## Extension operators from a geometric point of view

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

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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## Geometric analysis under integral curvature conditions

Talk for the Seminari informal de Matemàtiques de Barcelona (SIMBa), November 2020.

Abstract

Curvature is a local property. However, assumptions on curvature have implications on global quantities, like the diameter, the eigenvalues of the Laplacian or topological invariants. An n-manifold with Ricci curvature larger than (n−1)K can not have a diameter larger than the one of the n-sphere with constant sectional curvature K. Similarly, the first non-zero eigenvalue of such a manifold can not be smaller than the one of the sphere. In recent years there has been an increasing interest in weakening the curvature assumptions from pointwise lower bounds to integral conditions. Integral conditions are much more general, they

are more stable under perturbations of the metric, and can be more suitable for the study of geometric flows. We will discuss some classical and recent results in the fields of geometric analysis and comparison geometry with integral curvature conditions.

Abstract

Curvature is a local property. However, assumptions on curvature have implications on global quantities, like the diameter, the eigenvalues of the Laplacian or topological invariants. An n-manifold with Ricci curvature larger than (n−1)K can not have a diameter larger than the one of the n-sphere with constant sectional curvature K. Similarly, the first non-zero eigenvalue of such a manifold can not be smaller than the one of the sphere. In recent years there has been an increasing interest in weakening the curvature assumptions from pointwise lower bounds to integral conditions. Integral conditions are much more general, they

are more stable under perturbations of the metric, and can be more suitable for the study of geometric flows. We will discuss some classical and recent results in the fields of geometric analysis and comparison geometry with integral curvature conditions.