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Register a new userString geometry vs. spin geometry on loop spaces
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Authors
Konrad Waldorf
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In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply it to the construction of complete, noncompact maximal submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hypersurface to be of constant mean curvature. Finally, we study the strong $r$-stability of spacelike hypersurfaces of constant $r$-th mean curvature in a conformally stationary Lorentz manifold of constant sectional curvature, extending previous results in the current literature.
We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.
Let $X$ be a compact manifold, $G$ a Lie group, $P to X$ a principal $G$-bundle, and $mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_bullet$ we previously studied orientations on the real determinant line bundle over $mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X times S^1$. Thus, if $P to X$ and $Q to X times S^1$ are principal $G$-bundles with $Q|_{Xtimes{1}} cong P$, we relate spin structures on $(mathcal{B}_P,F_bullet)$ to orientations on $(mathcal{B}_Q,E_bullet)$ for a certain class of operators $F_bullet$ on $X$ and $E_bullet$ on $Xtimes S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.
We investigate the collapsing geometry of hyperkaehler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperkaehler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kaehler metric on the 2-sphere, or the unit interval. (2) Any complete hyperkaehler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.
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