scholarly journals Ancient solutions in Lagrangian mean curvature flow

2021 ◽  
pp. 31
Author(s):  
Ben Lambert ◽  
Jason D. Lotay ◽  
Felix Schulze
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Ancient Solutions

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

Collapsing ancient solutions of mean curvature flow

2021 ◽  
Vol 119 (2) ◽  
Author(s):  
Theodora Bourni ◽  
Mat Langford ◽  
Giuseppe Tinaglia
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Ancient Solutions

scholarly journals Pinched ancient solutions to the high codimension mean curvature flow

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Stephen Lynch ◽  
Huy The Nguyen
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
High Codimension ◽  
Ancient Solutions ◽  
Ancient Solution

AbstractWe study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker (J Differ Geom 85(3):357–395, 2010) have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.

ANCIENT SOLUTIONS OF CODIMENSION TWO SURFACES WITH CURVATURE PINCHING

2020 ◽  
Vol 102 (1) ◽  
pp. 162-171
Author(s):  
ZHENGCHAO JI
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Geometric Flows ◽  
Codimension Two ◽  
The Second Fundamental Form ◽  
Ancient Solutions ◽  
The Mean ◽  
Curvature Pinching

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces.

scholarly journals Convex Ancient Solutions to Mean Curvature Flow

2020 ◽  
pp. 50-74
Author(s):  
Theodora Bourni ◽  
Mat Langford ◽  
Giuseppe Tinaglia
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Ancient Solutions
Sign in / Sign up

Export Citation Format

Share Document