The singular structure and regularity of stationary varifolds

2020 ◽  
Vol 22 (10) ◽  
pp. 3305-3382 ◽  
Author(s):  
Aaron Naber ◽  
Daniele Valtorta
Keyword(s):  
Stationary Varifolds

scholarly journals Consequences of Strong Stability of Minimal Submanifolds

2018 ◽  
Vol 2020 (8) ◽  
pp. 2352-2360
Author(s):  
Jason D Lotay ◽  
Felix Schulze
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Stability Result ◽  
Minimal Submanifolds ◽  
Strong Stability ◽  
Curvature Flow ◽  
Dynamical Stability ◽  
Local Uniqueness ◽  
Flow Through ◽  
Stationary Varifolds

Abstract In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang [12] directly extends to the enhanced Brakke flows of Ilmanen [5]. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow, which are proved to occur by Neves [7].

scholarly journals An existence theorem for Brakke flow with fixed boundary conditions

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Salvatore Stuvard ◽  
Yoshihiro Tonegawa
Keyword(s):  
Boundary Conditions ◽  
Existence Theorem ◽  
Hausdorff Measure ◽  
Boundary Data ◽  
Dimensional Hausdorff Measure ◽  
Fixed Boundary ◽  
Plateau’S Problem ◽  
Strictly Convex ◽  
Dimensional Domain ◽  
Stationary Varifolds

AbstractConsider an arbitrary closed, countably n-rectifiable set in a strictly convex $$(n+1)$$ ( n + 1 ) -dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $$t \uparrow \infty $$ t ↑ ∞ , the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

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