TY - JOUR

T1 - Ricci curvature and orientability

AU - Honda, Shouhei

N1 - Funding Information:
Acknowledgements The author would like to express his appreciation to Luigi Ambrosio for his warm encouragement during the stay at SNS. He also thanks SNS for its warm hospitality and for giving him nice environment. He is grateful to Ayato Mitsuishi for informing us of [35,46]. I would like to thank Kota Hattori for discussing on [16]. He is also grateful to the referee for the very thorough reading and valuable suggestions. Moreover he wants to thank TSIMF, BICMR and MFO for their wonderful hospitality during his stays. Finally he acknowledges supports of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, of the Grant-in-Aid for challenging Exploratory Research 26610016 and of the Grantin-Aid for Young Scientists (B) 16K17585.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - In this paper we define an orientation of a measured Gromov–Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between L2-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani–Wenger, the pointed flat convergence by Lang–Wenger, and the Gromov–Hausdorff convergence, which is a generalization of a recent work by Matveev–Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.

AB - In this paper we define an orientation of a measured Gromov–Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between L2-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani–Wenger, the pointed flat convergence by Lang–Wenger, and the Gromov–Hausdorff convergence, which is a generalization of a recent work by Matveev–Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.

KW - 53C20

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U2 - 10.1007/s00526-017-1258-x

DO - 10.1007/s00526-017-1258-x

M3 - Article

AN - SCOPUS:85033722296

VL - 56

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 6

M1 - 174

ER -