On Manifolds with Lower Quadratic Curvature Decay and Volume Pinching

We give some conditions for a complete noncompact Riemannian manifold with lower quadratic curvature decay to have finite topological type. Most of our curvature conditions are much weaker than the assumptions in Lott [5] and in Sha–Shen [10], hence our results can be viewed as natural generalizations of those works.

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

For an orientable surfaceSof finite topological type with genusg≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph$\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger,J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger,Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

FINITE RIGID SETS IN CURVE COMPLEXES

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex 𝔛 of the curve complex [Formula: see text] such that every locally injective simplicial map [Formula: see text] is the restriction of an element of [Formula: see text], unique up to the (finite) pointwise stabilizer of 𝔛 in [Formula: see text]. Furthermore, if S is not a twice-punctured torus, then we can replace [Formula: see text] in this statement with the extended mapping class group Mod ±(S).

A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound and Its Applications

We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.

On manifolds with quadratic curvature decay

We give conditions which imply that a complete noncompact manifold with quadratic curvature decay has finite topological type. In particular, we find links between the topology of a manifold with quadratic curvature decay and some properties of the asymptotic cones of such a manifold.