Unique hyperplanes are the only translating solitons asymptotic to half-hyperplanes.
problem Characterizing translating solitons asymptotic to half-hyperplanes.
method Analyzing the geometry of translating solitons in Rn+1. result Hyperplanes are the only examples of translating solitons asymptotic to two half-hyperplanes.
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rn into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an (n−2)-dimensional plane at 120∘ …
Shows smoothness of varifolds with specific boundary angles.
problem Regularity of varifolds with prescribed contact angles.
method Analyzes varifolds with bounded first variation and prescribed contact angles, proving smoothness.
result Support of varifold is a C1,γ hypersurface near the boundary. A translating soliton is a hypersurface M in Rn+1 such that the family Mt=M−ten+1 is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point H=en+1⊥. In this paper we obtain a cha…
Proves regularity for stable varifolds near cones, expanding previous work.
problem Understanding stable varifolds near singularities.
method Proves C1,α regularity for varifolds in SQ. result Furthers understanding of local structure near singularities.
We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rn into q cells of prescribed (positive) Gaussian measure when 2≤q≤n+1, is to use a "simplicial cluster", obtained from the Voronoi cells of q equidistant points. Moreover, we prove that…
Proves regularity for stable varifolds near specific cones.
problem Regularity of stable codimension one integral varifolds near certain cones.
method Develops blow-up arguments and inductively performs finer blow-up procedures.
result Proves C1,α regularity for varifolds close to specific cones. The paper establishes structure theory for stable varifolds with applications to area minimising hypersurfaces.
problem Understanding the structure of stable codimension 1 integral varifolds.
method Develops a structure theory for stable codimension 1 stationary integral varifolds with no classical singularities.
result Establishes local structure properties of area minimising currents mod p, including the uniqueness of tangent cones at points with planar tangent cones.