scholarly journals Singularities of null mean curvature flow of null hypersurfaces in Lorentzian manifolds

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type I ◽  
Type Ii ◽  
Lorentzian Manifolds ◽  
Null Hypersurfaces ◽  
Geometric Conditions ◽  
Harnack Estimates

We classify two main singularities, as type I and type II, associated with null mean curvature flow of screen conformal null hypersurfaces in Lorentzian manifolds. We prove that the flow at a type I singularity is asymptotically self-similar, whereas at a type II singularity there is a blow-up solution which is an eternal solution. For further analysis of the above two singularities, we define null translating solitons and use them to prove some Harnack estimates for null mean curvature flow under certain geometric conditions.

scholarly journals Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

2019 ◽  
Vol 2019 (754) ◽  
pp. 225-251 ◽  
Author(s):  
James Isenberg ◽  
Haotian Wu
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type Ii ◽  
Matched Asymptotics ◽  
Curve Shortening Flow ◽  
Initial Hypersurface ◽  
Rate Dependent ◽  
Blowing Up

Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

scholarly journals Type-II singularities of two-convex immersed mean curvature flow

2016 ◽  
Vol 2 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Theodora Bourni ◽  
Mat Langford
Keyword(s):  
Mean Curvature ◽  
General Class ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Analogous Result ◽  
Curvature Flow ◽  
Type Ii ◽  
Rotationally Symmetric ◽  
The Mean ◽  
Mean Convex

AbstractWe show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of twoconvex hypersurfaces by the two-harmonic mean curvature.

scholarly journals Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

2020 ◽  
Vol 367 ◽  
pp. 107111
Author(s):  
James Isenberg ◽  
Haotian Wu ◽  
Zhou Zhang
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type Ii

ON THE EXTENSION OF THE MEAN CURVATURE FLOW IN ARBITRARY CODIMENSION

2010 ◽  
Vol 21 (11) ◽  
pp. 1429-1438 ◽  
Author(s):  
XIAOLI HAN ◽  
JUN SUN
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Integral Condition ◽  
Curvature Flow ◽  
Type I ◽  
Arbitrary Codimension ◽  
The Mean

In this paper we first give an integral condition under which the mean curvature flow can be extended in arbitrary codimension. Then we investigate some properties of Type I singularity.

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