Mark Haskins

G2-manifolds and associative submanifolds via semi-Fano 3-folds

We construct many new topological types of compact G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2. 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 3-folds built from semi-Fano 3-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 7-manifolds completely; we find that many 2-connected 7-manifolds can be realised as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G_2-metrics. Many of the G_2-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G_2-metrics. By varying the semi-Fanos used to build different G_2-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce.

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–folds starting with (almost) any deformation family of smooth weak Fano 3–folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. Semi-Fano 3–folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3–folds, but are far more numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain ℙ1s with normal bundle O(−1) ⊕O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi–Yau 3–folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi–Yau 3–folds studied here find application in [arXiv:1207.4470] where we construct many new compact G2–manifolds using Kovalev’s twisted connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds are particularly well-adapted for this purpose.

The geometric complexity of special Lagrangian T2-cones

We prove a number of results on the geometric complexity of special Lagrangian (SLG) T2-cones in ℂ3. Every SLG T2-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T2-cone gives a lower bound for its geometric complexity, i.e. the area, the stability index and the Legendrian index of any SLG T2-cone are all bounded below by explicit linearly growing functions of the spectral curve genus. We prove that the cone on the Clifford torus (which has spectral curve genus zero) in S5 is the unique SLG T2-cone with the smallest possible Legendrian index and hence that it is the unique stable SLG T2-cone. This leads to a classification of all rigid “index 1” SLG cone types in dimension three. For cones with spectral curve genus two we give refined lower bounds for the area, the Legendrian index and the stability index. One consequence of these bounds is that there exist S1-invariant SLG torus cones of arbitrarily large area, Legendrian and stability indices. We explain some consequences of our results for the programme (due to Joyce) to understand the “most common” three-dimensional isolated singularities of generic families of SLG submanifolds in almost Calabi-Yau manifolds.

Special Lagrangian cones with higher genus links

For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in

New G

\mathbb{C}^3

_2

over a closed Riemann surface of genus g, using a geometric PDE gluing method.

-holonomy cones and exotic nearly Kähler structures on

There is a rich theory of so-called (strict) nearly Kähler manifolds, almostHermitian manifolds generalising the famous almost complex structure on the 6–sphere induced by octonionic multiplication. Nearly Kähler 6–manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6–manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6–manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6–manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6–sphere and on the product of a pair of 3–spheres. We conjecture that these are the only (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions.

S^6

There is a rich theory of so-called (strict) nearly Kähler manifolds, almostHermitian manifolds generalising the famous almost complex structure on the 6–sphere induced by octonionic multiplication. Nearly Kähler 6–manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6–manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6–manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6–manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6–sphere and on the product of a pair of 3–spheres. We conjecture that these are the only (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions.

and

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.

S^3 \times S^3

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings of special Lagrangian cones is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h‐principle for Lagrangian immersions) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singularities; we view the (Maslov-zero) Lagrangian desingularization problem as the natural soft analogue of the special Lagrangian smoothing problem. 53D12, 53C38; 53C42