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.
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.
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