On manifolds with quadratic curvature decay

AbstractWe 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.

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On Manifolds with Lower Quadratic Curvature Decay and Volume Pinching

International Mathematics Research Notices ◽

10.1093/imrn/rny223 ◽

2018 ◽

Vol 2020 (21) ◽

pp. 7857-7872

Author(s):

Xiaosong Kang ◽

Xu Xu ◽

Dunmu Zhang

Keyword(s):

Riemannian Manifold ◽

Topological Type ◽

Finite Topological Type ◽

Noncompact Riemannian Manifold ◽

Curvature Decay

Abstract 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.

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EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

Glasgow Mathematical Journal ◽

10.1017/s0017089518000174 ◽

2018 ◽

Vol 61 (1) ◽

pp. 195-230 ◽

Cited By ~ 4

Author(s):

JESÚS HERNÁNDEZ HERNÁNDEZ

Keyword(s):

Topological Type ◽

Orientable Surface ◽

Finite Topological Type ◽

Curve Graph ◽

Finite Set ◽

Rigid Set

AbstractFor 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.

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Complete gradient shrinking Ricci solitons have finite topological type

Comptes Rendus Mathematique ◽

10.1016/j.crma.2008.03.021 ◽

2008 ◽

Vol 346 (11-12) ◽

pp. 653-656 ◽

Cited By ~ 11

Author(s):

Fu-quan Fang ◽

Jian-wen Man ◽

Zhen-lei Zhang

Keyword(s):

Topological Type ◽

Ricci Solitons ◽

Finite Topological Type

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Smooth Compactness for Spaces of Asymptotically Conical Self-Expanders of Mean Curvature Flow

International Mathematics Research Notices ◽

10.1093/imrn/rnz087 ◽

2019 ◽

Cited By ~ 1

Author(s):

Jacob Bernstein ◽

Lu Wang

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Topological Type ◽

Curvature Flow ◽

Natural Projection ◽

Asymptotic Cones ◽

Mean Convex ◽

Low Entropy ◽

Natural Families

Abstract We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all 2D self-expanders of a fixed topological type and, in all dimensions, for the set of self-expanders of low entropy and for the set of mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones is proper for these classes.

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FINITE RIGID SETS IN CURVE COMPLEXES

Journal of Topology and Analysis ◽

10.1142/s1793525313500076 ◽

2013 ◽

Vol 05 (02) ◽

pp. 183-203 ◽

Cited By ~ 13

Author(s):

JAVIER ARAMAYONA ◽

CHRISTOPHER J. LEININGER

Keyword(s):

Class Group ◽

Topological Type ◽

Orientable Surface ◽

Mapping Class ◽

Curve Complex ◽

Finite Topological Type ◽

Punctured Torus ◽

Curve Complexes ◽

Simplicial Map ◽

Pointwise Stabilizer

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).

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