scholarly journals The Number of Cusps of Complete Riemannian Manifolds with Finite Volume

2018 ◽  
Vol 22 (6) ◽  
pp. 1403-1425
Author(s):  
Thac Dung Nguyen ◽  
Ngoc Khanh Nguyen ◽  
Ta Cong Son
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Complete Riemannian Manifolds

Plateau-Stein manifolds

2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Misha Gromov
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Critical Points ◽  
Minimal Hypersurface ◽  
Proper Function ◽  
Stein Manifolds ◽  
Morse Functions ◽  
Complete Riemannian Manifolds

AbstractWe study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

scholarly journals Index theorems on manifolds with straight ends

2012 ◽  
Vol 148 (6) ◽  
pp. 1897-1968 ◽  
Author(s):  
Werner Ballmann ◽  
Jochen Brüning ◽  
Gilles Carron
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
Riemannian Manifolds ◽  
Dirac Operators ◽  
Important Class ◽  
Index Theorems ◽  
Complete Riemannian Manifolds ◽  
Index Formulas ◽  
Negative Sectional Curvature

AbstractWe study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

scholarly journals Piecewise straightening and Lipschitz simplicial volume

2017 ◽  
Vol 09 (01) ◽  
pp. 167-193
Author(s):  
Karol Strzałkowski
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
Riemannian Manifolds ◽  
Simplicial Volume ◽  
Product Inequality ◽  
Complete Riemannian Manifolds ◽  
Proportionality Principle

We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above.

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