Exam 2

Exam 2

Answer of Question No.1

Let Mn be a complete manifold with nonnegative Ricci curvature. Suppose x0 ? M is a fixed point and there exists a constant k > 0 such that the volume of the geodesic ball centered at x0 of radius r satisfies Vx0 (r) = O(rk) as r?8. Then the dimension of H1(M) satisfies the estimate dimH1(M) = k + 1. Note that the standard Bishop volume comparison theorem implies that k = n. Also a theorem of Yau [Y2] and Cheeger-Gromov-Taylor [CGT] implies that 1 = k. Let us also point out that H1(Rk) = k+1. Indeed, a basis for H1(Rk) is given by the constant function 1 and the standard rectangular coordinate functions. Using this, we can rewrite the Li-Tam estimate as dimH1(M) = dimH1(Rk).

There are some partialresul ts in the affirmative direction of this question. Tam and the author in [LT2] studied complete surfaces with finite total curvature. In particular, they gave relatively sharp upper and lower bounds for the dimension of Hp(M). When restricted to surfaces with nonnegative Gaussian curvature, their estimates confirmed inequality (1). In fact, the upper and lower bounds for the dimension of Hp(M) coincide when the surface has nonnegative Gaussian curvature outside a compact set. The interested reader should refer to [LT2] for a detailed statement of the theorem. We would also like to mention that Kasue [K1] also independently proved the upper bound for dimHp(M) on surfaces with finite total curvature. In some later work [K2,K3], Kasue also considered arbitrary n-dimensional complete manifolds that are in one of the following classes: (i) The sectionalcurv ature of M satisfies |KM|(x) = C1 r-2-_ x0 (x), and the volume growth at each end E satisfies V (E n Bx0 (r)) = C2 rn. (ii) M is a simply connected manifold with sectional curvature bounded by -C1 r-2 x0 (x) = KM(x) = 0, and the volume growth satisfies Vx0 (r) = C2 rn. (iii) The sectionalcurv ature satisfies 0 = KM, and the volume growth satisfies Vx0 (r) = C2 rn. (iv) The sectionalcurv ature is bounded by 0 = KM(x) = C1 r-2 x0 (x) for all x ? M. The constants above—C1, C2, and —are assumed to be nonnegative. In those situations, he showed that dimHp(M) < 8.

In fact, one can consider the metric ds2 = dr2 + f(r) r2 d?2 on Rn, where d?2 is the standard metric on the unit (n - 1)-sphere and f(r) is a smooth function satisfying f(r) = 1 when r is close to 0, and f(r) = a when r is sufficiently large. This metric is flat outside a compact set. However, using separation of variables, one can determine that for any fixed realn umber p > 0, dimHp(M) is a nondecreasing function of a. In fact, dimHp(M)?8 as a?8. The purpose of this paper is to study the equality case of Theorem A. Unfortunately, we still cannot characterize those manifolds with nonnegative Ricci curvature that has dimH1(M) = n ...