A modfied mean curvature flow in Euclidean space and soap bubbles in symmetric spaces

2017 ◽  
Vol 195 (1) ◽  
pp. 1-17
Author(s):  
Naoyuki Koike
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Symmetric Spaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Soap Bubbles

scholarly journals Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ruiwei Xu ◽  
Linfen Cao
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Indefinite Metric ◽  
Bernstein Theorem ◽  
Strictly Convex ◽  
Convex Solution

Letf(x)be a smooth strictly convex solution ofdet(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-fdefined on a domainΩ⊂Rn; then the graphM∇fof∇fis a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceRn2nwith the indefinite metric∑dxidyi. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graphM∇fis complete inRn2nand passes through the origin then it is flat.

scholarly journals Rigidity results and topology at infinity of translating solitons of the mean curvature flow

2017 ◽  
Vol 19 (06) ◽  
pp. 1750002 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Minimal Hypersurfaces ◽  
Rigidity Results ◽  
And Topology ◽  
The Mean ◽  
Translating Solitons

In this paper, we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of [Formula: see text]-minimal hypersurfaces.

scholarly journals A survey on Inverse mean curvature flow in ROSSes

2017 ◽  
Vol 4 (1) ◽  
pp. 245-262
Author(s):  
Giuseppe Pipoli
Keyword(s):  
Mean Curvature ◽  
Symmetric Spaces ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Unified Approach ◽  
Inverse Mean Curvature Flow ◽  
Rank One ◽  
Similarities And Differences ◽  
Mean Convex ◽  
Convex Hypersurfaces

AbstractIn this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.

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