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Differential Geometry: “A Mean Value Theorem for Riemannian Manifolds via the Obstacle Problem”

Dr. Brian Benson, UC Riverside

Friday, October 20, 2017
  11:10 a.m.–Noon


Location: Surge Building 268
  Parking Information

Category: Seminar

Description:

In joint work with Ivan Blank and Jeremy LeCrone, the speaker gave a new mean value theorem for the Laplace-Beltrami operator of a Riemannian manifold. The technique used to prove our mean value theorem uses an obstacle problem (a type of free boundary problem) depending on certain characteristics of the manifold, such as dimension, injectivity radius, and local Ricci curvature lower bounds. The idea of using an obstacle problem to prove mean values theorems for general divergence form elliptic operators in Euclidean space was briefly suggested by Caffarelli and later completed by Blank and Hao. 


We will discuss how our mean value theorem relates to others previously established in the literature in the setting of Riemannian manifolds. In particular, our average does not have any weights inside or outside of the integral, but much is still unknown about the geometry of the mean value sets. Finally, we will discuss the potential applications of this work to the study of harmonic manifolds.


 



Additional Information: Math Seminars

Open to: General Public
Admission: Free
Sponsor: Mathematics

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