| 1. |
- Ammann, Bernd, et al.
(författare)
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Harmonic spinors and local deformations of the metric
- 2011
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Ingår i: Mathematical Research Letters. - 1073-2780 .- 1945-001X. ; 18:5, s. 927-936
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Tidskriftsartikel (refereegranskat)abstract
- Let (M, g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.
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| 2. |
- Ammann, Bernd, et al.
(författare)
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Low-dimensional surgery and the Yamabe invariant
- 2015
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Ingår i: Journal of the Mathematical Society of Japan. - : Mathematical Society of Japan (Project Euclid). - 0025-5645 .- 1881-1167. ; 67:1, s. 159-182
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Tidskriftsartikel (refereegranskat)abstract
- Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k <= n - 3. The smooth Yamabe invariants sigma(M) and sigma(N) satisfy sigma(N) >= min(sigma(M), Lambda) for a constant Lambda > 0 depending only on n and k. We derive explicit positive lower bounds for A in dimensions where previous methods failed, namely for (n, k) is an element of {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.
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| 3. |
- Ammann, Bernd, et al.
(författare)
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Mass endomorphism, surgery and perturbations
- 2014
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Ingår i: Annales de l'Institut Fourier. - : Cellule MathDoc/CEDRAM. - 0373-0956 .- 1777-5310. ; 64:2, s. 467-487
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.
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| 4. |
- Ammann, Bernd, et al.
(författare)
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Smooth yamabe invariant and surgery
- 2013
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Ingår i: Journal of differential geometry. - : International Press of Boston. - 0022-040X .- 1945-743X. ; 94:1, s. 1-58
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Tidskriftsartikel (refereegranskat)abstract
- We prove a surgery formula for the smooth Yamabe invariant sigma(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Lambda(n), depending only on the dimension n of M, such that sigma(N) >= min{sigma(M), Lambda(n)}.
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| 5. |
- Ammann, Bernd, et al.
(författare)
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Square-integrability of solutions of the Yamabe equation
- 2013
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Ingår i: Communications in analysis and geometry. - 1019-8385 .- 1944-9992. ; 21:5, s. 891-916
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Tidskriftsartikel (refereegranskat)abstract
- We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds, which are bounded and L-p for p = 2n/(n -2) are also L-2. This L-p-L-2 implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in our paper [4]. As an application we see that the smooth Yamabe invariant of any two-connected compact seven-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions >= 11.
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| 6. |
- Ammann, Bernd, et al.
(författare)
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Surgery and harmonic spinors
- 2009
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 220:2, s. 523-539
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Tidskriftsartikel (refereegranskat)abstract
- Let M he a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.
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| 7. |
- Ammann, Bernd, et al.
(författare)
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Surgery and the Spinorial tau-Invariant
- 2009
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Ingår i: Communications in Partial Differential Equations. - : Informa UK Limited. - 0360-5302 .- 1532-4133. ; 34:10, s. 1147-1179
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Tidskriftsartikel (refereegranskat)abstract
- We associate to a compact spin manifold M a real-valued invariant (M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's sigma-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then (N) epsilon min{(M), n}, where n is a positive constant depending only on n=dim M. Various topological conclusions can be drawn, in particular that is a spin-bordism invariant below n. Also, below n the values of cannot accumulate from above when varied over all manifolds of dimension n.
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| 8. |
- Ammann, B., et al.
(författare)
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The conformal Yamabe constant of product manifolds
- 2013
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 141:1, s. 295-307
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Tidskriftsartikel (refereegranskat)abstract
- Let (V, g) and (W, h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V × W, g + h) in terms of the conformal Yamabe constants of (V, g) and (W, h).
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| 9. |
- Andersson, Lars, et al.
(författare)
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Boundary and lens rigidity of Lorentzian surfaces
- 1996
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 348, s. 2307-2329
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Tidskriftsartikel (refereegranskat)abstract
- Let g be a Lorentzian metric on the plane ℝ2 that agrees with the standard metric g0 = -dx2 + dy2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (ℝ2, g). Then (ℝ2, g) and (ℝ2, g0) are isometric. Further, if (M*, g*) and (M*, p*) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesies of (M, g) maximize the distances between their points and (M, g) and (M*, g*) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M*, g*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.
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| 10. |
- Andersson, L., et al.
(författare)
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On the geometry and topology of initial data sets with horizons
- 2018
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Ingår i: Asian Journal of Mathematics. - : International Press of Boston, Inc.. - 1093-6106 .- 1945-0036. ; 22:5, s. 863-882
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Tidskriftsartikel (refereegranskat)abstract
- We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M, g,K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M \ ∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension n = 3, to higher dimensions.
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