Monotonicity and its analytic and geometric implications

Monotonicity and its analytic and geometric implications

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 monotonicit...

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Veröffentlicht in:Proceedings of the National Academy of Sciences - PNAS 2013-11, Vol.110 (48), p.19233-19236
Hauptverfasser: Colding, Tobias Holck, Minicozzi, William P.
Format: Artikel
Sprache:eng
Schlagworte:
Cauchy Schwarz inequality
Cones
Curvature
Geometry
Heat equation
Infinity
Mathematical functions
Mathematical manifolds
Physical Sciences
QUANTITATIVE GEOMETRY SPECIAL FEATURE
Riemann manifold
Tangents
Uniqueness
Vector space
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Beschreibung
Zusammenfassung: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.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.1203856109