A class of Riemannian manifolds that pinch when evolved by Ricci flow

2000 ◽  
Vol 101 (1) ◽  
pp. 89-114 ◽  
Author(s):  
M. Simon
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Flow

GRADIENT ESTIMATES VIA TWO-POINT FUNCTIONS FOR PARABOLIC EQUATIONS UNDER RICCI FLOW

2020 ◽  
Vol 102 (2) ◽  
pp. 319-330
Author(s):  
MIN CHEN
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Parabolic Equations ◽  
Gradient Estimates ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

We derive estimates relating the values of a solution at any two points to the distance between the points for quasilinear parabolic equations on compact Riemannian manifolds under Ricci flow.

scholarly journals Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 927
Author(s):  
Josef Mikeš ◽  
Vladimir Rovenski ◽  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Flat Space ◽  
Conformally Flat ◽  
Space Forms ◽  
Bochner Technique ◽  
Locally Conformally Flat

In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.

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 Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

2020 ◽  
Author(s):  
Shaosai Huang ◽  
◽  
Xiaochun Rong ◽  
Bing Wang ◽  
◽  
...  
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Smoothing Techniques ◽  
Ricci Flat ◽  
Recent Developments ◽  
Open Question ◽  
Bounded Below

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

Ricci Flow on Quasi Einstein Manifold

2010 ◽  
Vol 0 (-1) ◽  
pp. 447-454
Author(s):  
A. Bhattacharyya ◽  
T. De
Keyword(s):  
Ricci Flow ◽  
Einstein Manifold
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