Xavier Ramos Olivé
  • Home
  • Research
    • Talks
  • Teaching
    • Previous quarters
    • A-term 2019
  • About Me
  • Blog

Extension operators from a geometric point of view

Talk for the 2020 Virtual Workshop on Ricci and Scalar Curvature in honor of Misha Gromov.

Abstract

Motivated by the study of the Neumann heat kernel on manifolds with integral Ricci curvature conditions, we study extension operators from a geometric point of view. By Extension Operator of a domain in M we mean a bounded linear operator E from H^1 of the domain to H^1(M) such that Eu restricted to the domain coincides with u, for any original function u. Although the existence of such operators is well known for reasonably regular domains, very little has been studied about the operator norms of E from a geometric point of view.

We discuss a quantitative estimate on the operator norm of a particular extension operator in terms of geometric parameters of a small tubular neighborhood of the boundary. The bound on the norm depends on sectional curvature bounds, bounds on the second fundamental form, and a global property called the rolling R-ball condition. These are uniform bounds in the class of domains satisfying these geometric conditions. As applications, we derive estimates on the Neumann heat kernel and the first non-zero Neumann eigenvalue of the domain for manifolds M with integral Ricci curvature conditions.

This is joint work with Olaf Post and Christian Rose.




If the video above doesn't work for you, you can also find it in the following platforms:

Bilibili: www.bilibili.com/video/BV16t4y1U7vA

Youtube: youtu.be/eFA0xaErPsw



Powered by Create your own unique website with customizable templates.