Let M(n, D) be the space of closed n-dimensional Riemannian manifolds (M, g) with diam(M) ≤ D and | sec M | ≤ 1. In this paper we consider sequencesconverging in the Gromov-Hausdorff topology to a compact metric space Y . We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotientcan be uniformly bounded from below by a positive constant C(n, r, Y ) for all points x ∈ M i . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y ) bounding the quotientuniformly from below for all x ∈ M i . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in M(n, D) with C ≤ vol(M) inj(M) .

Abstract. We study the behavior of the spectrum of the Dirac operator together with a symmetric W 1,∞ -potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin − structure on the limit space N , then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W 1,∞ -potential. In the case of an orientable limit space N , D is the spin Dirac operator D N on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D = D N ⊕ −D N . IntroductionThe structure of collapsing sequences of manifolds with bounded sectional curvature and diameter was studied in great detail by Cheeger et al. (see [6] and references therein). One of the next questions arising was how the spectrum of differential operators behaves in the limit of a collapsing sequence.As for the Laplacian on functions, Fukaya showed that if a sequence of manifolds with uniform bounded sectional curvature and diameter converges in the measured Gromov-Hausdorff-topology, then the eigenvalues of the Laplace operator converge to the eigenvalues of the Laplacian on the limit space with respect to a limit measure [8]. This is even the case if the limit space is a smooth manifold. Lott generalized this result to the Laplacian on p-forms [17,18]. Using the Bochner-type formula for Dirac operators on G-Clifford bundles on manifolds, where G ∈ {SO(n), Spin(n)}, Lott proved similar results for Dirac eigenvalues under collapse with bounded sectional curvature and diameter [16]. His results also include the Dirac operator acting on differential forms considering the measured Gromov-Hausdorff topology.In this paper, we consider sequences (M a , g a ) a∈N of (n + 1)-dimensional spin manifolds with bounded sectional curvature and diameter such that their GromovHausdorff limit (N , h) is n-dimensional. This already implies that N is a Riemannian orbifold [11, Proposition 11.5]. By restricting to the setting of spin manifolds we are able to show that the spectrum of the Dirac operator together with a uniform bounded symmetric W 1,∞ -potential converges again to the Dirac operator with S. Roos (B): Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: saroos@mpim-bonn.mpg.dehttps://doi.org/10.1007/s00229-018-1003-6 S. Roos symmetric W 1,∞ -potential on the limit space N . In this setting, the Dirac operator is taken with respect to the standard measure dvol(h). In particular, we do not need to consider the limit measure in the measured Gromov-Hausdorff topology as in [16]. This limit measure is in general different to the standard measure dvol(h). We only study the case of collapsing sequences losing one dimension in the limit because the situation is more complicated in the general case (see Remark 5.3).We use the techniques of [1], where Dirac operators on collapsing S 1 -principal bundles were studied. One of the main differences to ...

For a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots \times (M_l, g_l)$$ ( M , g ) = ( M 1 , g 1 ) × ⋯ × ( M l , g l ) , we define its multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] as the totality $$\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}$$ { f 1 2 g 1 ⊕ ⋯ ⊕ f l 2 g l } of all Riemannian metrics obtained from multiplying the metric $$g_i$$ g i of each factor $$M_i$$ M i by a positive function $$f_i$$ f i on the total space M. A multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] contains not only all warped product type deformations of g but also the whole conformal class $$[\tilde{g}]$$ [ g ~ ] of every $$\tilde{g}\in [\![ g ]\!]$$ g ~ ∈ [ [ g ] ] . In this article, we prove that $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of positive scalar curvature if and only if the conformal class of some factor $$(M_i, g_i)$$ ( M i , g i ) does, under the technical assumption $$\dim M_i\ge 2$$ dim M i ≥ 2 . We also show that, even in the case where every factor $$(M_i, g_i)$$ ( M i , g i ) has positive scalar curvature, $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of scalar curvature constantly equal to $$-1$$ - 1 and with arbitrarily large volume, provided $$l\ge 2$$ l ≥ 2 and $$\dim M\ge 3$$ dim M ≥ 3 .

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For a closed, connected direct product Riemannian manifold (M, g)of all Riemannian metrics obtained from multiplying the metric g i of each factor M i by a function f 2 i > 0 on the total space M . A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g] of every g ∈ [[g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (M i , g i ) does, under the technical assumption dim M i ≥ 2. We also show that, even in the case where every factor (M i , g i ) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l ≥ 2 and dim M ≥ 3. In this case, such negative scalar curvature metrics within [[g]] for l = 2 cannot be of any warped product type. 2010 Mathematics Subject Classification. Primary 53C21. Keywords and phrases. Positive scalar curvature, constant scalar curvature, the Yamabe problem, warped product, umbilic product, twisted product. 1 4 Scalar curvature computation à la Karcher 10 5 Integral and pointwise formulas 20 6 The sign of a multiconformal class 23 7 The infimum of the Yamabe constants 25 8 Multiconformal metrics of permutation type 27