47 citations · 277 across the 44 of their papers we have counts for
6 papers · 2 filters
Entropy and codimension bounds for generic singularities
Tobias Holck Colding, William P. Minicozzi
We show that all closed -dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subsp…
In search of stable geometric structures
Tobias Holck Colding, William P. Minicozzi
We will look for stable structures in four situations and discuss what is known and unknown.
Regularity of elliptic and parabolic systems
Tobias Holck Colding, William P. Minicozzi
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the m…
Complexity of parabolic systems
Tobias Holck Colding, William P. Minicozzi
We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy…
Liouville properties
Tobias Holck Colding, William P. Minicozzi
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it ho…
Optimal bounds for ancient caloric functions
Tobias Holck Colding, William P. Minicozzi
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the…