Abstract. We provide a significant extension of the twisted connected sum construction of G2-manifolds, ie Riemannian 7-manifolds with holonomy group G2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Our extension allows us to prove many new results about compact G2-manifolds and leads to new perspectives for future research in the area. Some of the main contributions of the paper are:(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in [18]. (iv) We obtain much more precise topological information about twisted connected sum G2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G2-manifolds. (v) We describe "geometric transitions" between G2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. The latter result leads to speculation that the moduli space of G2-metrics on a given 7-manifold may consist of many different connected components, and opens up many further questions for future study. For instance, the higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G2-metrics on a given 7-manifold that are not deformation equivalent.
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 P 1 s 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 [17] where we construct many new compact G 2 -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.
We define a very general "parametric connect sum" construction that can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used in the construction. Specifically, we prove uniform estimates related to (i) Sobolev Embedding Theorems, (ii) the invertibility of the Laplace operator and (iii) Poincaré and Gagliardo-NirenbergSobolev-type inequalities.Our main tools are the well-known theories of weighted Sobolev spaces and elliptic operators on "conifolds". We provide an overview of both, together with an extension of the former to general Riemannian manifolds.For a geometric application of our results we refer the reader to our paper [15] concerning desingularizations of special Lagrangian conifolds in C m .
Here, we propose a novel strategy that combines a typical ultra high performance liquid chromatography (UHPLC), data-independent mass spectrometry (MS(E)) workflow with traveling wave ion mobility (TWIM) and UV detection, to improve the characterization of carotenoids and chlorophylls in complex biological matrices. UV detection selectively highlighted pigments absorbing at specific wavelengths, while TWIM coupled to MS was used to maximize the peak capacity. We applied this approach for the analysis of pigments in different microalgae samples, including Chlorella vulgaris, Dunaliella salina, and Phaeodactylum tricornutum. Using UHPLC-UV-MS(E) information (retention time, absorbance at 450 nm, and accurate masses of precursors and product ions), we tentatively identified 26 different pigments (carotenes, chlorophylls, and xanthophylls). By adding TWIM information (collision cross sections), we further resolved 5 isobaric pigments, not resolved by UHPLC-UV-MS(E) alone. The characterization of the molecular phenotypes allowed us to differentiate the microalgae species. Our results demonstrate that a combination of TWIM and UV detection with traditional analytical approaches increases the selectivity and specificity of analysis, providing a new tool to characterize pigments in biological samples. We anticipate that such an analytical approach will be extended to other lipidomics and metabolomics applications.
Let M denote the space of probability measures on R D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced in [5]. In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D = 2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced in [3]. Throughout the paper we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of R D .It can be shown that (M, W 2 ) is a separable complete metric space, cf. e.g.[5] Proposition 7.1.5. It is an important result from Monge-Kantorovich theory thatis continuous and defines a left action of Diff c (R D ) on M. The map M → T M, µ → π µ (X) ∈ T µ M then defines the fundamental vector field associated to X in the sense of Section A.2. According to Section A.2, the orbit and stabilizer of any fixed µ ∈ M are: O µ := {ν ∈ M : ν = φ # µ, for some φ ∈ Diff c (R D )}, Diff c,µ (R D ) := {φ ∈ Diff c (R D ) : φ # µ = µ}.
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