On deformation of Riemannian metrics and manifolds with positive curvature operator

1986 ◽  
pp. 202-211 ◽  
Author(s):  
Seiki Nishikawa
Keyword(s):  
Positive Curvature ◽  
Riemannian Metrics ◽  
Curvature Operator

scholarly journals Ricci flow on manifolds with boundary with arbitrary initial metric

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tsz-Kiu Aaron Chow
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Manifolds With Boundary ◽  
Convex Boundary ◽  
Uniqueness Of The Solution ◽  
Positive Time ◽  
Natural Boundary Conditions ◽  
Short Time ◽  
Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

Manifolds with pinched 2-positive curvature operator

2012 ◽  
Vol 7 (5) ◽  
pp. 873-882
Author(s):  
Gang Peng ◽  
Hongliang Shao
Keyword(s):  
Positive Curvature ◽  
Curvature Operator

scholarly journals Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

2016 ◽  
Vol 19 (01) ◽  
pp. 1550092 ◽  
Author(s):  
Weimin Sheng ◽  
Haobin Yu
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Geodesic Sphere ◽  
Initial Hypersurface ◽  
Normalized Ricci Flow ◽  
Spherical Space Form

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

scholarly journals Weak stability of Ricci expanders with positive curvature operator

2016 ◽  
Vol 286 (3-4) ◽  
pp. 951-985 ◽  
Author(s):  
Alix Deruelle ◽  
Tobias Lamm
Keyword(s):  
Positive Curvature ◽  
Curvature Operator ◽  
Weak Stability
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