Study on Geometry and Analysis of Conformal Manifolds and Bubbling Trees

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We have studied the following:1.Study of cylindrical and orbifold Yamabe invariantsAs a generalization of the Yamabe constant/invariant of closed manifolds, we defined appropriately the orbifold Yamabe constant/invariant in terms of the cylindrical Yamabe constant/invariant.For a cylindrical 4-manifold with positive cylindrical Yamabe invariant, we also established a method for estimating its cylindrical Yamabe invariant from above, by means of the Atiyah-Patodi-Singer L^2-index theory. Moreover, we generalized the Kobayashi inequality for Yamabe invariants to cylindrical Yamabe invariants, and studied its applications.2.Study on the mass of compact conformal manifoldsThe mass is a geometric invariant for asymptotically flat manifolds. For a compact conformal manifold (M, C) with positive Yamabe invariant, a scalar-flat, asymptotically flat manifold (M-{p},g_<AF>) is defined naturally from each initial metric g in C, where [g_<AF>]=C. Then the mass m(g ; p) is non-negative. This mass m … More (g ; p) also depends on the choice of g and p. However, if we use the Habermann-Jost's canonical metric g_<HJ> as a initial metric, then the mass m(g_<HJ>;p) is now independent of the choice of p. By using this fact, we can define the mass mass(M ; C) of the conformal manifold (M, C) as a conformal invariant. Moreover, taking the infimum of it over all conformal classes, we can also define the mass invariant mass(M) as a differential-topological invariant of M. We studied on the Kobayashi-type inequality of the mass invariant for connected manifolds.3.Yamabe invariants of 3-manifoldsThe method of inverse mean curvature flow is the central technique for the resolution of the Riemannian Penrose Conjecture in Cosmology. By using this technique, Bray-Neves determined the value of the Yamabe invariant of RP^3. This result is the first affirmative answer to the Schoen's Conjecture for the Yamabe invariant of 3-manifolds with constant curvature. We also determined the Yamabe invariant of the connected manifold RP^3 # k(S^2 x S^1), by means of the inverse mean curvature flow technique. This is also one of the open problems proposed by Bray-Neves.For the above study, the support by the 'Grant-in-Aid for Sci. Res. (C)(2),14540072' was very important. Less