherical limit are analysed. All $\sorth{p} \times\sorth{q}$-invariant SL cones arise from a more general construction ofindependent interest which we call the special Legendrian twisted productconstruction. Using this twisted product construction and simple variants of itwe can construct a constellation of new special Lagrangian and Hamiltonianstationary cones in $\C^{n}$. We prove the following theorems: A. there areinfinitely many topological types of special Lagrangian and Hamiltonianstationary cones in $\C^{n}$ for all $n\ge 4$, B. for $n\ge 4$ specialLagrangian and Hamiltonian stationary torus cones in $\C^{n}$ can occur incontinuous families of arbitrarily high dimension and C. for $n\ge 6$ there areinfinitely many topological types of special Lagrangian and Hamiltonianstationary cones in $\C^{n}$ that can occur in continuous families ofarbitrarily high dimension.
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