Shouhei Honda

A weakly second-order differential structure on rectifiable metric measure spaces

We give the definition of angles on a Gromov-Hausdorff limit space of a sequence of complete n-dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a weakly second order differential structure on these spaces and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

On low-dimensional Ricci limit spaces

We call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space . Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

Ricci curvature and orientability

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

Lp-Spectral Gap and Gromov-Hausdorff Convergence

L^2

Ricci curvature and Lp-convergence

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.

Bishop-Gromov type inequality on Ricci limit spaces

In this note we prove in the nonlinear setting of

Ricci curvature and

{{\mathrm{CD}}}(K,\infty )