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The round sphere minimizes entropy among closed self-shrinkers

https://doi.org/10.4310/jdg/1375124609

Colding, Tobias Holck; Ilmanen, Tom; Minicozzi, William P; White, Brian (September 2013, Journal of Differential Geometry)

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On uniqueness of tangent cones for Einstein manifolds

https://doi.org/10.1007/s00222-013-0474-z

Colding, Tobias Holck; Minicozzi, William P (June 2013, Inventiones mathematicae)

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Ricci curvature and monotonicity for harmonic functions

https://doi.org/10.1007/s00526-013-0610-z

Colding, Tobias Holck; Minicozzi, William P (February 2013, Calculus of Variations and Partial Differential Equations)

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Monotonicity and its analytic and geometric implications

https://doi.org/10.1073/pnas.1203856109

Colding, Tobias Holck; Minicozzi, William P (August 2012, Proceedings of the National Academy of Sciences)

In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of function spaces and the geometry of the underlining spaces are intertwined. After briefly discussing some of the well-known analytical applications of monotonicity for parabolic operators, we turn to their elliptic counterparts, their geometric meaning, and some geometric consequences.
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