On the Dirichlet-to-Neumann operator for the Connection Laplacian

TL;DR

This paper investigates how the Dirichlet-to-Neumann operator's symbol for a connection Laplacian encodes boundary geometry and derivatives, revealing that boundary geometric data are uniquely determined by this symbol.

Contribution

It establishes a link between the symbol of the Dirichlet-to-Neumann operator and boundary geometric information for connection Laplacians, including higher normal derivatives in higher dimensions.

Findings

Boundary geometric data are determined by the symbol.

Normal derivatives at the boundary are also determined in higher dimensions.

The relationship between the symbol and boundary geometry is explicitly characterized.

Abstract

We study the relationship between the symbol of the Dirichlet-to-Neumann operator associated with a connection Laplacian, and the geometry on and near the boundary. As a consequence, we show that the geometric data on the boundary, and when the dimension of the base is greater than two all corresponding normal derivatives, are determined by the symbol.