Monotonicity - analytic and geometric implications
Tobias Holck Colding, William P. Minicozzi II
TL;DR
This paper explores the analytical and geometric implications of monotonicity formulas for parabolic and elliptic operators, highlighting their interconnectedness and applications in understanding function spaces and geometric structures.
Contribution
It provides an expository overview of monotonicity formulas, linking analytical applications with geometric interpretations for elliptic and parabolic operators.
Findings
Monotonicity formulas are crucial in analyzing parabolic and elliptic operators.
The analysis of function spaces is deeply connected to the geometry of underlying spaces.
Geometric consequences of monotonicity formulas are discussed in detail.
Abstract
In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of the 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Advanced Harmonic Analysis Research