論文の公開元へRicci flow with bounded curvature integrals and Decompositions of the space of Riemannian metrics on a compact manifold with boundary
概要
In this thesis, we treat two themes on differential geometry.
In the Part I, we study the Ricci flow on a closed manifold and finite time interval [0, T) (T < ∞) on which certain integral curvature energies are finite. A Ricci flow on a manifold M is given by a smooth family g(t) (t ∈ [0, T)), of Riemannian metrics satisfying the evolution equation
∂tg(t) = −2 Ricg(t) .
The Ricci flow equation was introduced by Hamilton [17] in 1982. In the same paper, he stated the existence and uniqueness of the Ricci flow on a closed manifold. Moreover, he proved that a Ricci flow on a closed manifold develops a singularity at a finite time T (i.e., T is the maximal existence time of the flow) if and only if the maximum of the norm of the Riemannian curvature tensor blows up at T. On the other hand, Wang [29] characterized the maximal existence time of the flow by certain geometric energies which consist of integral bounds rather than point-wise ones. Later, Di Matteo [14] generalized Wang’s results using mixed integral norms which are parametrized by α, β ∈ (1, ∞) with α ≥ n 2 β β−1 . In the Part I, we study the case that (α, β) = (n/2, ∞) and (∞, 1). Under some stronger assumptions, we prove that in dimension four, such flow converges to a smooth Riemannian manifold except for finitely many orbifold singularities. We also show that in higher dimensions, the same assertions hold for a closed Ricci flow satisfying another conditions of integral curvature bounds. Moreover, we show that such flows can be extended over T by an orbifold Ricci flow.
In the Part II, for a compact manifold M with non-empty boundary ∂M, we give a Koiso-type decomposition theorem, as well as an Ebin-type slice theorem, for the space of all Riemannian metrics on M endowed with a fixed conformal class on ∂M. In the case that ∂M = ∅, Ebin [13] particularly has proved a slice theorem for the pullback action of the diffeomorphism group on the space M, of all Riemannian metrics on M. In [25], Koiso has extended it to an Inverse Limit Hilbert (ILH for brevity)- version. Moreover, he has also studied the conformal action on M, and consequently has proved certain decomposition theorem for M. We generalize these results to the case that ∂M ̸= ∅ with some suitable boundary conditions. As a corollary, we give a characterization of relative Einstein metrics. Moreover, we also give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric.
